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+Optimal search reach for heavy neutral leptons at a muon collider
+Krzysztof Mękała∗
+Deutsches Elektronen-Synchrotron DESY,
+Notkestr.
+85, 22607 Hamburg, Germany
+and
+Faculty of Physics, University of Warsaw,
+Pasteura 5, 02-093 Warszawa, Poland
+Jürgen Reuter†
+Deutsches Elektronen-Synchrotron DESY,
+Notkestr.
+85, 22607 Hamburg, Germany
+Aleksander Filip Żarnecki‡
+Faculty of Physics, University of Warsaw,
+Pasteura 5, 02-093 Warszawa, Poland
+(Dated: January 9, 2023)
+Neutrinos are the most elusive particles known. Heavier sterile neutrinos mixing with the standard
+neutrinos might solve the mystery of the baryon asymmetry of the universe. In this letter, we show
+that among all future energy frontier accelerators, muon colliders will provide the farthest search
+reach for such neutrinos for mass ranges above the Z pole into the multi-TeV regime. We compare
+the performance of muon with electron colliders of the same machine energy and briefly discuss the
+complementarity in flavor space between the two types of accelerators.
+PACS numbers: 13.35.Hb, 13.66.Lm. 14.60.Pq, 14.60.St
+Introduction
+Massive neutrinos are considered the first established building blocks of physics beyond the Standard
+Model (SM) of particle physics. Their tiny masses are believed to originate from seesaw-like mixing with heavier
+sterile neutrinos whose masses could be all the way from the electroweak (EW) to the unification scale. While long-
+distance neutrino oscillation experiments like DUNE or Hyper-Kamiokande will shed more light on the mass hierarchy
+and the mixing parameters, heavier neutrinos can be directly searched for at hadron colliders such as the LHC and
+future lepton colliders [1–17]. For collider searches, three different regimes can be considered: light neutrinos which
+are long-lived and result in displaced vertices or decay outside the detectors, intermediate-mass neutrinos that decay
+promptly and are dominantly produced in Z (and W or Higgs) decays, and heavy neutrinos with masses Mν ≳ MH.
+In this paper, building upon an analysis framework similar to earlier studies for searches at linear e+e− machines, we
+focus on the third case and show that the most sensitive searches for direct heavy neutrino production are possible at
+high-energy muon colliders. Lepton colliders are, in general, sensitive to much smaller mixing parameters and hence
+to much higher scales of UV completions. In this paper, we will consider a muon collider setup with energies of 3 and
+10 TeV, and integrated luminosities of 1 and 10 ab−1, respectively [18–20].
+Model setup and simulation framework
+In this letter, we consider the Phenomenological Type I Seesaw Mech-
+anism [21, 22], implemented within the HeavyN model with Dirac neutrinos [5, 23], i.e.
+we assume it just as a
+representative model candidate without any prejudice (our findings are quite generic, though specific model setups
+like artificial flavor mixings could of course lead to singular cases; Refs. [24, 25] provide an example where such heavy
+neutrinos appear even at a multi-TeV scale UV completion). This effective extension of the SM introduces three
+flavors of right-handed neutrinos (denoted as Nk) that are singlets under the SM gauge groups. The Lagrangian of
+the model reads:
+L = LSM + LN + LW Nℓ + LZNν + LHNν
+(1)
+where LN is a sum of kinetic and mass terms for heavy neutrinos (in 4-spinor notation, which combines terms with
+spinors of dotted and undotted indices):
+LN = ¯Nki/∂Nk − mNk ¯NkNk
+for k = 1, 2, 3,
+(2)
+LW Nℓ yields neutrino interactions with the W boson:
+LW Nℓ = − g
+√
+2W +
+µ
+3
+�
+k=1
+τ
+�
+l=e
+¯NkV ∗
+lkγµPLℓ− + h.c.,
+(3)
+arXiv:2301.02602v1  [hep-ph]  6 Jan 2023
+
+2
+0
+2000
+4000
+6000
+8000
+10000
+ [GeV]
+qql
+m
+1
+−
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+ Events
+ee bg
+sig_3000
+FIG. 1: qqℓ mass distribution for a reference scenario assuming the existence of one Dirac neutrino with a mass of
+3 TeV, at a 10 TeV muon collider. The black solid line stand for the µ+µ− background and the thick green one for
+the signal scenario.
+LZNν interactions with the Z boson:
+LZNν = −
+g
+2 cos θW
+Zµ
+3
+�
+k=1
+τ
+�
+l=e
+¯NkV ∗
+lkγµPLνl + h.c.,
+(4)
+and LHNν interactions with the Higgs boson:
+LHNν = − gmN
+2MW
+h
+3
+�
+k=1
+τ
+�
+l=e
+¯NkV ∗
+lkPLνl + h.c.
+(5)
+The UFO library of the model contains 12 free parameters in addition to the SM parameters, which are three masses
+of the heavy neutrinos: mNk and nine real (no CP violation expected) mixing parameters Vlk, where l = e, µ, τ and
+k = N1, N2, N3. For the purpose of this analysis, we considered a scenario with only one heavy Dirac neutrino N1 ≡ N
+with a mass below O(10 TeV) and equal couplings to all SM leptons (|VeN1|2 = |VµN1|2 = |VτN1|2 ≡ V 2
+lN). For sample
+generation, the mixing parameter V 2
+lN has been set to 0.0003. Other values for the mixing parameters in the analysis
+below were accessed via rescaling with the corresponding cross section. Although there are many different possible
+signatures of such particles at future colliders, for center-of-mass energies above the Z pole, the t-channel W exchange
+resulting in the production of a light-heavy neutrino pair (µ+µ− → N ν) is dominant [11] and the production cross
+section is of the order of 1-10 fb for masses of the neutrinos up to the collision energy. For our choice of the parameter
+space, the heavy neutrino has a microscopic lifetime (cτ ≪ 1 nm) and no displaced vertices are expected. Among
+the possible decay channels of such particles, the signature of two jets and a lepton (N → qqℓ) is the most promising
+because it allows for direct reconstruction of the mass of the heavy state.
+In the first step, we generated event samples with Whizard 3.0.2 [26–28] at leading order (LO) in the SM coupling
+constants (though recently higher-order corrections have become available in an automated manner [29] and simulated
+detector response with Delphes 3.5.0 [30] using built-in Muon Collider detector cards. At the generator level, a
+set of cuts was applied to remove possible singularities.
+They included 10-GeV cuts on the energy of produced
+jets and leptons, the invariant mass of quark and lepton pairs, and the four-momentum transfer from the incoming
+muons. Furthermore, it was required that at least one lepton could be detected in the central detector (we assumed
+5◦ < θ < 175◦, where θ is the lepton polar angle). For the detector simulation, the VLC clustering algorithm in the
+
+3
+1
+−
+0.5
+−
+0
+0.5
+1
+BDT response
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+Events
+ee
+sig
+FIG. 2: Distribution of the BDT response for the reference scenario (Dirac neutrino, mN = 3 TeV) with electrons in
+the final state at a 10 TeV muon collider. The red line denotes the background, and the green line the signal.
+exclusive two-jet mode (R = 1.5, β = 1, γ = 1 – see [31]) was applied. Since the considered Delphes model cannot
+generate fake lepton tracks, only 4- and 6-fermion background processes with at least one lepton in the final state
+(qqℓν, qqℓℓ, ℓℓℓℓ, qqqqℓν, qqqqℓℓ, qqℓνℓν, qqℓννν) were generated. The most important channels in terms of cross
+section (O(1 ab) at both energy stages) were qqℓν and ℓℓℓℓ; the latter could be, however, easily reduced by lepton
+identification. Background channels induced by photons from collinear initial-state splittings were neglected, as their
+impact on the final results was found to be marginal.
+Analysis procedure
+In the next step, a set of selection cuts was applied to reject events incompatible with the
+expected topology of two jets and one lepton. To exclude events with significant contributions of forward deposits
+assigned to the beam jets, an upper limit of 20 GeV was applied on the transverse momentum of objects not included
+in the reconstructed final state. In Figure 1, we show a distribution of the invariant mass of two jets and a lepton for
+a reference scenario (a 3 TeV neutrino at a 10 TeV muon collider). A peak corresponding to the mass of the heavy
+neutrino is clearly visible. The left tail is due to events with leptonic τ decays, for which the escaping neutrinos
+reduce the reconstructed invariant mass. On the right-hand side, the tail is an effect of finite detector resolution.
+Subsequently, we applied the Boosted Decision Tree (BDT) method implemented in the TMVA package [32] to
+discriminate between signal and background events. A set of eight variables describing event kinematics was chosen
+to optimize the classification:
+• mqqℓ – invariant mass of the dijet-lepton system,
+• α – angle between the dijet system and the lepton,
+• αqq – angle between the two jets,
+• Eℓ – lepton energy,
+• Eqqℓ – energy of the dijet-lepton system,
+• pT
+ℓ – lepton transverse momentum,
+• pT
+qq – dijet transverse momentum,
+• pT
+qqℓ – transverse momentum of the dijet-lepton system.
+Due to the considerable difference between the numbers of expected background events, the algorithm was implemented
+separately for events with reconstructed electrons and muons in the final state. The BDT response for the reference
+scenario is shown in Figure 2. The two distributions are partially separated and thus, they were used to extract
+expected limits on the coupling parameter V 2
+lN within the CLs method, implemented in the RooStats package [33].
+
+4
+3
+10
+4
+10
+ [GeV]
+N
+m
+7
+−
+10
+6
+−
+10
+5
+−
+10
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+2
+lN
+lim. V
+CMS
+HL-LHC
+HE-LHC
+FCC-hh
+ILC 1 TeV
+CLIC 3 TeV
+Muon Collider 10 TeV
+Muon Collider 3 TeV
+FIG. 3: Limits on the coupling V 2
+ℓN for different Muon Collider setups (solid lines: 3 TeV – turquoise, 10 TeV –
+orange). Dashed lines indicate limits from current and future hadron [1, 5] machines, dashed-dotted for e+e−
+colliders [16]. See text for details.
+This allowed for combining the electron and muon channels. The impact of systematic uncertainties has been neglected
+at this stage, as they are not expected to significantly affect the final conclusions.
+Results
+In Figure 3, limits on the coupling V 2
+lN for the two Muon Collider setups are presented and compared
+with the current limits coming from the CMS experiment (Majorana neutrinos, Fig. 2 in [1]), as well as with the
+results obtained for future hadron colliders (Dirac neutrinos, Fig. 25b in [5]) and e+e− colliders (Dirac neutrinos,
+Fig. 12 in [16]). It should be noted that in the hadron collider analyses, heavy neutrino decays into taus were not
+considered, and thus their sensitivity is enhanced relative to the results presented for the lepton colliders, where the
+tau-channel decays are included. As shown in Figure 3, limits expected from the e+e− colliders, ILC running at 1 TeV
+and CLIC running at 3 TeV, are more stringent for masses of the heavy neutrinos up to about 700 GeV. The fact that
+the results for CLIC and a Muon Collider operating at the same energy of 3 TeV do not coincide may be surprising.
+However, several effects must be taken into account for a proper comparison: the most important factors are different
+integrated luminosities and beam polarizations. In addition, the beam spectra and the beam-induced background
+channels cannot be neglected for e+e− colliders, while their impact is significantly reduced for µ+µ− machines due
+to the larger mass of the muon.
+It was verified that, for the same generation setup (no beam polarization, no
+beam spectrum, no beam-induced background channels, but different initial-state particles and detector designs), the
+expected CLIC limits are consistent with the Muon Collider ones, giving the analysis precision. The discrepancy
+visible in Figure 3 could then be explained as follows: at lower neutrino masses, the expected limits from CLIC are
+more stringent due to the higher integrated luminosity and electron beam polarization, and at higher masses, they
+are worse because of the impact of the luminosity spectra and beam-induced backgrounds.
+In the analysis, we assumed that all the mixing parameters VlN have the same value. It is important to note that
+this approach is not unique. Using data from both electron-positron and muon colliders, one could potentially loosen
+this assumption and constrain the parameters VeN and VµN separately, by either excluding taus from the physical
+model or implementing a proper tau tagging procedure. Such a method would give limits not only on the couplings
+themselves but also on their product in the framework where couplings are treated independently, possibly hinting at
+a flavor-universality violation. The details are, however, beyond the scope of this letter.
+Conclusions
+Extensions of the Standard Model introducing heavy neutrinos offer interesting solutions to several
+of its open questions, e.g. the baryon asymmetry of the universe, dark matter and flavor. If such particles are at mass
+scales well above a GeV, they can be efficiently searched for at future lepton colliders. Due to the highest achievable
+energies and the clean experimental environments, muon colliders would provide the furthest discovery reach for
+
+5
+this kind of particles and models, vastly surpassing high-energy hadron colliders. By employing the synergy of both
+different types of lepton machines, electron-positron and muon colliders, different paths in the flavor parameter space
+of the models could be pursued.
+Acknowledgments
+The work was partially supported by the National Science Centre (Poland) under the OPUS research project no.
+2021/43/B/ST2/01778. KM and JRR acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, Ger-
+man Research Association) under Germany’s Excellence Strategy-EXC 2121 "Quantum Universe"-3908333. This work
+has also been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 491245950.
+∗ [email protected]
+† [email protected]
+‡ fi[email protected]
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+

1NE0T4oBgHgl3EQfuQFR/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf,len=343
+page_content='Optimal search reach for heavy neutral leptons at a muon collider  Krzysztof Mękała∗  Deutsches Elektronen-Synchrotron DESY,  Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  85, 22607 Hamburg, Germany  and  Faculty of Physics, University of Warsaw,  Pasteura 5, 02-093 Warszawa, Poland  Jürgen Reuter†  Deutsches Elektronen-Synchrotron DESY,  Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  85, 22607 Hamburg, Germany  Aleksander Filip Żarnecki‡  Faculty of Physics, University of Warsaw,  Pasteura 5, 02-093 Warszawa, Poland  (Dated: January 9, 2023)  Neutrinos are the most elusive particles known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Heavier sterile neutrinos mixing with the standard  neutrinos might solve the mystery of the baryon asymmetry of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' In this letter, we show  that among all future energy frontier accelerators, muon colliders will provide the farthest search  reach for such neutrinos for mass ranges above the Z pole into the multi-TeV regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' We compare  the performance of muon with electron colliders of the same machine energy and briefly discuss the  complementarity in flavor space between the two types of accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  PACS numbers: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='Hb, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='Pq, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='St  Introduction  Massive neutrinos are considered the first established building blocks of physics beyond the Standard  Model (SM) of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Their tiny masses are believed to originate from seesaw-like mixing with heavier  sterile neutrinos whose masses could be all the way from the electroweak (EW) to the unification scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' While long-  distance neutrino oscillation experiments like DUNE or Hyper-Kamiokande will shed more light on the mass hierarchy  and the mixing parameters, heavier neutrinos can be directly searched for at hadron colliders such as the LHC and  future lepton colliders [1–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' For collider searches, three different regimes can be considered: light neutrinos which  are long-lived and result in displaced vertices or decay outside the detectors, intermediate-mass neutrinos that decay  promptly and are dominantly produced in Z (and W or Higgs) decays, and heavy neutrinos with masses Mν ≳ MH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  In this paper, building upon an analysis framework similar to earlier studies for searches at linear e+e− machines, we  focus on the third case and show that the most sensitive searches for direct heavy neutrino production are possible at  high-energy muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Lepton colliders are, in general, sensitive to much smaller mixing parameters and hence  to much higher scales of UV completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' In this paper, we will consider a muon collider setup with energies of 3 and  10 TeV, and integrated luminosities of 1 and 10 ab−1, respectively [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Model setup and simulation framework  In this letter, we consider the Phenomenological Type I Seesaw Mech-  anism [21, 22], implemented within the HeavyN model with Dirac neutrinos [5, 23], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  we assume it just as a  representative model candidate without any prejudice (our findings are quite generic, though specific model setups  like artificial flavor mixings could of course lead to singular cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' [24, 25] provide an example where such heavy  neutrinos appear even at a multi-TeV scale UV completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' This effective extension of the SM introduces three  flavors of right-handed neutrinos (denoted as Nk) that are singlets under the SM gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The Lagrangian of  the model reads:  L = LSM + LN + LW Nℓ + LZNν + LHNν  (1)  where LN is a sum of kinetic and mass terms for heavy neutrinos (in 4-spinor notation, which combines terms with  spinors of dotted and undotted indices):  LN = ¯Nki/∂Nk − mNk ¯NkNk  for k = 1, 2, 3,  (2)  LW Nℓ yields neutrino interactions with the W boson:  LW Nℓ = − g  √  2W +  µ  3  �  k=1  τ  �  l=e  ¯NkV ∗  lkγµPLℓ− + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=',  (3)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='02602v1  [hep-ph]  6 Jan 2023  2  0  2000  4000  6000  8000  10000  [GeV]  qql  m  1  −  10  1  10  2  10  3  10  4  10  5  10  6  10  Events  ee bg  sig_3000  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 1: qqℓ mass distribution for a reference scenario assuming the existence of one Dirac neutrino with a mass of  3 TeV, at a 10 TeV muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The black solid line stand for the µ+µ− background and the thick green one for  the signal scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  LZNν interactions with the Z boson:  LZNν = −  g  2 cos θW  Zµ  3  �  k=1  τ  �  l=e  ¯NkV ∗  lkγµPLνl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=',  (4)  and LHNν interactions with the Higgs boson:  LHNν = − gmN  2MW  h  3  �  k=1  τ  �  l=e  ¯NkV ∗  lkPLνl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  (5)  The UFO library of the model contains 12 free parameters in addition to the SM parameters, which are three masses  of the heavy neutrinos: mNk and nine real (no CP violation expected) mixing parameters Vlk, where l = e, µ, τ and  k = N1, N2, N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' For the purpose of this analysis, we considered a scenario with only one heavy Dirac neutrino N1 ≡ N  with a mass below O(10 TeV) and equal couplings to all SM leptons (|VeN1|2 = |VµN1|2 = |VτN1|2 ≡ V 2  lN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' For sample  generation, the mixing parameter V 2  lN has been set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Other values for the mixing parameters in the analysis  below were accessed via rescaling with the corresponding cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Although there are many different possible  signatures of such particles at future colliders, for center-of-mass energies above the Z pole, the t-channel W exchange  resulting in the production of a light-heavy neutrino pair (µ+µ− → N ν) is dominant [11] and the production cross  section is of the order of 1-10 fb for masses of the neutrinos up to the collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' For our choice of the parameter  space, the heavy neutrino has a microscopic lifetime (cτ ≪ 1 nm) and no displaced vertices are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Among  the possible decay channels of such particles, the signature of two jets and a lepton (N → qqℓ) is the most promising  because it allows for direct reconstruction of the mass of the heavy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  In the first step, we generated event samples with Whizard 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='2 [26–28] at leading order (LO) in the SM coupling  constants (though recently higher-order corrections have become available in an automated manner [29] and simulated  detector response with Delphes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='0 [30] using built-in Muon Collider detector cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' At the generator level, a  set of cuts was applied to remove possible singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  They included 10-GeV cuts on the energy of produced  jets and leptons, the invariant mass of quark and lepton pairs, and the four-momentum transfer from the incoming  muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Furthermore, it was required that at least one lepton could be detected in the central detector (we assumed  5◦ < θ < 175◦, where θ is the lepton polar angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' For the detector simulation, the VLC clustering algorithm in the  3  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='5  1  BDT response  1  10  2  10  3  10  4  10  5  10  Events  ee  sig  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 2: Distribution of the BDT response for the reference scenario (Dirac neutrino, mN = 3 TeV) with electrons in  the final state at a 10 TeV muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The red line denotes the background, and the green line the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  exclusive two-jet mode (R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='5, β = 1, γ = 1 – see [31]) was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Since the considered Delphes model cannot  generate fake lepton tracks, only 4- and 6-fermion background processes with at least one lepton in the final state  (qqℓν, qqℓℓ, ℓℓℓℓ, qqqqℓν, qqqqℓℓ, qqℓνℓν, qqℓννν) were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The most important channels in terms of cross  section (O(1 ab) at both energy stages) were qqℓν and ℓℓℓℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' the latter could be, however, easily reduced by lepton  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Background channels induced by photons from collinear initial-state splittings were neglected, as their  impact on the final results was found to be marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Analysis procedure  In the next step, a set of selection cuts was applied to reject events incompatible with the  expected topology of two jets and one lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' To exclude events with significant contributions of forward deposits  assigned to the beam jets, an upper limit of 20 GeV was applied on the transverse momentum of objects not included  in the reconstructed final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' In Figure 1, we show a distribution of the invariant mass of two jets and a lepton for  a reference scenario (a 3 TeV neutrino at a 10 TeV muon collider).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' A peak corresponding to the mass of the heavy  neutrino is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The left tail is due to events with leptonic τ decays, for which the escaping neutrinos  reduce the reconstructed invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' On the right-hand side, the tail is an effect of finite detector resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Subsequently, we applied the Boosted Decision Tree (BDT) method implemented in the TMVA package [32] to  discriminate between signal and background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' A set of eight variables describing event kinematics was chosen  to optimize the classification:  mqqℓ – invariant mass of the dijet-lepton system,  α – angle between the dijet system and the lepton,  αqq – angle between the two jets,  Eℓ – lepton energy,  Eqqℓ – energy of the dijet-lepton system,  pT  ℓ – lepton transverse momentum,  pT  qq – dijet transverse momentum,  pT  qqℓ – transverse momentum of the dijet-lepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Due to the considerable difference between the numbers of expected background events, the algorithm was implemented  separately for events with reconstructed electrons and muons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The BDT response for the reference  scenario is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The two distributions are partially separated and thus, they were used to extract  expected limits on the coupling parameter V 2  lN within the CLs method, implemented in the RooStats package [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  4  3  10  4  10  [GeV]  N  m  7  −  10  6  −  10  5  −  10  4  −  10  3  −  10  2  −  10  1  −  10  2  lN  lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' V  CMS  HL-LHC  HE-LHC  FCC-hh  ILC 1 TeV  CLIC 3 TeV  Muon Collider 10 TeV  Muon Collider 3 TeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 3: Limits on the coupling V 2  ℓN for different Muon Collider setups (solid lines: 3 TeV �� turquoise, 10 TeV –  orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Dashed lines indicate limits from current and future hadron [1, 5] machines, dashed-dotted for e+e−  colliders [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  This allowed for combining the electron and muon channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The impact of systematic uncertainties has been neglected  at this stage, as they are not expected to significantly affect the final conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Results  In Figure 3, limits on the coupling V 2  lN for the two Muon Collider setups are presented and compared  with the current limits coming from the CMS experiment (Majorana neutrinos, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 2 in [1]), as well as with the  results obtained for future hadron colliders (Dirac neutrinos, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 25b in [5]) and e+e− colliders (Dirac neutrinos,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' 12 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' It should be noted that in the hadron collider analyses, heavy neutrino decays into taus were not  considered, and thus their sensitivity is enhanced relative to the results presented for the lepton colliders, where the  tau-channel decays are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' As shown in Figure 3, limits expected from the e+e− colliders, ILC running at 1 TeV  and CLIC running at 3 TeV, are more stringent for masses of the heavy neutrinos up to about 700 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The fact that  the results for CLIC and a Muon Collider operating at the same energy of 3 TeV do not coincide may be surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  However, several effects must be taken into account for a proper comparison: the most important factors are different  integrated luminosities and beam polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' In addition, the beam spectra and the beam-induced background  channels cannot be neglected for e+e− colliders, while their impact is significantly reduced for µ+µ− machines due  to the larger mass of the muon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  It was verified that, for the same generation setup (no beam polarization, no  beam spectrum, no beam-induced background channels, but different initial-state particles and detector designs), the  expected CLIC limits are consistent with the Muon Collider ones, giving the analysis precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The discrepancy  visible in Figure 3 could then be explained as follows: at lower neutrino masses, the expected limits from CLIC are  more stringent due to the higher integrated luminosity and electron beam polarization, and at higher masses, they  are worse because of the impact of the luminosity spectra and beam-induced backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  In the analysis, we assumed that all the mixing parameters VlN have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' It is important to note that  this approach is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Using data from both electron-positron and muon colliders, one could potentially loosen  this assumption and constrain the parameters VeN and VµN separately, by either excluding taus from the physical  model or implementing a proper tau tagging procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Such a method would give limits not only on the couplings  themselves but also on their product in the framework where couplings are treated independently, possibly hinting at  a flavor-universality violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' The details are, however, beyond the scope of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Conclusions  Extensions of the Standard Model introducing heavy neutrinos offer interesting solutions to several  of its open questions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' the baryon asymmetry of the universe, dark matter and flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' If such particles are at mass  scales well above a GeV, they can be efficiently searched for at future lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' Due to the highest achievable  energies and the clean experimental environments, muon colliders would provide the furthest discovery reach for  5  this kind of particles and models, vastly surpassing high-energy hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' By employing the synergy of both  different types of lepton machines, electron-positron and muon colliders, different paths in the flavor parameter space  of the models could be pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  Acknowledgments  The work was partially supported by the National Science Centre (Poland) under the OPUS research project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  2021/43/B/ST2/01778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' KM and JRR acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, Ger-  man Research Association) under Germany’s Excellence Strategy-EXC 2121 "Quantum Universe"-3908333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content=' This work  has also been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 491245950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='  ∗ krzysztof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='mekala@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
+page_content='de  † juergen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'}
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+Atomic Transition Probabilities for Transitions of Si I and Si II and the Silicon Abundances of 
+Several Very Metal-Poor Stars1
+ 
+E. A. Den Hartog2, J. E. Lawler2, C. Sneden3, I. U. Roederer4,5 & J. J. Cowan6 
+2Department of Physics, University of Wisconsin-Madison, 1150 University Ave, Madison, WI 
+53706; [email protected]; [email protected] 
+3Department of Astronomy and McDonald Observatory, University of Texas, Austin, TX 78712; 
+[email protected] 
+4Department of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 
+48109, [email protected] 
+5Joint Institute for Nuclear Astrophysics – Center for the Evolution of the Elements (JINA-CEE) 
+6Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 
+73019; [email protected] 
+ 
+ 
+ 
+ORCIDS: 
+E. A. Den Hartog: 
+0000-0001-8582-0910 
+ 
+J. E. Lawler:  
+0000-0001-5579-9233 
+ C. Sneden: 
+ 
+0000-0002-3456-5929 
+ 
+ 
+I. U. Roederer  
+0000-0001-5107-8930 
+J. J. Cowan 
+ 
+0000-0002-6779-3813 
+ 
+                                                            
+1 Based on observations made with the NASA/ESA Hubble Space Telescope (HST), obtained at the Space Telescope 
+Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA 
+contract NAS 5‐26555.  Other data have been obtained from the European Southern Observatory (ESO) Science 
+Archive Facility; and the Keck Observatory Archive, which is operated by the W. M. Keck Observatory and the NASA 
+Exoplanet Science Institute, under contract with NASA.  These data are associated with HST programs GO-7402, 
+GO-14161, and GO-14232; ESO programs 66.D-0636(A), 073.D-0024(A), and 095.D-0504(A); and Keck program 
+H41aH.  
+
+Abstract 
+We report new measurements of branching fractions for 20 UV and blue lines in the spectrum of 
+neutral silicon (Si I) originating in the 3s23p4s 3Po1,2, 1Po1 and 3s3p3 1Do1,2 upper levels.  
+Transitions studied include both strong, nearly pure LS multiplets as well as very weak spin-
+forbidden transitions connected to these upper levels.  We also report a new branching fraction 
+measurement of the 4P1/2 – 2Po1/2,3/2 intercombination lines in the spectrum of singly-ionized 
+silicon (Si II).  The weak spin-forbidden lines of Si I and Si II provide a stringent test on recent 
+theoretical calculations, to which we make comparison.  The branching fractions from this study 
+are combined with previously reported radiative lifetimes to yield transition probabilities and 
+log(gf)s for these lines.  We apply these new measurements to abundance determinations in five 
+metal-poor stars. 
+ 
+ 
+
+1. 
+Introduction 
+Silicon is one of the most abundant elements in the solar system and plays an important 
+role in many astrophysical environments.  With its high abundance and relatively low ionization 
+potential it is a significant source of electrons in the interior of cool stars and contributes 
+significantly to the interior opacity in solar-type stars (Amarsi & Asplund 2017).  Because 
+silicon is abundant and nonvolatile, it is often used as a reference element to reconcile the 
+absolute scales of meteoritic (e.g. Lodders, Palme & Gail 2009) and solar photospheric 
+abundances (e.g. Asplund et al. 2009).  Emission line ratios of Si II, and in particular the ratio of 
+weak resonance lines (3s23p 2Po – 3s3p2 2D) and weak intercombination lines (3s23p 2Po – 3s3p2 
+4P), are potentially useful as a plasma diagnostic because of their sensitivity to temperature and 
+density (e.g. Bautista et al. 2009).  Silicon-burning, in which 28Si is converted to 56Ni in a series 
+of successive alpha captures, is the final phase of fusion reactions in the interior of massive stars.  
+Fusion reactions involving elements heavier than 56Ni are endothermic and thus not 
+spontaneous.  After a brief period (approximately one earth day) of Silicon-burning, the core of a 
+massive star collapses and may explode to release more energy as a Type II supernova. 
+Motivation for the current study lies in the desire to better understand stellar 
+nucleosynthesis.  Records of the “means of production” by which the elements came into being 
+in the earliest epoch of our Galaxy are written into the abundance patterns of the oldest, metal-
+poor stars in the halo of the Milky Way.  Here can be found evidence of the early births, short 
+lives and violent deaths of the first massive stars.  Before these abundance patterns can be 
+decoded to gain deeper understanding of the history of nucleosynthesis, we must first be able to 
+determine the abundances of the elements with accuracy and precision.  This requires both 
+accurate atomic data and realistic stellar models. As an α–capture element7, trends of abundance 
+ratios such as [Si/Fe]8 with metallicity yield insight into stellar nucleosynthesis and the chemical 
+evolution of the Galaxy.  In an earlier study of the heaviest α–element, Ca, we made detailed 
+comparison between new and published experimental transition probabilities for Ca I and 
+modern theory (Den Hartog et al. 2021).   In the present study we make similar comparison with 
+improved transition probabilities for lines of Si I and Si II. 
+In §2 below, we present a discussion of our measurement method including a description 
+of a new radiometric calibration technique for our high-resolution spectrometer.  We present our 
+transition probabilities for 20 lines of Si I and two Si II intercombination lines in §3 along with 
+comparison to the best experimental and theoretical results from the literature.  In §4 we apply 
+the new data to derive Si abundances in five warm, very metal-poor main-sequence stars.  
+                                                            
+7 Formally an α-element is one whose dominant isotope is composed of multiple 4He nuclei. The major natural isotopes 
+of Si (Z = 14) are 28Si (92.191% in the solar system), 29Si (4.645%), 30Si (3.037) (Meija et al. 2016). For astrophysical 
+purposes, Si is pure 28Si. Since the minor isotopes of Si collectively contribute only 7.6% to the Si elemental 
+abundance, they will not contribute significantly in solar and stellar optical spectra. 
+8 We use standard abundance notations. For elements X and Y, the relative abundances are written [X/Y] = 
+log10(NX/NY)star − log10(NX/NY)Ÿ. For element X, the “absolute” abundance is written log10 ε(X) = log10(NX/NH) + 
+12.  Metallicity is defined as the stellar [Fe/H] value.  We adopt the Solar reference abundances from Asplund 
+(2009). 
+
+ 
+2.  Emission Branching Fractions 
+  The technique of combining radiative lifetimes from laser-induced fluorescence 
+measurements with emission branching fractions (BFs) measured using high-resolution 
+spectrometers is now the standard method for measuring transition probabilities, or Einstein A-
+values, with efficiency and accuracy (e.g. Lawler et al. 2009).  The BF for a transition between 
+an upper level u and a lower level l is given by the ratio of its A-value to the sum of the A-values 
+for all transitions associated with u, which is the inverse of the radiative lifetime, u. Thus the 
+radiative lifetime, u, provides the absolute scale when converting a BF to an A-value.  For the 
+purposes of measuring BFs, it can also be expressed as the ratio of relative emission intensities I 
+(in any units proportional to photons/time) for these transitions: 
+𝐵𝐹�� � 𝐴��
+∑ 𝐴��
+�
+� 𝐴��𝜏� � 𝐼��
+∑ 𝐼��
+�
+.                                                                                            �1� 
+BFs, by definition, sum to unity.  In order to assure the correct normalization, it is therefore 
+important when measuring BFs to account for all possible decay paths from an upper level. If 
+some weak transitions cannot be measured, these “residual” BFs need to be estimated from 
+theory and accounted for in the total decay rate.  If the sum is over significantly less than the full 
+complement of lines, then one has a branching ratio (BR). 
+In order to avoid line blends, a high-resolution spectrometer is usually required to 
+measure the emission branching fractions unless the spectrum is very sparse.  Often a Fourier 
+transform spectrometer (FTS) is used as these instruments have many advantages, including 
+high-resolution, broad spectral coverage and excellent absolute wavenumber accuracy.  FTS 
+instruments have one significant disadvantage in that the quantum noise in the spectrum gets 
+spread evenly throughout the spectrum.  This “multiplex” noise results in weak lines being 
+swamped in the noise from the strong lines in the spectrum.  To overcome the multiplex noise 
+the lamp current is often increased to the point that strong lines in the spectrum are affected by 
+optical depth, which in turn results in inaccurate BFs.  Corrections for optical depth can be made, 
+but if the corrections are large they lead to increased uncertainties.   
+In the current study, BFs in Si I and II have been determined from spectra recorded with 
+the University of Wisconsin (UW) high-resolution echelle spectrograph.  This instrument is 
+described in detail in Wood & Lawler (2012).  As a dispersive instrument, it does not have 
+multiplex noise and is much better-suited than an FTS for measurement of weak lines while 
+keeping source currents low and avoiding significant self-absorption on the strong transitions.  It 
+is a 3-m cross-dispersed echelle spectrograph with broad spectral coverage, resolving power R ¥ 
+250,000 and a 4 Mpixel CCD detector.  The spectra are two-dimensional CCD images containing 
+multiple grating orders, with the high-resolution of each grating order running in one direction 
+and the orders arranged side-by-side in the other dimension.  The cross-disperser utilizes a prism 
+
+to separate the orders, so the orders are further apart at lower wavelength and get increasingly 
+closer together at higher wavelengths.  In the far-ultraviolet (far-UV) one CCD frame covers 
+approximately 150 nm in the low resolution direction and three overlapping frames are required 
+to capture an entire grating order in the high-resolution direction.  The usual mode of operation 
+would be to acquire five overlapping frames for each UV spectrum, to provide some redundancy 
+and check for source drifts.  However, the wavelengths of transitions from the upper levels in the 
+current study are such that all transitions from each level can be studied with a single grating 
+setting.  This serendipitous coincidence of line placement means that there is no need to combine 
+frames with different grating settings, eliminating the contribution to the uncertainty that such 
+combining generates. 
+The optical sources used for generating the Si I, II spectra are commercially manufactured 
+Si-Ne and Si-Ar hollow cathode lamps (HCLs).  Each CCD frame recorded is accompanied by a 
+continuum lamp spectrum recorded after the frame, from which a relative radiometric calibration 
+for that frame is determined.  In the current study a deuterium (D2) lamp is used as the 
+calibration light source.  The only change made between these two recordings is the angle of a 
+steering mirror on a kinematic mount.  Beyond this mirror light from each lamp encounters the 
+same optical path.  Table 1 lists all spectra recorded for the current study of Si II and Si I BFs.  
+The spectra are analyzed by taking a numerical integral of each line across the width of the 
+grating order in which it is found and dividing that by an integral of the D2 lamp intensity at the 
+same CCD position.  The relative irradiance of the D2 lamp can be used to put all lines on the 
+same relative scale. These radiometrically calibrated intensities are then converted to BFs using 
+Equation 1.   
+Multiple spectra are taken of our primary source, the Si-Ne HCL, over a range of currents 
+between 3 mA and 32 mA. A range of lamp currents is used to check for evidence of self-
+absorption on the strongest lines of Si I.  Self-absorption becomes apparent by studying the BR 
+of a weaker line from the same upper level compared to a strong line that connects to the lowest 
+term.  If self-absorption is present on the strong transition this BR will increase with increasing 
+lamp current.  We see some evidence of minor self-absorption on three strong Si I lines that 
+connect to the ground term.  These have small corrections applied based on the extrapolation of 
+the BR to zero current.  The largest of these extrapolations is only 2% lower than the BR 
+measured on the lowest current spectrum. 
+2.1 Detector-based Radiometric Calibration 
+A continuum lamp is required for the calibration of the echelle spectrometer in order to 
+capture the rapidly changing instrument sensitivity along the grating orders due to the sinc2 blaze 
+envelope of the grating.  However, the calibration in the low resolution direction, which changes 
+slowly as a function of wavelength, can be achieved by some other means and then transferred 
+onto the D2 source.  For this project we have chosen to use a National Institute of Standards and 
+Technology (NIST) calibrated photodiode detector as our standard.  Switching to a detector-
+
+based standard from a source-based standard has the advantage that the detector will remain 
+stable for many years, whereas lamp sources age both with shelf life and with usage.  UV 
+damage to the window causes changes to the radiant output, particularly in the far-UV.  The 
+irradiance of the lamp has to be periodically checked against another little-used lamp and then 
+corrections applied, or the lamp must be sent out to be recalibrated at considerable expense.  
+Another motivation for switching to the detector-based calibration is that D2 lamps are only 
+calibrated between 200 nm and 400 nm and the current project required a calibration out to 410 
+nm.  Even the calibrated irradiance above 370 nm requires careful correction in order to use the 
+lamp at high resolution.  This is because above 370 nm there are increasing numbers of lines in 
+the D2 lamp spectrum in addition to the continuum radiation.  The original irradiance calibration 
+of our lamp was made with a 4 nm bandpass,9 effectively smoothing over the increasing forest of 
+lines.  At high resolution these lines are resolved and care must be taken to use only continuum 
+radiation when calibrating the metal line intensities.  For past studies we have estimated 
+corrections such that the corrected irradiance gave the irradiance of the continuum only rather 
+than an average of continuum plus lines, but such corrections introduce additional uncertainty in 
+the calibration. 
+The detector used in this calibration is a Hamamatsu S2281 silicon photodiode calibrated 
+at NIST over the wavelength range 200 – 1100 nm.  The accuracy of this calibration is 1.2 - 0.34 
+% over the 200 – 410 nm range of the present study.  A line source is also required and we have 
+chosen a Hg pen lamp because it has a spectrum sparse enough that only one to a few lines are 
+transmitted through each of the narrowband optical filters employed, as described below.  It is 
+also necessary that the source has short term stability over the period of several hours which is 
+the case for the Hg pen lamp.  It does not need to have long term stability.  Also required for this 
+calibration are several narrowband optical filters which allow a subset of Hg lines through each 
+filter.  We have used filters centered at wavelengths of 250 nm, 296 nm, 313 nm, 365 nm, 405 
+nm and 436 nm.  In addition we have used a sharp-cut colored glass filter (Corning 0-56) to 
+block the strong 254 nm light from leaking through the 296 and 313 nm filters.  The narrowband 
+filters are ½ inch diameter, and are mounted in a ten position filter wheel for ease and 
+reproducibility of switching from one to the next.  One position in the filter wheel is left open 
+with no filter installed to allow unfiltered light from the D2 lamp through. 
+Figure 1 shows a schematic of the measurement layout.  Two lamps are employed, the 
+Hg pen lamp and the D2 lamp, each mounted at one of the positions viewed by the steering 
+mirror on a kinematic mount.  Light from either lamp is imaged on the entrance pinhole of the 3-
+m echelle spectrometer with a focusing mirror. The Hg pen lamp is rotated in its holder such that 
+the pair of capillaries are viewed side-on rather than front on, to limit structure in the image.  
+Light from the source passes through an iris, which limits the cross section of the beam, and then 
+                                                            
+9 private communication from Optronics Laboratories 
+
+through the filter wheel before reaching the pinhole.  When the filter wheel is set to either the 
+296 nm or 313 nm filter, a two inch square colored glass filter (Corning 0-56 sharp-cut filter) is 
+mounted just in front of the iris (not shown in Figure 1).  The calibrated photodiode is moved 
+into the path between the filter wheel and the entrance pinhole to measure the power of light 
+transmitted by each filter.  This is done at both the beginning of measurement and then again at 
+the end, to make sure the lamp has remained stable.  The photodiode is removed for echelle 
+measurements.  A full UV spectrum (three frames) is recorded for light passing through each 
+filter.  An unfiltered D2 spectrum is recorded on each frame.  Calibrated line intensities are 
+determined for all lines getting through each filter by dividing integrated line intensities by the 
+D2 continuum intensity, using the same analysis software and method as for the Si I,II BFs, as 
+described above.  We use the unfiltered D2 spectrum to determine the filtered line intensities so 
+that the D2 intensity removes the sinc2 dependence of the grating order envelope from the 
+intensities but does not remove the effect of the filter bandpass.  The calibration of the 
+photodiode is transferred onto the D2 lamp relative irradiance by insisting that the sum of line 
+intensities through each filter be proportional to the photodiode measurement for each filter (in 
+Amps) divided by the responsivity of the photodiode (in Amp/Watt) and divided by the 
+wavenumber of the transition(s) to convert Watts into something proportional to photons/s.  The 
+level of reproducibility for this calibration can be seen in Figure 2 which shows two such 
+measurements of the relative D2 irradiance made approximately one month apart.  Since the new 
+Figure 1.  Schematic of the set up for the Hg pen lamp + NIST calibrated photodiode calibration technique. 
+
+steering mirror on
+kinematicmount
+focusing
+mirrorpinhole
+for3-m
+echelle
+irisPhotodiode:
+moved outof
+opticalpath
+D2lamp
+filterwheel
+Hgpen lamp
+duringechelle
+measurementscalibration only extends down to 250 nm, we use a calibration from our windowless Ar mini-arc 
+lamp (λ < 232nm) and our little-used D2 lamp transferred to our everyday D2 lamp to bridge the 
+gap between these two calibrations. 
+It should be mentioned that the Hg pen lamp is not a pure line source but also has a weak 
+continuum component.  The paper by Reader, Sansonetti & Bridges (1996) drew our attention to 
+this problem.  The weak continuum peaks around 405 nm, but there is also significant continuum 
+associated with the self-absorption on the strong 254 nm line.  This continuum contributes to the 
+power measured with the photodiode, but is not accounted for in the filtered line intensity 
+measurements.  The problem can be mitigated to some extent by choosing a narrower bandpass 
+for the filter.  In the current study we have employed mostly 10 nm bandpass filters, but used a 5 
+nm bandpass filter at 405 nm where the continuum was strongest.  The narrower bandpass 
+reduces the contribution of the continuum relative to the lines.  The residual continuum was 
+accounted for by making a measurement of the ratio of line intensity to line+continuum intensity 
+for each filtered spectrum that had some continuum contribution (these were the 250 nm, 365 
+nm, 405 nm and 436 nm filters).  This ratio was then applied as a correction to the photodiode 
+readings in the measurements described above.   
+We estimate the uncertainty of the calibration to be ~3 – 5% at each point of the curve 
+shown in Figure 2.  However, because the D2 irradiance changes smoothly and gradually with 
+wavelength, the uncertainty of the relative calibration between two points on the curve will be 
+less than this estimate and depends on the spacing of the lines being calibrated.  A BR for two 
+closely spaced lines, such as the Si II doublet discussed below, will have little contribution to the 
+Figure 2.  Relative D2 lamp irradiance between 250 nm and 436 
+nm as measured on two separate dates using the Hg pen lamp + 
+NIST calibrated photodiode calibration method as described in 
+the text. 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+250
+300
+350
+400
+450
+Relative D2 Irraddiance (W)
+Wavelength (nm)
+19‐Mar‐22
+16‐Apr‐22
+
+uncertainty from the calibration whereas lines that are widely separated in wavelength will have 
+a higher contribution to the BR uncertainty.  We include a systematic uncertainty of 0.001% per 
+cm-1 of wavenumber difference between the line and the dominant line from the upper level as a 
+conservative estimate of uncertainty in the radiometric calibration.  This is then added in 
+quadrature to the statistical uncertainty.  We estimate the statistical uncertainty as the larger of 
+twice the standard deviation of the weighted mean branching ratio and the inverse of the 
+weighted average signal to noise ratio.  The uncertainties of the BRs are then combined using an 
+appropriate error propagation formula to determine the final BF uncertainties. 
+ 
+3. Results and Discussion 
+3.1  
+Si I results 
+The experimental work on Si I transition probabilities to date has been limited.  Garz et 
+al. (1973) determined relative f-values for 51 lines between 250 and 800 nm from emission 
+measurements on a wall stabilized arc.  They tied these to an absolute scale using radiative 
+lifetimes of Marek (1972).  These were later renormalized with new radiative lifetime 
+measurements by Becker et al (1980).  Smith et al. (1987; hereafter Sm87) reported experimental 
+BFs or BRs and log(gf)s (the log of the level degeneracy multiplied by the oscillator strength) for 
+108 lines of Si I between 163 and 410 nm.  They used a combination of techniques including 
+emission and absorption (Hook) measurements that they tied together using the bowtie method to 
+produce a set of self-consistent relative f-values.  They chose the beam-foil lifetime 
+measurements of Bashkin et al. (1980) to establish their absolute scale.  O’Brian & Lawler 
+(1991, hereafter OL91) measured radiative lifetimes to 5% accuracy for 47 odd-parity levels of 
+Si I and then combined their lifetimes with the BFs of Sm87 for 36 lines originating in 13 of the 
+lower-lying levels that Sm87 studied.  Levels above the 3s23p3d 1Po1 level at 53387 cm-1 were 
+deemed by OL91 to have strong infrared branches, and the BFs of Sm87, having only estimated 
+the strength of these transitions, were thought to be less reliable. 
+There have been a number of theoretical investigations of Si I.  Recent studies include the 
+work of Froese Fischer (2005) who used the Breit-Pauli approximation for all levels in Si I up to 
+3s23p3d 3Do.  Savukov (2016; hereafter Sav16) used the configuration-interaction plus many-
+body-perturbation-theory (CI+MBPT) method to determine transition probabilities, log(gf)s and 
+lifetimes for levels of Si I up to the 3s23p5s 1Po1 level.  Wu et al. (2016) used the multi-
+configuration Dirac-Hartree-Fock (MCDHF) and active space approach to determine levels, 
+hyperfine structure and transition probabilities in Si I up through the 3s23p4d 3Do levels.  Finally, 
+the thesis work of Pehlivan Rhodin (2018; hereafter PR18) used MCDHF method using the 
+GRASP2K package to determine transition probabilities in Si I up through the 3s23p7s and in Si 
+II up through the 3s27f configuration. 
+
+Our measured BFs of Si I are presented in Table 2 organized by upper level.10  Also in 
+this table we compare to a subset of the experimental BFs of Sm87.  Note that for several of the 
+weak, spin-forbidden transitions Sm87 only report an upper bound (although what is meant by 
+<0.000 for the 3Do1,2 – 1D2 BFs is unclear).  In this study, we report the first measurements of 
+these very weak BFs.  For lines in common between the two studies, we see an average 
+fractional difference (in the sense (Sm87 – UW)/UW) of +6.0% with a standard deviation of 
+10.3%.  For lines with BFs > 0.01 the average fractional difference is +1.7% with standard 
+deviation of 5.6%.  
+As a point of reference, we also compare to BFs calculated from LS coupling (also 
+known as Russell-Saunders coupling) theory for the triplet multiplets in Table 2.  The upper 3p4s 
+3Po1 and 3Po2 levels at 39760 and 39955 cm-1 are nearly pure, with NIST ASD giving the leading 
+percentages as 98 and 99%, respectively.  The J=1 level has ~1% mixing with the 3p4s 1Po1 level 
+resulting in weak decays to 1D2 and 1S0 lower levels.  The upper 3s3p3 3Do1 and 3Do2 levels at 
+45276 and 45294 cm-1 are listed in the NIST ASD as 56% from that configuration and 39% 3pnd 
+3Do, but probably have some mixing with nearby 1Po1 and 1Do2 levels, respectively, since both 
+have weak decay to the 3s23p2 1Do2 level at 6299 cm-1.  The LS BFs are calculated from relative 
+line strengths tabulated in Appendix I of Cowan (1981).  Frequency-cubed scaling is included, 
+and the LS BFs are renormalized to the total multiplet strength as measured in the current study. 
+Our measured BFs are converted to A-values and log(gf)s following the relations in 
+Thorne et al. (1988),  
+𝐴�� � 𝐵𝐹��
+𝜏�
+   ;  log�𝑔𝑓� �  log �1.499𝑔�𝐴��
+𝜎�
+�                  ,                                �2� 
+ 
+where Aul is the transition probability in s-1, u is the radiative lifetime of the upper level in s, gu 
+is the degeneracy of the upper level, and  is the transition wavenumber in cm-1.  We use the 
+radiative lifetimes measured previously in our group by OL91 to establish the absolute scale for 
+our BFs.  The uncertainty of the A-value is the uncertainty of the BF and that of the lifetime 
+added in quadrature.  We present A-values and log(gf)s in Table 3.  Also in Table 3 we compare 
+to two of the recent theoretical calculations, those by Sav16 and PR18.11   
+ 
+Sav16 determined transition probabilities, log(gf)s and lifetimes only for the low-lying 
+levels of Si I up to the 3s23p5s 1Po1 levels at ~54870 cm-1.  As such, that study is limited in scope, 
+                                                            
+10 Throughout this paper and accompanying tables, Ritz wavelengths and energy levels are taken from the National 
+Institute of Standards and Technology Atomic Spectra Database (NIST ASD; Kramida, Ralchenko & Reader 2021). 
+11 We do not make comparison to the best experimental measurements in Table 3.  NIST ASD references the results 
+of OL91 (for all but the weakest lines) which combine new lifetime measurements with BFs from Sm87.  Our results 
+are not independent from OL91 as we use their lifetimes.  We would like to alert the reader that there appears to be an 
+error in the A-values and log(gf)s in the NIST ASD for two of the transitions included in this study: 2443.365 Å and 
+2452.118 Å.  NIST ASD log(gf)s are +0.32 and -0.53 dex different, respectively, from those found in OL91.  This 
+discrepancy is also found in the critical compilation on Silicon by Kelleher & Podobedova (2008). 
+
+but achieves relatively high precision on the transitions that it covers by fine-tuning the cavity 
+size, which in turn reduced the basis needed for the lowest states.  Sav16 makes detailed 
+comparison to earlier theory of Froese-Fischer (2005) and the experimental A-values and 
+radiative lifetimes reported in OL91.  We find that we are in good agreement with Sav16 for the 
+20 transitions studied here even for the weakest transitions down to log(gf) < -4.  The average 
+fractional difference between our A-values (in the sense (Sav16 – UW)/UW) is +1.7% with a 
+standard deviation of 9.7%. 
+ 
+We also compare in Table 3 to the MCDHF calculations of PR18 who determined 
+transition probabilities for Si I belonging to the even 3s23p2, 3s23pnp (n ≤ 7), and 3s23pnf (n ≤ 6) 
+configurations and to the odd 3s3p3, 3s23pns (n ≤ 8), and 3s23pnd (n ≤ 6) configurations.  Here 
+we find that the agreement with our measured transition probabilities is also very good, with 
+average fractional difference (in the sense (PR18 – UW)/UW) of -8.5% with a standard deviation 
+of 13.5%.  This improves to an average of -3.9% and standard deviation of 10.4% for lines with 
+log(gf)>-3.    Unlike Sav16, the PR18 study is a comprehensive calculation involving over 100 
+levels up to 61936 cm-1 and more than 1300 transitions ranging in wavelength from 6333 nm in 
+the infrared to 161 nm in the vacuum-UV.  As such, it will prove a very valuable resource for 
+astronomers. 
+ 
+The comparisons made in Table 2 and Table 3 are visualized in Figure 3, where we 
+present logarithmic differences (in the sense log(other) – log(UW) versus log(UW)) of the 
+experimental BFs of Sm87 in panel (a) and the log(gf)s of PR18 and Sav16 in panels (b) and (c), 
+respectively.  In panels (a) and (b) the error bars represent the combined uncertainties added in 
+quadrature.  (The uncertainties reported in PR18 are the relative difference between the length 
+and velocity gauges.)  Sav16 did not give uncertainties for their A-values so no error bars are 
+plotted in panel (c). In panel (a) the point with an arrow beside it is the upper bound quoted in 
+Sm87 for the transition at 4102 Å.  The weakest, spin-forbidden transitions in these comparisons 
+are very difficult to measure and to calculate.   The level of agreement with recent theory, both 
+with the limited-in-scope but high precision calculations of Sav16, and with the comprehensive 
+calculations of PR18, is very satisfactory. 
+ 
+ 
+3.2 
+Si II results 
+We have remeasured the BF of the very weak spin-forbidden 4P1/2 - 2Po1/2,3/2 doublet of Si 
+II at 2334.407 Å and 2350.172 Å, respectively, using the first eight spectra listed in Table 1.  
+Optical depth is not a concern in this measurement because of the weakness of the transitions.  
+This BF had previously been measured in our group and reported in Calamai, Smith & Bergeson 
+(1993, hereafter CSB93).  That paper had reported the measurement of the radiative lifetimes of 
+the 4P1/2,3/2,5/2 levels as well as the BFs of the 4P1/2 level.  We use the radiative lifetime of CSB93 
+to convert our BFs to A-values.  These are reported in Table 4 along with comparison to the 
+CSB93 measurement.  CSB93 report that these lines had signal-to-noise ratios of 10-15 in their
+
+‐1.0
+‐0.8
+‐0.6
+‐0.4
+‐0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+‐4.5
+‐4.0
+‐3.5
+‐3.0
+‐2.5
+‐2.0
+‐1.5
+‐1.0
+‐0.5
+0.0
+log(BF)Sm87 ‐ log(BF)UW
+log(gf)UW
+‐1.0
+‐0.8
+‐0.6
+‐0.4
+‐0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+‐4.5
+‐4.0
+‐3.5
+‐3.0
+‐2.5
+‐2.0
+‐1.5
+‐1.0
+‐0.5
+0.0
+log(gf)PR18 ‐ log(gf)UW
+log(gf)UW
+‐1.0
+‐0.8
+‐0.6
+‐0.4
+‐0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+‐4.5
+‐4.0
+‐3.5
+‐3.0
+‐2.5
+‐2.0
+‐1.5
+‐1.0
+‐0.5
+0.0
+log(gf)Sav16 ‐ log(gf)UW
+log(gf)UW
+c 
+a 
+b 
+Figure 3.  Comparison of log(BF)s or log(gf)s of Si I in the present 
+work to those of a) the experimental results of Sm87, b) theoretical 
+MCDHF calculations of PR18, and c) theoretical CI+MBPT 
+calculations of Sav16 versus log(gf) from this study.  In each figure 
+the horizontal line at 0.0 represents perfect agreement.  Error bars 
+represent combined uncertainties where available.  See text for 
+further discussion. 
+
+spectra whereas we have signal-to-noise ratios ranging from 45 to 200.  The radiometric 
+calibration does not significantly contribute to the uncertainty of our BF because of the small 
+wavelength span between the doublet, resulting in an uncertainty that is primarily statistical.  The 
+superior signal-to-noise in our spectra explains why our uncertainties are lower than those of 
+CSB93.  We also compare to recent theoretical results of PR18 and Wu et al. 2020 in Table 4.  
+CSB93 appear to have used the theoretical BF of Nussbaumer (1977) to convert their 
+lifetime for the 4P3/2 level to A-values for the 4P3/2 - 2Po1/2,3/2 doublet at 2328.517 Å and 2344.202 
+Å.  This is not stated clearly in their paper, and in fact they state “Thirty-four measurements of 
+the 4P3/2 branching fraction were made.  The total uncertainty (systematic and statistical) was 
+about 10% at the 90% level of confidence.”  This appears to be a typo, and refers to the 
+measurement and uncertainty of the 4P1/2 BF.  It is stated clearly elsewhere in the paper that a BF 
+was measured for only one level, the 4P1/2 level, and the 10% uncertainty mentioned in the quote 
+is not consistent with the 50% uncertainty on the weak branch of the 4P3/2 level. We attempted a 
+BF measurement of the 4P3/2 - 2Po1/2,3/2 doublet at 2328.517 Å and 2344.202 Å, but were 
+unsuccessful.  The weaker 2328 Å line of this pair is estimated by the theory of Nussbaumer 
+(1977) and that of Dufton et al. (1991) to be a ~1% branch. Although we saw a weak feature at 
+this wavelength in our higher current Si-Ne spectra, we decided that this feature was a blend with 
+a very weak neon line.  There is no observed transition listed at this wavelength in the NIST 
+ASD neon spectrum, but there is a possible Ne II electric dipole transition nearby that obeys 
+parity and J selection rules.  Our analysis software looks for these possibilities based on known 
+energy levels of both the metal and buffer gas first and second spectra.  We investigated this 
+further by looking at this wavelength in high current Hf-Ne and Hf-Ar spectra taken for a 
+different study.  In these spectra we also saw a very weak feature in the Hf-Ne spectra but not in 
+the Hf-Ar spectra, suggesting a neon blend.  Unfortunately, switching to a Si-Ar lamp does not 
+help in this case because the other line in the doublet pair, 2344.202 Å, has a known argon blend.  
+We attempted to procure a third commercial HCL with krypton buffer gas which has no potential 
+blends on either line, but were unsuccessful.  The most we can say regarding the weak line at 
+2328.517 Å is that it is less than a 4.5% branch with an upper bound of log(gf) < -6.7.  
+The 4P - 2Po intercombination lines have been part of numerous theoretical investigations 
+of Si II.  These lines are allowed E1 transitions due to the mixing of the 3s3p2 4P levels with 
+doublets from the same configuration.  The accuracy of calculated radiative rates depend on the 
+accuracy to which the mixing coefficients and the multiplet energy splittings are calculated.  
+Nussbaumer (1977) used the SUPERSTRUCTURE code to calculate radiative parameters from 
+sophisticated configuration interaction wavefunctions.  Dufton et al. (1991) significantly 
+improved on those results by including a more extensive set of configurations.  These lines were 
+included in the calculations of Froese Fischer (2006) and Tayal (2007) using the MCHF method.  
+Bautista (2009) calculated radiative rates between many configurations using several different 
+approximations and generated a list of recommended log(gf)s for transitions among the 15 lowest 
+levels in Si II.  Aggarwal & Keenan (2014) used the General-purpose Relativistic Atomic 
+
+Structure Package (GRASP()) and the Flexible Atomic Code (FAC) to calculate a large number 
+of radiative parameters and collision strengths in Si II, but estimate ~20% uncertainty on the 
+strong transitions with weak transitions such as these intercombination lines being much more 
+uncertain.  PR18 calculate A-values for these intercombination lines using the MCDHF method 
+and GRASP2K package with uncertainties based on the relative difference between the length 
+and velocity gauges of ~19% and 12% for the 2334.407 Å and 2350.172 Å lines, respectively.  
+Finally Wu et al. (2020) also used the MCDHF method and the GRASP2K package in their 
+study of Si II.  In Figure 4 we make comparison to the experimental results for the BR (4P1/2 - 
+2Po3/2)/(4P1/2 - 2Po1/2) of CSB93 and to the above-mentioned theoretical studies, with the exception 
+of the Aggarwal & Keenan (2014) study.  The BR from that study lies significantly off-scale on 
+Figure 4 at 1.37.  In this figure the horizontal line is simply a guide to the eye, and lies at the 
+experimental value determined in this study.  It can be seen from this figure that the general level 
+of agreement between experiment and theory has improved dramatically over recent decades, 
+undoubtedly owing, at least in part, to rapid increase in computing power.  We see particularly 
+excellent agreement between our study and the recent theoretical results of Wu et al. (2020) and 
+PR18 as well as that of Froese Fischer et al. (2006). 
+4. Silicon Abundances in Very Metal-Poor Stars 
+All but two of the transitions studied here lie in the ultraviolet (UV) spectral domain 
+below the atmospheric absorption cutoff, i.e. 𝜆 < 3000 Å.  This limits high-resolution stellar 
+spectroscopy to the Space Telescope Imaging Spectrograph (STIS; Kimble et al. 1998; 
+Figure 4.  Experimental and theoretical values determined for the BR of the 
+(4P1/2 - 2Po3/2)/(4P1/2 - 2Po1/2) doublet of Si II.  The two experimental 
+measurements are leftmost followed by the theoretical values in reverse 
+chronological order left to right.  The horizontal line lies at the BR as 
+measured in this work as a guide for the eye. 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+BR (4P1/2 ‐ 2P3/2)/(4P1/2 ‐ 2P1/2)
+This Expt. 
+ CSB93  
+Wu et al. 2020  
+PR18  
+Tayal 2007  
+Froese Fischer 2006  
+Dufton et al. 1991  
+Nussbaumer 1977  
+Bautista 2009
+‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Theory ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 
+‐‐‐ Expt. ‐‐‐ 
+
+Woodgate et al. 1998), on board the Hubble Space Telescope (HST).  Additionally, the UV 
+spectrum is crowded with strong absorption lines of light and Fe-group elements, making reliable 
+abundance analyses difficult to execute.  The UV spectral region of cool stars features complex 
+blends of transitions with various pedigrees, ranging from prominent well-known lines that have 
+well-documented laboratory histories to many moderate and weak lines with poor or completely 
+unknown atomic parameters.   
+The UV lines of neutral Si studied here are almost all very strong, having low excitation 
+energies (𝜒 < 6299 cm-1 or < 0.8 eV) and relatively large transition probabilities (17 out of 20 
+lines in Table 3 have log(gf) > –3).  The problem here is not in identifying Si I lines; it is in 
+finding stars with lines that are weak enough for abundance analysis.  With this unusual 
+constraint we concentrated on metal-poor (Fe/H] < –2) halo stars that have been observed by 
+HST/STIS.  The list is small:  7 stars are considered in the metallicity study of Roederer et al. 
+(2018); the bright main sequence star HD 84937 ([Fe/H] �  –2.2) has been featured in previous 
+papers in this series (Den Hartog et al. 2021, and references therein); the famous warm low 
+metallicity stars HD 19445 and HD 140283 (Chamberlain & Aller 1951) have been featured in 
+several UV line identification contributions (e.g., Peterson et al. 2020 and references therein); the 
+mildly metal-poor warm giant HD 222925 ([Fe/H] = -1.5) has been recently studied by Roederer 
+et al. (2022) to produce a nearly complete abundance set for 63 elements.  A few other such stars 
+can be found but do not change the basic results which we will discuss here. 
+We employed HST/STIS spectra of seven of the stars included in the papers cited above 
+in order to explore if Si abundances derived from UV spectra could be more trustworthy than the 
+few optical-wavelength lines treated in the literature.  We supplemented our HST/STIS spectra 
+with blue spectra collected using the High Resolution Echelle Spectrometer (Vogt et al. 1994) at 
+the Keck I telescope, and the Ultraviolet and Visual Echelle Spectrograph (Dekker et al. 2000) at 
+the Very Large Telescope.  We accessed these data through the Keck Observatory Archives and 
+European Southern Observatory Archives, respectively, and Table 1 of Roederer et al. (2018) 
+presents a description of these data. 
+We derived Si abundances using synthetic/observed spectrum matches.  The synthetic 
+spectra were computed with the plane-parallel LTE (local thermodynamic equilibrium) line 
+analysis code MOOG (Sneden 1973)12.  Atomic line lists for these syntheses were generated with 
+the linemake facility (Placco et al. 2021)13, which emphasizes laboratory transition data on Fe-
+group and neutron-capture neutral and singly-ionized species from the Wisconsin atomic physics 
+group and on molecular species from the Old Dominion University group (e.g., Brooke et al. 
+2016, and references therein).  We adopted the atmosphere parameters of Roederer et al. 2018, 
+2022) to produce model atmospheres interpolated from the ATLAS grid (Kurucz 2011, 2018)14.  
+                                                            
+12 Available at https://www.as.utexas.edu/~chris/moog.html 
+13 https://github.com/vmplacco/linemake 
+14 http://kurucz.harvard.edu/grids.html 
+
+For almost all stars the lower wavelength boundary of our HST/STIS spectra was 𝜆 �  2300 Å, 
+thus ruling out work on the five lowest-wavelength Si I transitions.   
+Our initial synthetic spectrum tests yielded results that further narrowed the range of 
+stellar parameters that are useful for this abundance exercise.  For stars that have [Fe/H] > –2.5 
+and effective temperatures Teff < 6000 K, many of the promising Si I lines simply are too strong 
+and/or too blended with other strong neutral and ionized species features to yield reliable 
+abundances.  In particular, we discarded the giant star HD 222925 (Teff/log(g)/[M/H]/vt = 
+5636K/2.54/–1.5/2.20km s-1; Roederer et al. 2022) and the subgiant HD 140283 (5600K/3.66/-
+2.6/1.15km s-1; Roederer et al. 2018).  We report here on five very metal-poor main sequence 
+turnoff stars that have Teff � 6050 K. 
+In Table 5 we list the model parameters, individual line abundances, and final species 
+abundances for both Si I and Si II transitions in the program stars.  The mean abundances are 
+based on 10-11 Si I lines and 2 Si II lines, all in the vacuum UV spectral domain, whereas in 
+previous studies the Si abundances of these kinds of stars have come almost exclusively from the 
+optical Si I transitions at 3905.5 and 4102.9 Å.  We derive <[Si/Fe]I> = 0.43 (𝜎 = 0.11).  The 
+inclusion of the ionized species in Si abundance studies is a rarity, and for our program stars the 
+abundance agreement between neutral and ion is excellent.  From Table 5 we find <[Si/Fe]II –
+[Si/Fe]I> = +0.03 (𝜎 = 0.05).  In Figure 5 we show small spectral regions around both Si II lines 
+and around six representative Si I lines in the program star BD+03º 740.  For this star and the 
+other two lowest metallicity stars BD-13º 3442 and CD-33º 1173 the Si II lines are essentially on 
+the weak-line linear part of the curve of growth.  They are easy to detect, and to employ in 
+abundance analyses. Many Si I lines are also reliable abundance indicators.  However, the 2516, 
+2519, and 2881 Å transitions are clearly saturated and thus less sensitive to abundance changes.  
+In cooler, higher metallicity stars such as HD 19445 and HD 84937 these and other lines become 
+so strong that they are untrustworthy for abundance determinations.  Some caution should be 
+used in interpreting the Si abundances of those stars. 
+We also derived abundances for the Si I 3905 Å line and list them in Table 5.  The 4102.9 
+Å Si I line was too weak and too blended with the strong H𝛿 4101.75 Å feature in our stars.  
+However we did not include the 3905 Å line in the mean abundance calculations because this 
+transition is known to yield temperature-dependent abundances in LTE calculations.  Si in metal-
+poor giants from the 𝜆3905 line is almost uniformly overabundant, <[Si/Fe]> ~ +0.4 ± 0.1 (e.g, 
+Cayrel et al. 2004), but is much less abundant in main sequence stars near the turnoff region, 
+<[Si/Fe]> ~ +0.1 ± 0.1 (e.g, Cohen et al. 2004).  The sample of horizontal-branch stars 
+investigated by Preston et al. (2006) covers a large temperature range and shows this effect 
+clearly in their Figure 8.  A summary of the observational issues in LTE abundances was 
+discussed by Sneden & Lawler (2008).  From Table 5 we compute <[Si/Fe]> = +0.28 (𝜎 = 0.11) 
+from the 3905 Å line, clearly lower than the mean from the UV Si I lines discussed above.  
+Amarsi & Asplund (2017) computed NLTE corrections for optical-wavelength Si I transitions in
+
+      
+Figure 5:  Observed and synthesized spectra for both Si II lines (the 2 bottom panels) 
+and for representative lines of Si I (the 6 upper panels) in the star BD+03º 740.  In each 
+panel, the filled circles are the observations.  The red line is a synthesis without any 
+contribution from Si.  The best fit to a line is given by the black line, and the blue and 
+green lines show the synthetic spectra for Si abundances 0.4 dex lesser and greater than 
+the best match. 
+
+the solar photosphere, and have published on-line tables of NLTE corrections for many 
+(Teff/log(g)/[Fe/H]/vt) combinations.15 Their suggested correction for the 3905 Å line in stars with 
+parameters (6000 K/4.0/-3.0/1-2 km s-1) is D[Si/Fe] @ +0.1 dex.  Applying this adjustment to the 
+abundances from this line for our stars would bring the 3905 Å line into better agreement with 
+our abundances derived from the UV Si I transitions.  Abundances from the UV lines should be 
+preferred. 
+5. Discussion 
+  In Figure 6 we illustrate the Galactic Chemical Evolution (GCE) trends of [Si/Fe] as a 
+function of metallicity ([Fe/H]).  Silicon is synthesized in explosive oxygen burning, and thus is 
+formed in core-collapse supernovae early in the history of the Galaxy and then ejected into the 
+gas that eventually forms the halo stars. (Curtis et al. 2019)  We show a compilation of 
+                                                            
+15 Anish Amarsi - Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University - Astronomy 
+and Space Physics Theoretical astrophysics Department of Physics and Astronomy Uppsala University Box 516, 
+75120 Uppsala Sweden; Email: [email protected] ; www.astro.uu.se 
+Figure 6. The [Si/Fe] abundance ratios as a function of metallicity ([Fe/H]) 
+for metal-poor stars from Roederer et al. (2014) (blue open squares) and this 
+paper (red filled circles).   
+
+口
+口
+0.5
+口口口
+I/Fe]
+口
+0
+中
+S口
+Roederer et al. (2014)
+Thispaper
+3
+2
+[Fe I /H]abundance data, [Si/Fe], from an earlier survey of low-metallicity Galactic stars (Roederer et al. 
+2014; shown as open squares). The values of [Si/Fe] exhibit significant scatter over the observed 
+metallicity range. This could be the result of comparing different types of stars (i.e., dwarfs with 
+giants) or due to the choice of the atomic lines used for the abundance determinations and/or the 
+source of the log(gf)s employed.  Employing our new experimental silicon data (discussed 
+above, see Tables 3 and 4) leads to a more consistent pattern with less scatter.  For the five stars 
+in this study (shown as filled red circles in Figure 6) the average value of [Si/Fe] = 0.44, 
+significantly higher than the solar value of 0.  This value can serve as a constraint on GCE 
+models and, in particular, on supernovae nucleosynthesis model predictions for early Galactic 
+times.  
+It would be expected that the [Si/Fe] values illustrated in Figure 6 would begin to exhibit 
+a downward pattern at metallicities closer to [Fe/H] = -1 with the onset of Type Ia supernovae 
+(the main producer of iron) throughout the Galaxy.  The abundance data from Roederer et al. 
+(2014) does hint at such a downward trend, but clearly more studies employing the new precise 
+atomic data in somewhat more metal-rich stars will be needed to confirm such a trend. 
+ 
+6. Conclusions 
+We have made new BF measurements for 20 UV and blue lines of Si I as well as the 
+4P1/2 intercombination lines of Si II.  Comparisons are made to earlier experiment as well as 
+theory.  These BF have been combined with radiative lifetimes measured previously to determine 
+A-values and log(gf)s for these transitions.  The current study represents a significant 
+improvement in measurement of the very weak spin-forbidden lines of both Si I and Si II.  These 
+new data have been applied to abundance determinations in five metal-poor main sequence 
+turnoff stars.  We find that many of the Si I UV transitions can be used as reliable abundance 
+indicators in very metal-poor stars and we obtain excellent agreement between abundances 
+determined using Si I transitions and the Si II intercombination lines.  
+ 
+ACKNOWLEDGEMENTS 
+This work is supported by NSF grant AST-1814512 and AST-2206050 (E.D.H. and J.E.L).  
+I.U.R. acknowledges support from NSF grants AST 2205847 and PHY 14-30152 (Physics 
+Frontier Center/JINA-CEE), and NASA grants GO-14232, GO-15657 and AR-16630 from the 
+Space Telescope Science Institute, which is operated by the Association of Universities for 
+Research in Astronomy, Incorporated, under NASA contract NAS5-26555. We are grateful to 
+Hampus Nilsson for sharing the Si I, II data from the Pehlivan Rhodin (2018) thesis prior to its 
+publication, and to Karen Lind for helpful discussions.    
+Facilities: HST (STIS), Keck I (HIRES), VLT (UVES). 
+Software: LINEMAKE (Placco et al. 2021), MOOG (Sneden 1973). 
+ 
+
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+thesis, Lund University, Lund, Sweden. 
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+
+Table 1.  Echelle spectra of commercial HCLs used in the study of Si II and Si I BFs.a 
+Indexb 
+Date 
+Serial 
+Number 
+Buffer  
+Gas 
+Lamp 
+Current 
+(mA) 
+Spectral Coverage 
+(Å) 
+Coadds 
+Total 
+Exposure 
+(min) 
+11 
+2021 Jul 31 
+2 
+Neon 
+12 
+2090 - 2955 
+120 
+360 
+12 
+2021 Aug 03 
+2 
+Neon 
+12 
+2090 - 2955 
+5 
+150 
+13 
+2021 Aug 12 
+1 
+Neon 
+12 
+2090 - 2955 
+6 
+180 
+14 
+2021 Aug 27 
+1 
+Neon 
+20 
+2090 - 2955 
+15 
+150 
+15 
+2021 Aug 29 
+1 
+Neon 
+22 
+2040 - 2790 
+18 
+180 
+16 
+2021 Sept 04 
+1 
+Neon 
+15 
+2040 - 2790 
+3 
+180 
+17 
+2022 Apr 02 
+1 
+Neon 
+25 
+2040 - 2790 
+4 
+120 
+18 
+2022 Apr 05 
+1 
+Neon 
+25 
+2040 - 2790 
+16 
+160 
+31 
+2021 Dec 17 
+1 
+ Neon 
+12 
+2150 - 3245 
+80 
+120 
+32 
+2021 Dec 17 
+3 
+ Neon 
+12 
+2350 - 4300 
+144 
+120 
+33 
+2022 Jan 06 
+3 
+ Neon 
+18 
+2150 - 3245 
+90 
+60 
+34 
+2022 Jan 06 
+1 
+ Neon 
+18 
+2350 - 4300 
+120 
+60 
+35 
+2022 Jan 06 
+5 
+ Neon 
+6 
+2150 - 3245 
+15 
+75 
+36 
+2022 Jan 06 
+7 
+ Neon 
+6 
+2350 - 4300 
+37 
+74 
+37 
+2022 Jan 08 
+1 
+ Neon 
+12 
+2350 - 4300 
+720 
+120 
+38 
+2022 Jan 08 
+3 
+ Neon 
+18 
+2350 - 4300 
+90 
+90 
+39 
+2022 Jan 08 
+5 
+ Neon 
+6 
+2350 - 4300 
+60 
+60 
+40 
+2022 Jan 13 
+1 
+ Neon 
+24 
+2350 - 4300 
+120 
+60 
+41 
+2022 Jan 13 
+3 
+Neon 
+29 
+2350 - 4300 
+120 
+60 
+42 
+2022 Mar 29 
+1 
+ Neon 
+24 
+2350 - 4300 
+240 
+60 
+43 
+2022 Mar 29 
+3 
+ Neon 
+28 
+2350 - 4300 
+240 
+60 
+44 
+2022 Apr 05 
+3 
+Neon 
+25 
+2350 - 4300 
+180 
+60 
+45 
+2022 Apr 07 
+1 
+Neon 
+12 
+2280 – 4200 
+60 
+60 
+46 
+2022 Apr 07 
+3 
+ Neon 
+18 
+2280 - 4200 
+90 
+60 
+47 
+2022 Apr 07 
+5 
+ Neon 
+25 
+2150 – 3245 
+103 
+60 
+48 
+2022 May 14 
+1 
+ Argon 
+18 
+2050 – 2800 
+100 
+100 
+49 
+2022 May 19 
+1 
+ Argon 
+20 
+2050 – 2800 
+60 
+60 
+50 
+2022 May 19 
+3 
+ Argon 
+20 
+2150 – 3245 
+60 
+60 
+51 
+2022 May 19 
+5 
+ Neon 
+28 
+2150 – 3245 
+144 
+60 
+52 
+2022 May 21 
+1 
+ Neon 
+30 
+2150 - 3245 
+240 
+60 
+53 
+2022 May 21 
+3 
+ Argon 
+19 
+2350 - 4300 
+240 
+60 
+54 
+2022 May 22 
+1 
+ Neon 
+21 
+2350 -4300  
+120 
+120 
+55 
+2022 May 30 
+1 
+ Neon 
+3 
+2350 - 4300 
+24 
+120 
+56 
+2022 May 30 
+3 
+Neon 
+3 
+2280 - 4200 
+24 
+120 
+Note: 
+ aAll echelle spectra were taken from commercially manufactured Si-Ne or Si-Ar HCLs, and have a spectral 
+resolving power of ~250,000 although the effective resolving power is somewhat lower due to line broadening.  
+Each of the spectra were calibrated with a D2 lamp spectrum, which was recorded immediately following the 
+completion of each HCL spectrum.  Each spectrum listed is a single CCD frame, and does not cover an entire 
+echelle grating order, but is sufficient coverage to determine branching fractions of all transitions from one or more 
+upper levels studied. 
+bThe first eight spectra list (indices 11 – 18) were used to study the BF of the Si II intercombination lines.  The 
+remaining spectra (indices 31 – 56) were used in the Si I BF study. 
+
+Table 2.   
+Branching Fractions of Si I 
+Upper levela
+ 
+Lower levela
+lair 
+svac 
+This Expt.
+ 
+Other Expt.b
+LSc
+Configuration 
+and Term 
+Ek (cm-1) 
+Termd 
+Ei (cm-1) 
+(Å) 
+(cm-1) 
+BF 
+(±%) 
+BF 
+(±%) 
+BF 
+3s23p4s 3Po1 
+39760.285 
+3P0 
+0.000 
+2514.316 
+39760.20 
+0.337 
+(1) 
+0.333 
+(0.9) 
+0.333 
+ 
+ 
+3P1 
+77.115 
+2519.202 
+39683.17 
+0.244 
+(1) 
+0.247 
+(1.6) 
+0.248 
+ 
+ 
+3P2 
+223.157 
+2528.508 
+39537.11 
+0.409 
+(1) 
+0.407 
+(1.0) 
+0.409 
+ 
+ 
+1D2 
+6298.850 
+2987.643 
+33461.42 
+0.0103 
+(6) 
+0.012 
+(8) 
+ 
+ 
+ 
+1S0 
+15394.370 
+4102.936 
+24365.91 
+0.00056 
+(17) 
+<0.0020 
+(30) 
+ 
+3s23p4s  3Po2 
+39955.053 
+3P1 
+77.115 
+2506.897 
+39877.90 
+0.243 
+(1) 
+0.246 
+(1.2) 
+0.252 
+ 
+ 
+3P2 
+223.157 
+2516.112 
+39731.88 
+0.757 
+(1) 
+0.754 
+(0.4) 
+0.748 
+ 
+ 
+1D2 
+6298.850 
+2970.353 
+33656.18 
+0.00020 
+(10) 
+0.00027 
+(13) 
+ 
+3s23p4s  1Po1 
+40991.884 
+3P0 
+0.000 
+2438.768 
+40991.80 
+0.0030 
+(7) 
+0.0034 
+(5.9) 
+ 
+ 
+ 
+3P1 
+77.115 
+2443.365 
+40914.80 
+0.0024 
+(7) 
+0.0027 
+(7.4) 
+ 
+ 
+ 
+3P2 
+223.157 
+2452.118 
+40768.70 
+0.0022 
+(7) 
+0.0025 
+(8.7) 
+ 
+ 
+ 
+1D2 
+6298.850 
+2881.578 
+34693.02 
+0.940 
+(0.5) 
+0.934 
+(0.2) 
+ 
+ 
+ 
+1S0 
+15394.370 
+3905.523 
+25597.51 
+0.052 
+(9) 
+0.057 
+(2.2) 
+ 
+3s3p3 3Do1 
+45276.188 
+3P0 
+0.000 
+2207.978 
+45276.10 
+0.566 
+(1) 
+0.577 
+(1.4) 
+0.557 
+ 
+ 
+3P1 
+77.115 
+2211.745 
+45199.20 
+0.409 
+(1) 
+0.398 
+(2.3) 
+0.415 
+ 
+ 
+3P2 
+223.157 
+2218.916 
+45053.10 
+0.025 
+(4) 
+0.023 
+(13) 
+0.027 
+ 
+ 
+1D2 
+6298.850 
+2564.825 
+38977.34 
+0.00044 
+(26) 
+<0.000 
+(15) 
+ 
+3s3p3  3Do2 
+45293.629 
+3P1 
+77.115 
+2210.892 
+45216.60 
+0.763 
+(0.5) 
+0.760 
+(0.4) 
+0.751 
+ 
+ 
+3P2 
+223.157 
+2218.057 
+45070.40 
+0.236 
+(1) 
+0.240 
+(1.3) 
+0.248 
+ 
+ 
+1D2 
+6298.850 
+2563.679 
+38994.78 
+0.00053 
+(19) 
+<0.000 
+(5) 
+ 
+Notes: 
+a Upper and lower levels are taken from NIST ASD and are ordered by term. 
+b Sm87: Smith et al. 1987, ApJ 322, 573.  The BFs of weak lines at 2438.768 Å and 2443.365 Å have only one significant digit listed in 
+Sm87 Table 1.  We have calculated to two significant digits from their log(gf)s. 
+c LS BFs within the triplet multiplets calculated from the relative line strengths in Appendix I of Cowan (1981) and with frequency-cubed 
+scaling.  They are renormalized to the total multiplet strength from the current measurements. 
+d The configuration of all lower levels is 3s23p2 
+
+Table 3.  
+A-values and log(gf)s for 20 transitions of Si I 
+lair 
+Eupper 
+Jupper
+Elower 
+Jlower 
+This Expt.
+Sav16a
+ 
+PR18
+(Å) 
+(cm-1) 
+ 
+(cm-1) 
+ 
+Aki  (s-1) 
+(±%) 
+log(gf) 
+log(gf)
+log(gf) 
+(±%) 
+2207.978 
+45276.188 
+1 
+0.000
+0 
+2.57E+07
+(5) 
+-1.248 
+-1.229 
+-1.318 
+(6.5) 
+2210.892 
+45293.629 
+2 
+77.115
+1 
+3.47E+07
+(5) 
+-0.895 
+-0.876 
+-0.965 
+(7.5) 
+2211.745 
+45276.188 
+1 
+77.115
+1 
+1.86E+07
+(5) 
+-1.388 
+-1.372 
+-1.459 
+(5.8) 
+2218.057 
+45293.629 
+2 
+223.157
+2 
+1.07E+07
+(5) 
+-1.402 
+-1.392 
+-1.477 
+(6) 
+2218.916 
+45276.188 
+1 
+223.157
+2 
+1.13E+06
+(6) 
+-2.603 
+-2.586 
+-2.670 
+(4.4) 
+2438.768 
+40991.884 
+1 
+0.000
+0 
+7.06E+05
+(9) 
+-2.723 
+-2.684 
+-2.705 
+(5.6) 
+2443.365 
+40991.884 
+1 
+77.115
+1 
+5.52E+05
+(9) 
+-2.828 
+-2.788 
+-2.805 
+(0.7) 
+2452.118 
+40991.884 
+1 
+223.157
+2 
+5.01E+05
+(9) 
+-2.868 
+-2.829 
+-2.850 
+(3.4) 
+2506.897 
+39955.053 
+2 
+77.115
+1 
+5.39E+07
+(5) 
+-0.595 
+-0.566 
+-0.578 
+(0.9) 
+2514.316 
+39760.285 
+1 
+0.000
+0 
+7.48E+07
+(5) 
+-0.672 
+-0.667 
+-0.679 
+(0.8) 
+2516.112 
+39955.053 
+2 
+223.157
+2 
+1.68E+08
+(5) 
+-0.098 
+-0.088 
+-0.101 
+(0.9) 
+2519.202 
+39760.285 
+1 
+77.115
+1 
+5.42E+07
+(5) 
+-0.810 
+-0.793 
+-0.805 
+(0.9) 
+2528.508 
+39760.285 
+1 
+223.157
+2 
+9.08E+07
+(5) 
+-0.583 
+-0.567 
+-0.579 
+(0.8) 
+2563.679 
+45293.629 
+2 
+6298.850
+2 
+2.43E+04
+(20) 
+-3.922 
+-3.953 
+-4.078 
+(25.2) 
+2564.825 
+45276.188 
+1 
+6298.850
+2 
+2.00E+04
+(26) 
+-4.228 
+-4.298 
+-4.389 
+(28.7) 
+2881.578 
+40991.884 
+1 
+6298.850
+2 
+2.19E+08
+(5) 
+-0.088 
+-0.044 
+-0.061 
+(1.4) 
+2970.353 
+39955.053 
+2 
+6298.850
+2 
+4.44E+04
+(11) 
+-3.531 
+-3.577 
+-3.613 
+(6.4) 
+2987.643 
+39760.285 
+1 
+6298.850
+2 
+2.30E+06
+(8) 
+-2.035 
+-2.082 
+-2.113 
+(3) 
+3905.523 
+40991.884 
+1 
+15394.370
+0 
+1.22E+07
+(10) 
+-1.077 
+-0.999 
+-1.018 
+(3.5) 
+4102.936 
+39760.285 
+1 
+15394.370
+0 
+1.24E+05
+(18) 
+-3.026 
+-3.126 
+-3.154 
+(3) 
+Notes 
+a log(gf)s calculated from A-values presented in Sav16 using equation 2.
+
+Table 4. 
+Branching Fractions, A-values and log(gf)s for the 4P1/2 - 2P1/2,3/2 doublet of Si II from experiment and recent theory. 
+ 
+3s3p2 4P1/2 – 3s23p 2Po1/2 ; 
+lair = 2334.407 Å
+3s3p2 4P1/2 – 3s23p 2Po3/2 ;  
+lair = 2350.172 Å
+ 
+BF 
+A (s-1) 
+log(gf) 
+BF 
+A (s-1) 
+log(gf) 
+This Expt:a 
+0.519 ≤ 1% 
+4990 ± 16% 
+-5.088 
+0.481 ≤ 1% 
+4630 ± 16% 
+-5.116 
+Other Expt: CSB93 
+0.541 ≤ 10%
+5200 ± 19% 
+-5.070 
+0.459 ≤ 10%
+4410 ± 21% 
+-5.136 
+Theory: PR18b 
+0.520 
+5280 ± 18.9%
+-5.064 
+0.480 
+4882 ± 11.7%
+-5.092 
+Theory: Wu20b 
+0.514 
+5230 
+-5.068 
+0.486 
+4940 
+-5.087 
+Notes: 
+a our BFs are combined with the lifetime of CSB93 (104 ± 16 μs) to determine our A-value and log(gf) 
+b PR18 and Wu20 do not report BFs.  We calculate BFs from their A-values to show the excellent agreement with the BFs measured in this study. 
+ 
+ 
+ 
+ 
+
+Table 5. 
+Line-by-line abundances from Si I and Si II lines for the five metal-poor stars investigated. 
+Stellar Parameters 
+ 
+ 
+star 
+BD+03º 740 
+BD-13º 3442 
+CD-33º 1173 
+HD 19445 
+HD 84937 
+ 
+ 
+Teff 
+6351 
+6405 
+6625 
+6055 
+6300 
+ 
+ 
+log(g) 
+3.97 
+4.04 
+4.29 
+4.49 
+4.00 
+ 
+ 
+vt 
+1.7 
+1.6 
+1.6 
+1.2 
+1.5 
+ 
+ 
+[Fe I/H] 
+-2.89 
+-2.84 
+-2.98 
+-2.14 
+-2.24 
+ 
+ 
+[Fe II/H] 
+-2.78 
+-2.73 
+-2.90 
+-2.17 
+-2.26 
+ 
+ 
+source 
+Cowan20 
+Cowan20 
+Cowan20 
+Roederer18 
+Sneden16 
+Line-by-Line Abundances – Si I  
+λ (Å) 
+χ (eV) 
+log(gf) 
+[Si I/Fe] 
+[Si I/Fe] 
+[Si I/Fe] 
+[Si I/Fe] 
+[Si I/Fe] 
+2438.768 
+0.000 
+-2.723 
+0.34 
+0.29 
+0.48 
+0.64 
+0.57 
+2443.365 
+0.010 
+-2.828 
+0.37 
+0.34 
+0.48 
+0.59 
+0.47 
+2452.118 
+0.028 
+-2.868 
+0.41 
+0.34 
+0.43 
+… 
+0.42 
+2506.897 
+0.010 
+-0.595 
+0.44 
+0.46 
+0.58 
+0.64 
+0.52 
+2514.316 
+0.000 
+-0.672 
+0.49 
+0.46 
+0.58 
+0.59 
+0.52 
+2516.112 
+0.028 
+-0.098 
+0.49 
+0.49 
+0.63 
+0.49 
+0.52 
+2519.202 
+0.010 
+-0.810 
+0.34 
+… 
+0.58 
+… 
+0.57 
+2528.508 
+0.028 
+-0.583 
+0.34 
+0.49 
+0.58 
+… 
+0.57 
+2564.825 
+0.780 
+-3.922 
+… 
+… 
+… 
+0.39 
+… 
+2881.578 
+0.780 
+-0.088 
+0.24 
+0.29 
+0.33 
+0.39 
+0.32 
+2970.353 
+0.780 
+-3.531 
+0.41 
+0.24 
+… 
+0.29 
+0.37 
+2987.643 
+0.780 
+-2.035 
+0.19 
+0.24 
+0.38 
+0.29 
+0.17 
+3905.523a 
+1.907 
+-3.026 
+0.14 
+0.24 
+0.28 
+0.44 
+0.32 
+ 
+ 
+meana 
+0.37 
+0.36 
+0.51 
+0.48 
+0.46 
+ 
+ 
+ 
+0.09 
+0.10 
+0.10 
+0.14 
+0.13 
+Line-by-Line Abundances – Si II 
+λ (Å) 
+χ (eV) 
+log(gf) 
+[Si II/Fe] 
+[Si II/Fe] 
+[Si II/Fe] 
+[Si II/Fe] 
+[Si II/Fe] 
+2334.407 
+0.036 
+-5.088 
+0.43 
+0.48 
+0.50 
+0.57 
+0.52 
+2350.172 
+0.036 
+-5.116 
+0.36 
+0.43 
+0.65 
+0.32 
+0.37 
+ 
+ 
+mean 
+0.40 
+0.46 
+0.58 
+0.45 
+0.45 
+ 
+ 
+ 
+0.05 
+0.04 
+0.11 
+0.18 
+0.11 
+Note 
+aThe mean and standard deviation of [Si I/Fe] are calculated without the λ3905 line data as this this transition is known to yield 
+temperature-dependent abundances in LTE calculations.  See text for further discussion. 
+ 
+

3dFKT4oBgHgl3EQf8i5H/content/tmp_files/2301.11949v1.pdf.txt

@@ -0,0 +1,2162 @@
+ 
+1 
+Magnetic Amplification at Yb3+ "Designer Defects"  
+in the van der Waals Ferromagnet, CrI3 
+ 
+Kimo Pressler, Thom J. Snoeren, Kelly M. Walsh, Daniel R. Gamelin* 
+Department of Chemistry, University of Washington, Seattle, WA 98195, United States 
+Email: [email protected]  
+ 
+ 
+Abstract. The two-dimensional (2D) van der Waals ferromagnet CrI3 has been doped with the 
+magnetic optical impurity Yb3+ to yield materials that display sharp multi-line Yb3+ 
+photoluminescence (PL) controlled by the magnetism of CrI3. Magneto-PL shows that Yb3+ 
+magnetization is pinned to the magnetization of CrI3. An effective internal field of ~10 T at Yb3+ 
+is estimated, attributed to strong in-plane Yb3+-Cr3+ superexchange coupling. The anomalously 
+low energy of Yb3+ PL in CrI3 reflects relatively high Yb3+-I- covalency, contributing to Yb3+-
+Cr3+ superexchange coupling. The Yb3+ PL energy and linewidth both reveal the effects of 
+spontaneous zero-field CrI3 magnetic ordering within 2D layers below TC, despite the absence of 
+net magnetization in multilayer samples. These results illustrate the use of optical impurities as 
+"designer defects" to introduce unique functionality to 2D magnets. 
+Keywords: 2D Ferromagnet, Lanthanide Doping, Molecular Field, Chromium Triiodide, 
+Photoluminescence  
+ 
+ 
+Defects have the power to transform the physical properties of crystals, imparting new and 
+potentially useful functionalities from conductivity to quantum photon emission.1-6 In magnetic 
+materials, defects can strongly affect spin-wave propagation, magnetic domain-wall propagation, 
+skyrmion dynamics, and magnetic vortex pinning.7-9 Recently, the layered van der Waals 
+ferromagnet CrI3 has emerged as a promising platform for exploring strongly correlated spin 
+physics, magnetic proximity effects, and next-generation spin-based device architectures in the 
+
+ 
+2 
+two-dimensional (2D) limit,10-14 but the potential to expand CrI3 functionality through 
+introduction of defects remains untapped. Here, we report that doping CrI3 with Yb3+ as a 
+"designer point defect" transforms its normally broad and featureless d-d photoluminescence 
+(PL) into narrow-line sensitized f-f emission, without compromising its attractive magnetic 
+properties. We further show that Yb3+ in CrI3 experiences a large internal effective field that 
+makes it extremely sensitive to small external magnetic fields. Using this property, we 
+demonstrate magnetically saturated circular polarization of Yb3+ emission at anomalously small 
+applied fields. Strikingly, the internal effective field also transmits magnetic information to Yb3+ 
+even in the absence of any applied field, making Yb3+ a unique embedded luminescent probe of 
+spontaneous zero-field magnetic ordering within the 2D monolayers of bulk CrI3. These 
+discoveries establish optical impurity doping as an effective strategy for expanding the 
+functionality of 2D magnets, with potential ramifications for both basic science and future spin-
+photonic technologies. 
+CrI3 has become a model system for exploring magnetic exchange in 2D van der Waals 
+structures,10-14 stimulated by recent discoveries of Ising-like hard ferromagnetism in exfoliated 
+monolayer CrI3 and layer- and stacking-dependent magnetism in multi-layer CrI3.15,16 Layering 
+CrI3 with non-magnetic 2D materials introduces magnetic functionality to the non-magnetic 
+material via inter-layer exchange coupling, allowing magnetic manipulation of properties such as 
+WSe2 valley polarization and valley Zeeman splittings.17 Extension from few to many (bulk) 
+layers preserves the strong Ising-like intralayer ferromagnetic ordering, but facile motion of 
+domain walls unblocks demagnetization.18 Despite its rich magnetic properties, CrI3 itself has not 
+garnered much attention as an optical material. Bulk CrI3 has been investigated for its very large 
+Kerr and Faraday rotation strengths in relation to optical isolators and associated 
+
+ 
+3 
+technologies.19,20 PL of bulk CrI3 has apparently not been reported, and few-layer CrI3 shows17 
+only the very broad d-d PL characteristic of weak-field pseudo-octahedral Cr3+.21 Circular 
+polarization of this d-d PL was used to probe the magnetism of few-layer CrI3,17 but the 
+emission's breadth limits its further utility for fundamental studies or in spin-photonics, 
+stimulating efforts to narrow the band via cavity coupling.22 Doping CrI3 with optically active 
+impurities has also not been reported, either in bulk or exfoliated samples.  
+To investigate intralayer "proximity" effects resulting from magnetic exchange coupling, we 
+have prepared CrI3 doped with luminescent and spin-bearing Yb3+ ions. Large-diameter single-
+crystal flakes of CrI3 were prepared by chemical vapor transport. Yb3+ was introduced by adding 
+Yb(0) to the precursor mix. The Yb3+ concentration in the resulting Yb3+:CrI3 crystals is 
+controllable, and samples with up to ~5% Yb3+ (cation mole fraction, [Yb3+]/([Cr3+]+[Yb3+])) are 
+described here. Further experimental details are provided in the Supporting Information (SI). 
+Figure 1a shows a photograph of representative Yb3+:CrI3 flakes in their growth tube. The flakes 
+are between 5 and 10 mm across, with typical thicknesses of 5-20 µm (see SI). Figure 1b plots 
+XRD data collected on undoped and 4.9% Yb3+-doped CrI3 single-crystal flakes using a powder 
+diffractometer. Only (00l) peaks are observed, corresponding to the interlayer lattice spacing and 
+reflecting the flake's alignment. Figure 1c highlights the shift to smaller angle of the 001 peak 
+upon doping. From fitting the XRD peak positions of undoped and 4.9% Yb3+-doped CrI3 
+samples, the interlayer lattice parameter was found to increase 0.24% from 6.996 ± 0.002 to 
+7.013 ± 0.002 Å, attributed to the larger ionic radius of Yb3+ than Cr3+ (87 vs 62 pm, 
+respectively) (see SI). These data suggest that the local strain of doping is relieved by distorting 
+the lattice along its softest dimension, as expected. Substitutional incorporation of Yb3+ at the 
+Cr3+ site is verified by single-crystal XRD measurements (see SI), which also show the increased 
+
+ 
+4 
+interlayer spacing. The single-crystal data show no detectable electron density between layers, 
+ruling out Yb3+ intercalation. 
+ 
+ 
+Figure 1. (a) Photograph of 4.9% Yb3+:CrI3 crystals prepared by chemical vapor 
+transport. The scale bar shows 5 mm. All experiments were performed on 
+individual single-crystal flakes from such a reaction tube. (b) XRD data collected 
+on undoped and Yb3+-doped CrI3 single crystals using a powder diffractometer. 
+Only (00l) peaks are observed, indicating an oriented sample. Reference peaks for 
+c-oriented CrI3 diffraction are included (black, ICSD Coll. Code 251654). (c) 
+Magnified view of the 001 reflection for the same samples, displaying an increase 
+in the interlayer lattice spacing upon Yb3+ doping. The x axis in (c) was 
+determined as described in the SI. 
+ 
+ 
+Figure 2a plots the PL spectra of CrI3 and Yb3+:CrI3 single flakes measured at several 
+temperatures between 4 and 200 K. The CrI3 spectrum broadens and decreases in intensity with 
+increasing temperature, eventually reaching only 7.5% of its 4 K intensity at 200 K (see SI). 
+Although the broadening to higher energies is expected from thermal hot bands, the broadening 
+to lower energies is abnormal and suggests an additional feature. Upon introduction of Yb3+, the 
+broad featureless d-d emission of Cr3+ disappears and is replaced by a series of sharp f-f 
+transitions of Yb3+ around 1.15 eV. Assignment of the PL fine structure is discussed later. In 
+some samples, Yb3+ doping also reveals another broad emission band centered at ~0.95 eV, 
+which is responsible for the red tail of the CrI3 PL here and in some literature spectra. This 
+
+a
+b
+C
+Yb3+: Crl3
+Intensity (rel.)
+Intensity (rel.)
+Crl3
+001
+002
+ref.
+001
+003
+004
+005
+006
+I x10
+[x10
+20
+40
+60
+80
+13.0
+14.0
+2θ (deg.)
+20 (deg.) 
+5 
+feature has been traced to Ni2+ impurities (<0.4%) found in some Cr(0) precursors, and it can be 
+mostly eliminated by using 5N Cr(0) precursors (Fig. 2a, bottom). The Yb3+ PL is not influenced 
+by this Ni2+ impurity (see SI). 
+ 
+ 
+Figure 2. (a) Variable-temperature PL spectra of CrI3 (top) and 4.9% Yb3+:CrI3 
+(bottom), measured from 4 to 200 K under 1.88 eV CW excitation at 4 mW/cm2. 
+(b) Single-configurational-coordinate diagram (A1g coordinate) describing 
+vibronic broadening of the absorption and luminescence bands associated with 
+transitions between the 4A2g and 4T2g ligand-field states of pseudo-octahedral 
+Cr3+. In Yb3+-doped CrI3, energy transfer from the Cr3+ 4T2g excited state to Yb3+ 
+yields sensitized 2F5/2 ! 2F7/2 f-f luminescence. 
+ 
+ 
+Figure 2b illustrates the photophysics of Yb3+:CrI3 schematically. The lowest-energy excited 
+state of CrI3 is the Cr3+ 4T2g ligand-field state, involving excitation of a t2g electron into a σ-
+antibonding eg orbital (in idealized Oh symmetry). The resulting change in equilibrium geometry 
+is described by the single-configurational-coordinate (SCC) diagram of Fig. 2b, which illustrates 
+the totally symmetric distortion coordinate. This 4T2g excited state also distorts along a 
+symmetry-breaking Jahn-Teller coordinate (not illustrated).21 These distortions lead to extensive 
+
+b
+a
+1.0
+Crls Undoped
+, Intensity (norm.)
+4 - 200 K
+0.8
+Cr3+
+g
+0.6
+ET
+0.4-
+ 0.2
+Cr3+
+Cr3+
+1.0
+Yb3+:Crl3
+Abs
+PL
+ Intensity (norm.)
+0.8
+4 - 200 K
+Yb3+
+0.6.
+A
+12g
+PL
+0.4-
+0.2
+2
+7/2
+0.0
+1.2
+1.1
+1.0
+0.9
+Energy (eV) 
+6 
+vibronic progressions in the absorption and PL spectra associated with this transition, and cause 
+a large PL Stokes shift. Doping CrI3 with Yb3+ introduces a set of 2F5/2 states just below the Cr3+ 
+4T2g excited state, favorably positioned for efficient Cr3+ ! Yb3+ energy transfer. At 4.9% Yb3+ 
+doping, the Cr3+ 4T2g PL is entirely quenched and strong Yb3+ 2F5/2 emission is observed in its 
+place (Fig. 2a). Because both Cr3+ and Yb3+ states are localized at single ions, energy migration 
+within the CrI3 lattice is required for this complete quenching. In undoped CrI3, energy migration 
+among equivalent Cr3+ sites may occur but is not readily apparent. In Yb3+:CrI3, this energy 
+migration is interrupted when energy is captured by Yb3+ dopants. In 4.9% Yb3+:CrI3, the 
+average Cr3+ ion has only ~14% probability of having a neighboring Yb3+, and ~50% probability 
+of having at least one Yb3+ within its first two cation shells. Energy must therefore migrate over 
+at least a few lattice sites within the 4T2g lifetime to fully quench the Cr3+ emission as observed in 
+Fig. 2a. 
+Figure 3a shows the anticipated electronic structure of Yb3+ in CrI3. In the free ion, spin-orbit 
+coupling splits the 2F term into 2F5/2 (excited) and 2F7/2 (ground) states by an amount ΔE = 7/2ζ, 
+where ζ = 361.8 meV is the free-ion spin-orbit coupling constant.23 In crystals, each of these 
+states is further split by the crystal field. Figure 3b shows circularly polarized PL spectra of 4.9% 
+Yb3+:CrI3 measured in a 0.5 T field applied parallel to the crystal's c axis (vide infra). Three 
+zero-phonon electronic origins are observed and assigned to the Γ8 ! Γ6, Γ8, and Γ7 transitions 
+anticipated from Fig. 3a using idealized Oh notation. The actual cation site symmetry in CrI3 is 
+lower (Fig. 3a, right),24 but the expected low-symmetry splitting of the Γ8 origin is not clearly 
+identifiable. The Γ6 peak is broad with observable structure on its high-energy shoulder, thus 
+making the precise energy of this origin unclear within ~20 cm-1 (~2.5 meV). Analysis of these 
+PL energies within the Angular Overlap Model (AOM)25 reproduces the 2F7/2 splittings well, 
+
+ 
+7 
+predicting a 2F5/2 splitting of ~34 meV and splittings of the two Γ8 levels by <0.5 meV each (see 
+SI). Additional satellite features are observed ~127 cm-1 (15.7 meV) below the Γ8 and Γ7 
+electronic origins and assigned as phonon sidebands. Raman spectra show a totally symmetric 
+lattice breathing mode of CrI3 at this energy (ν = 127 cm-1).26  
+A striking aspect of this Yb3+:CrI3 PL is its very low energy relative to other Yb3+ PL. This 
+energy is primarily determined by spin-orbit coupling (Fig. 3a). Yb3+ spin-orbit coupling can be 
+reduced from that in the free ion by covalent expansion of the f-electron wavefunctions 
+(nephelauxetic effect),27,28 but f-orbital covalency in trivalent lanthanides is typically very small 
+and this effect is usually considered negligible at ambient pressure. A survey of Yb3+-doped 
+crystals shows that the energy gap between Yb3+ 2F5/2 and 2F7/2 barycenters remains very near the 
+free-ion value of ΔE ~ 1.266 eV across doped oxide, fluoride, chloride, bromide, sulfide, and 
+phosphide lattices (see SI).29-33 We note that we have been unable to find any reports of PL from 
+other Yb3+-doped iodide crystals, perhaps because Yb3+ is easily reduced to Yb2+ under common 
+iodide crystal-growth conditions. Yb3+:CrI3 deviates from this typical behavior substantially: ΔE 
+is only ~1.163 eV, or ~9% smaller than in the free ion, representing the smallest spin-orbit 
+coupling yet reported for Yb3+. Covalency in Yb3+:CrI3 is certainly enhanced by the large ionic 
+radius and polarizability of the iodides, but this consideration alone likely cannot explain the 
+anomaly. The atomic spin-orbit coupling of I is also much greater than those of other common 
+ligands for Yb3+, and should contribute to the spectroscopic spin-orbit splitting via covalency. 
+Furthermore, the large ionic radius of Yb3+ compared to Cr3+ means that Yb3+ experiences an 
+internal pressure imposed by the surrounding lattice, which may also increase covalency. 
+Importantly, Yb3+-I- covalency is essential for strong Yb3+-Cr3+ superexchange coupling. 
+ 
+
+ 
+8 
+ 
+ 
+ 
+ 
+ 
+Figure 3. (a) Splitting of the Yb3+ free-ion 2F term due to spin-orbit (ζ) and 
+crystal-field (Oh, <D3d) interactions. The colored down arrows indicate the three 
+crystal-field transitions anticipated in the low-temperature PL spectrum in the 
+idealized Oh site symmetry. The actual site symmetry is reduced to <D3d, e.g., to 
+C2, splitting each Γ8 level into two Kramers doublets. (b) Magnetic circularly 
+polarized luminescence (MCPL) spectra of 4.9% Yb3+:CrI3 measured at 5 K with 
+an applied magnetic field of 0.5 T. The σ- (red) and σ+ (black) spectra were 
+collected using unpolarized 1.88 eV CW excitation at 40 mW/cm2 and have 
+different amplitudes. The three electronic origins in idealized Oh symmetry are 
+indicated below the spectra, assigned to the Γ8 ! Γ6, Γ8, and Γ7 transitions 
+illustrated in panel (a). The dashed black lines indicate vibronic sidebands with a 
+characteristic energy spacing of ~127 cm-1 (15.7 meV), consistent with the A1g 
+lattice mode of CrI3. (c) False-color plot of the MCPL polarization ratio, ρ = (σ- –
+ σ+)/(σ- + σ+), for the full Yb3+ PL spectrum, measured from -2 to +2 T at 5 K. (d) 
+ρ of the Γ8 ! Γ7 electronic origin (1.117 eV) plotted as a function of magnetic 
+field from -6 to 6 T. The black (red) trace corresponds to the positive (negative) 
+field sweep direction. Inset: Expanded plot of ρ between -0.4 and +0.4 T, showing 
+a coercive field of ~55 mT. For both field-sweep measurements, the sample was 
+excited with linearly polarized 1.96 eV excitation, but with different powers (see 
+Methods). (e) False-color plot of the polarization ratio vs temperature, measured 
+at 0.5 T. The dashed black line indicates the Curie temperature of bulk CrI3 (TC = 
+61 K). (f) Plot of the Γ8 ! Γ7 polarization ratio at the peak maximum measured at 
+0.5 T as a function of temperature. The red curve is a guide to the eye. Inset: 
+
+a
+c
+d
+2
+Oh
+<D3d
+5 K
+1.0-
+5 K
+0.2
+Iz
+(wou)
+1
+0.5-
+0.1
+I:
+E
+E
+7/2 
+0.0 Φ
+0.0
+Iz
+-0.1
+-0.5 -
+-1-
+-0.4
+0.0
+0.4
+Fieid (T)
+F12
+-0.2
+-1.0
+Free Ion + Spin-Orbit
++
+Crystal Field
+-2 -
+1.18
+1.16
+1.14
+1.12
+1.10
+1.08
+-6 
+-4
+-2 
+0
+2
+4
+6
+Energy (eV)
+Energy (eV)
+Field (T)
+b
+e
+f
+1.16
+1.14
+1.12
+1.10
+1.08
+125
+0.20
+0.5 T
+g-
+0.5 T
+0.5 T
+5 K
+0.2
+0.15-
++0
+0.1
+Ip/dp
+75
+←
+Tc
+0.0 Φ
+0.10-
+I Tc = 61 K
+50 -
+-0.1
+0
+50
+100
+150
+[7
+0.05 -
+T(K)
+-
+25-
++-0.2
+A1g vibration
+9264 cm
+9010 cm
+Tci
+9410cm
+~127cm
+0.00 -
+5-
+9400
+9200
+9000
+8800
+8600
+1.18
+1.16
+1.14
+1.12
+1.10
+1.08
+0
+20
+40
+60
+80
+100
+120
+Wavenumber (cm")
+Energy (eV)
+Temperature (K) 
+9 
+Derivative of ρ as a function of temperature. The extracted Curie temperature is 
+61 K, indistinguishable from that of the undoped crystal.  
+ 
+ 
+From Fig. 3a, all features show circularly polarized PL, with the Γ8 ! Γ7 origin showing the 
+greatest polarization ratio (ρ = (σ- – σ+)/(σ- + σ+) = 19%). ρ is independent of excitation power 
+but its maximum value varies somewhat between samples (see SI). Figure 3c plots ρ across the 
+entire PL spectrum as a function of magnetic field. All Yb3+ transitions are influenced by the 
+applied field in the same way, consistent with all PL arising from the same excited state (Γ8). 
+Figure 3d plots ρ for the Γ8 ! Γ7 peak as a function of applied field. ρ increases rapidly at very 
+low fields and saturates at only ~0.2 T. Increasing the field from 0.2 to 6.0 T does not change ρ 
+further, consistent with complete magnetization of Yb3+ by 0.2 T. On an expanded scale, these 
+data show a hysteresis with coercivity of ~55 mT, comparable to that found in magnetic 
+measurements of bulk CrI3.18,34 We note that these ρ values are generally small compared to 
+those in cubic Yb3+:InP (~70% at 10 T, 4.2 K),33 possibly suggesting an in-plane or canted Yb3+ 
+anisotropy. Figure 3e summarizes the temperature dependence of ρ, measured at 0.5 T, and Fig. 
+3f highlights the temperature dependence for Γ8 ! Γ7 individually. All spectral features behave 
+similarly, showing a pronounced drop in polarization at the Curie temperature of bulk CrI3 (~61 
+K, see Fig. 3f, inset). These magneto-optical data agree well with magnetic susceptibility data 
+(see SI), and both indicate that Yb3+ doping causes no significant change in the magnetic 
+characteristics of CrI3 in these samples. This MCPL field and temperature dependence is highly 
+unusual for Yb3+, which generally shows simple paramagnetism of a pseudo-spin 1/2. For 
+example, our AOM crystal-field analysis (see SI) predicts gavg ~ 2.7 for the lowest 2F7/2 Kramers 
+doublet. Overall, the anomalous magnetism seen in the Yb3+ MCPL reflects magnetic integration 
+of Yb3+ with ferromagnetic CrI3. 
+
+ 
+10 
+Magnetic ordering was originally explained by Weiss in terms of a huge internal "molecular 
+field"35 exerted upon individual ions by their surrounding magnetic matrix, and this model 
+provides a useful heuristic for estimating the effective field experienced by Yb3+ within CrI3. In 
+this model, the effective field is given by the sum of external and molecular fields, as in eq 1. 
+ 
+ 
+Heff = Hext + Hmol 
+ 
+ 
+ 
+ 
+(1) 
+In Fig. 3c,d, CrI3 reaches magnetic saturation at very small Hext (<0.2 T). At such low fields, Hext 
+<< Hmol, and hence Heff ~ Hmol. In the molecular-field model, Hmol in CrI3 is given by eq 2, 
+!!"# =
+!!" !
+!!!  
+ 
+ 
+ 
+ 
+ 
+(2) 
+where, J is the nearest-neighbor exchange coupling constant, z = 3 in CrI3, g is the Landé g factor 
+(2.00 for Cr3+ in CrI3), µB is the Bohr magneton, and !  is the spin expectation value for Cr3+ in 
+CrI3, whose absolute value equals 3/2 at saturation. TC in this model is determined by J according 
+to eq 3, 
+!! =
+!!"#(!!!)
+!!!
+   
+ 
+ 
+ 
+ 
+ 
+(3) 
+where S = 3/2 for Cr3+, and kB is the Boltzmann constant. From TC = 61 K, eq  
+3 yields a value of J = 0.70 meV in CrI3. Entering this J value into eq 2 yields Hmol = ~54 T in 
+CrI3. Hmol is dominated by superexchange coupling, since dipolar contributions cannot account 
+for the high TC of CrI3.36 For Yb3+ in CrI3, J is reduced by the shielding of the 4f orbitals. 
+Cr3+(d)-Yb3+(f) superexchange coupling has received relatively little experimental or theoretical 
+attention,37-39 but relevant experimental data are found in inelastic neutron scattering analyses of 
+Cs3Yb1.8Cr0.2Br9, where Yb3+-Cr3+ exchange splittings are ~1/4 those for Cr3+-Cr3+.37 This 
+scaling factor is approximate because of the different lattice structure, but Cs3Yb1.8Cr0.2Br9 is the 
+most similar halide-bridged Yb3+-Cr3+ system for which reliable exchange-coupling strengths 
+could be found. This rough scaling reduces Hmol to ~14 T. Accounting for the larger g value of 
+
+ 
+11 
+Yb3+ (~2.7, see SI), our best estimate is Hmol ~ 10 T for Yb3+ ions within CrI3. Future 
+spectroscopic measurements (e.g., inelastic neutron scattering, Mössbauer, etc.) and calculations 
+will be needed to refine this estimate, but the central conclusion drawn from both the 
+experimental data and this analysis is clear: Yb3+ magnetization in Yb3+:CrI3 is effectively 
+pinned to the magnetic ordering of the CrI3 lattice through strong Yb3+-Cr3+ superexchange 
+coupling. The large Hmol in Yb3+:CrI3 is attributable in large part to the Yb3+-I- covalency 
+discussed above. For comparison, exchange fields of 1.7 and ~1.1 T are reported for Yb3+ in 
+ferrimagnetic hexagonal YbFeO3
+40 and distorted orthorhombic YbCrO3.41 At these values, Yb3+ 
+magnetization is not pinned to the ordered TM3+ spin sublattices.  
+A further remarkable aspect of Yb3+:CrI3 is that the effects of Hmol are evident even at zero 
+magnetic field (Hext = 0). Figure 4a plots zero-field Yb3+ PL spectra as a function of temperature 
+from 4 to 200 K. Viewing the data starting from high temperature, the peak positions appear 
+nearly constant until roughly TC. Below TC, the peaks all shift to lower energy together. This 
+redshift is also evident in Fig. 3e. Figure 4b highlights the temperature dependence of the Γ8 ! 
+Γ7 transition energy. From 120 K to ~TC, the transition energy increases gradually by only ~2 
+meV. Such temperature dependence has been variously modeled in terms of Raman scattering of 
+non-resonant phonons or direct absorption/emission of phonons resonant with a crystal-field 
+splitting.42,43 For example, both models reproduce the 2F7/2 ! 2F5/2 transition energies of 
+Yb3+:YAG well, whereas the resonant phonon model reproduces absorption linewidths 
+marginally better.43 As such, we apply the resonant phonon model here. The PL energies above 
+TC are thus described by eq 4,42,43 
+ 
+ 
+ 
+!(!) = !! + 
+!!
+!! !!!!! 
+ 
+ 
+T > TC  
+(4) 
+
+ 
+12 
+where E0 is the energy at 0 K, αs describes the electron-phonon interaction strength, and Δ is the 
+energy of the activating phonon mode, fixed at Δ = 127 cm-1 (15.7 meV, Fig. 3b).  
+The solid curve in the high-temperature portion of Fig. 4b (>TC) shows a fit to the high-
+temperature data using eq 4, floating E0 and αs and yielding best-fit values of 1.1242 eV and -6.3 
+meV, respectively. Eq 4 plateaus at E0 in the limit of 0 K (dashed line < TC in Fig. 4b), but the 
+experimental peak energy shows a discontinuity at TC, dropping sharply and decreasing with 
+decreasing temperature until reaching ~7 meV below E0 in the low-temperature limit. With its 
+link to TC and its characteristic curvature, this trend in Yb3+ PL energy is associated with the 
+spontaneous magnetization of individual CrI3 monolayers, even though there is no net 
+magnetization in these samples. 
+Spontaneous ferromagnetic ordering is classified as a second-order phase transition and, 
+within the theory of universal scaling laws, is characterized by the order parameter β shown in eq 
+5 describing the magnetization temperature dependence.44 
+!(!) = !! −
+!!!!
+!!
+!
+ 
+ 
+ 
+ 
+ 
+(5) 
+M0 is the saturation moment per magnetic ion and equals 3.1 µB for CrI3.18 The precise value of β 
+depends on the underlying spin physics, but it is commonly around 1/3.12 Previous examination 
+of bulk CrI3 found a critical exponent of β = 0.284, between that expected from the 3D Ising 
+model (β = 0.325) and that of the tri-critical mean-field model (β = 0.250).34 Accordingly, the 
+data in Fig. 4b below TC were simulated using eq 6 (sum of eq 4 and eq 5, with eq 4 parameters 
+fixed by the high-temperature data). The scaling parameter (γ) in eq 6 relates magnetization to 
+PL energy shift. The data are reproduced well using fixed values of β = 1/3, TC = 60 K, and Δ = 
+127 cm-1 (15.7 meV), with γ as the only adjustable parameter. Relating eqs 5 and 6, these results 
+indicate an Yb3+ PL energy shift of -2.2 meV/µB during spontaneous CrI3 intralayer 
+
+ 
+13 
+magnetization. We stress that the zero-field PL data in Fig. 4 are not magnetic data, but highlight 
+the strong influence of CrI3 spontaneous magnetization on the Yb3+ PL. Because TC in these 
+samples is indistinguishable from that of bulk CrI3 (Figs. 3f, S15), we tentatively attribute the 
+small apparent broadening of the PL energy discontinuity around TC in Fig. 4b to additional PL 
+hot bands that are not spectrally resolved. 
+!(!) = !! + 
+!!
+!! !!!!! +  ! −
+!!!!
+!!
+!
+ 
+ 
+T < TC  
+(6) 
+ 
+ 
+Figure 4. (a) False-color plot of the Yb3+ PL intensities vs temperature measured 
+for 4.9% Yb3+:CrI3 from 4 to 150 K at zero external magnetic field. The 
+horizontal dashed line indicates TC = 61 K. (b) Peak position of the Γ8 ! Γ7 
+transition plotted vs temperature. The solid red curve shows the behavior 
+predicted from the combination of resonant phonon interactions (eq 4) and 
+spontaneous magnetization (below TC, eq 6). The dashed red curve shows the 
+behavior predicted from eq 4 alone below TC. The solid curve was obtained using 
+eqs 4 and 6 with fixed parameters of Δ = 127 cm-1 (15.7 meV), TC = 60 K, and β = 
+1/3, adjusting only the amplitude scaling. (c) Plot of the Γ8 ! Γ7 PL linewidth vs 
+temperature, from the same VTPL measurements. 
+ 
+ 
+
+a
+b
+150
+1124
+(meV)
+6
+Peak
+1.0
+125-
+1122
+Position 
+ Position
+4
+Peak Position
+1120.
+0.8
+2
+100.
+Temperature (K)
+Peak
+PL 
+(meV)
+1118
+. Intensity (
+0.6
+0
+1116
+75
+C
+9.
+FWHM (meV)
+8
+50.
+0.2
+Peak FWHM
+7
+6.
+25.
+-0.0
+5.
+5 .
+4
+1.181.161.141.121.101.08
+0
+20
+40
+60
+80
+100
+120
+Energy (ev)
+Temperature (K) 
+14 
+Figure 4c plots the temperature dependence of the Γ8 ! Γ7 linewidth (full-width-at-half-
+maximum, FWHM). These data show similar trends as observed in the peak energies of Fig. 4b. 
+Below TC, the FWHM decreases from ~9 meV to ~4.5 meV in the low-temperature limit, 
+attributed to the reduction in spin disorder around Yb3+. These data thus also reflect spontaneous 
+magnetic ordering in monolayers of CrI3. Although distinct low-energy shoulders are not 
+resolved in these data, we hypothesize that the energy and linewidth changes below TC both 
+ultimately stem from loss of hot-magnon sideband intensity as CrI3 monolayers order 
+magnetically.45 It will be an interesting future direction to explore magnon coupling to f-f 
+transitions in these and related doped 2D magnetic materials. 
+In summary, doping Yb3+ into the 2D van der Waals ferromagnet CrI3 transforms this 
+material's PL from broad-band to sharp multi-line, while retaining its key magnetic functionality. 
+The f-f PL of Yb3+:CrI3 is anomalously low in energy, reflecting relatively covalent Yb3+-I- 
+bonding. Yb3+ magnetization is pinned to CrI3 by strong superexchange interactions, which 
+contribute an effective internal field of ~10 T that is greater than the field required for magnetic 
+saturation of paramagnetic Yb3+ and much greater than the field required for full CrI3 
+magnetization at low temperature (~0.2 T). Flipping the magnetization of CrI3 with a small 
+external field thus also flips the Yb3+ magnetization and inverts its PL circular polarization. 
+Magnetic pinning is maintained up to the TC of CrI3, but is rapidly lost above TC. We further 
+showed that the Yb3+ PL energy and linewidth both sense this internal field even at zero applied 
+field, mapping spontaneous intralayer magnetic ordering below TC despite the absence of net 
+magnetization. Because each Yb3+ ion is a local lattice defect within an individual CrI3 
+monolayer, we expect these induced functionalities to persist down to the monolayer, prompting 
+future studies on exfoliated Yb3+:CrI3 and associated stacked van der Waals heterostructures and 
+
+ 
+15 
+layered devices. These results demonstrate the power of designer defects to add functionality to 
+2D magnetic materials, enrich their fundamental physics, and create new materials of potential 
+utility for future spin-photonics applications. 
+ 
+Acknowledgments. Support of this project by the US NSF (DMR-1807394) is gratefully 
+acknowledged. Initial stages of this work were performed as part of Programmable Quantum 
+Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), 
+Office of Science, Basic Energy Sciences (BES), under award DESC0019443. Additional 
+support was received from the UW Clean Energy Institute (graduate fellowships to T.J.S. and 
+K.M.W.). Part of this work was conducted at the Molecular Analysis Facility, a National 
+Nanotechnology Coordinated Infrastructure (NNCI) site at the University of Washington that is 
+supported in part by the National Science Foundation (NNCI-1542101 and NNCI-2025489), the 
+University of Washington, the Molecular Engineering & Sciences Institute, the Clean Energy 
+Institute, and the National Institutes of Health. The authors thank Dr. Werner Kaminsky and 
+Paige M. Gannon for single-crystal XRD measurements, Dr. Xi Wang for assistance with optical 
+microscope measurements, Prof. Jiun-Haw Chu and Dr. Zhaoyu Liu for VSM measurements, 
+and Prof. Robert Glaum, Maximilian Jähnig, and Julia Spitz for provision of and assistance with 
+the BonnMag code. 
+ 
+ 
+Author Information 
+ 
+Corresponding Author 
+Daniel R. Gamelin - Department of Chemistry, University of Washington, 
+Seattle, Washington 98195-1700, United States; orcid.org/0000-0003-2888-9916;  
+Email: [email protected] 
+ 
+ 
+Authors 
+Kimo Pressler - Department of Chemistry, University of Washington, Seattle, 
+Washington 98195-1700, United States; orcid.org/0000-0003-2788-1592 
+Thom J. Snoeren - Department of Chemistry, University of Washington, Seattle, 
+Washington 98195-1700, United States; orcid.org/0000-0001-8055-3710 
+Kelly M. Walsh - Department of Chemistry, University of Washington, Seattle, 
+Washington 98195-1700, United States; orcid.org/0000-0001-5349-8816 
+ 
+ 
+ 
+Supporting Information  
+
+ 
+16 
+The Supporting Information is available free of charge at https://pubs.acs.org/doi/XXXX 
+Additional experimental details, including about sample preparation and characterization. 
+Additional variable-temperature PL data, PL polarization vs magnetic field data, 
+excitation-power-dependence data, results from Yb3+ crystal-field calculations, and 
+comparison of Yb3+ crystal-field barycenter energies in various lattices (PDF). 
+ 
+ 
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+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+19 
+Table of Contents Graphic 
+ 
+ 
+ 
+ 
+ 
+
+q+
+q
+g+
+++++个
++++++
+Yb3+:Crl3	
+Supporting Information for 
+	
+ 
+Magnetic Amplification at Yb3+ "Designer Defects"  
+in the van der Waals Ferromagnet, CrI3 
+ 
+Kimo Pressler, Thom J. Snoeren, Kelly M. Walsh, Daniel R. Gamelin* 
+Department of Chemistry, University of Washington, Seattle, WA 98195, United States 
+Email: [email protected]  
+ 
+Experimental Methods 
+General Considerations. All sample preparation and manipulation was performed in a 
+glovebox under an atmosphere of purified dinitrogen.  
+Chemicals. Chromium metal powder (200 mesh, 99.94%, lot X15E028) was purchased from 
+Alfa Aesar. According to the manufacturer's certificate of analysis, the majority of the impurity 
+in this sample lot was Ni at 343 ppm. A chromium chip (99.995%, lot MKCH4484) was also 
+purchased from Sigma Aldrich as a higher-purity Cr source. The Cr chip was ground to a powder 
+using a mortar and pestle and used in an analogous manner as the powder precursor. I2 
+(≥99.99%) was purchased from Sigma Aldrich. Ytterbium metal powder 40 mesh (99.9%) was 
+purchased from BeanTown chemical. All chemicals were used as received without further 
+purification.  
+Synthesis of CrI3 and Yb3+-Doped CrI3 Single Crystals. Single crystals of the doped and 
+undoped CrI3 were grown by chemical vapor transport in a manner similar to that described in 
+previous literature reports.1 For undoped CrI3, Cr(0) metal and I2 were loaded as a 1:3 
+stochiometric ratio into a quartz tube and sealed under an evacuated atmosphere. For Yb3+-doped 
+CrI3, additional Yb(0) metal was loaded along with the other starting materials. The quartz tubes 
+were 15 cm long with inner and outer diameters of 14 and 16 mm, respectively. Sealed tubes 
+were placed in an open-ended horizontal tube furnace with the starting materials in the hot zone 
+set at 650 ˚C and the other end at a temperature of ca. 500 ˚C. Samples were heated for 5 days 
+and then allowed to slowly cool to room temperature. Once cooled, the tubes were brought into a 
+glove box and cracked open to yield shiny dark plate-like crystals that had formed at the cold end 
+of the quartz tube. Elemental analysis of the Yb3+-doped samples was performed by inductively 
+coupled plasma mass spectrometry (ICP-MS) using a PerkinElmer NexION 2000B. Samples 
+were prepared by digesting single crystals in concentrated nitric acid with sonication and then 
+further diluted in ultrapure H2O. Yb3+ doping levels are reported as cation mole fraction, 
+[Yb3+]/([Cr3+]+[Yb3+]), in percentage, with an estimated uncertainty of ±0.1%. Crystal thickness 
+was measured by mounting a representative flake to a glass slide using double-sided tape and 
+imaging the flake with an optical microscope in a glovebox at various magnifications. The edge 
+length was calculated in ImageJ2 using known pixel resolutions. 
+X-ray Diffraction (XRD) Characterization. Samples were prepared for XRD on the 
+powder diffractometer by placing single crystals onto silicon substrates and sealing under Kapton 
+films to reduce exposure to air. Data were collected using a Bruker D8 Discover powder 
+diffractometer with a high-efficiency IµS microfocus x-ray source for Cu Kα radiation (50 kV, 1 
+mA). For single-crystal XRD, a crystal measuring 0.10 x 0.05 x 0.01 mm3 was mounted on a 
+loop with oil. Data were collected at 263 K on a Bruker APEX II single-crystal X-ray 
+diffractometer using Mo-radiation, equipped with a Miracol X-ray optical collimator. The data 
+were integrated and scaled using SAINT, SADABS within the APEX2 software package by 
+
+	
+S-2 
+Bruker.3 Solution by direct methods (SHELXT4, 5 or SIR976, 7) produced a complete heavy-atom 
+phasing model consistent with the proposed structure. The structure was completed by difference 
+Fourier synthesis with SHELXL.8, 9 Scattering factors are from Waasmair and Kirfel.10 All atoms 
+were refined anisotropically by full-matrix least-squares. 
+Including intrinsic disorder, a least squares refinement optimization of the data yields the 
+lattice structure that we report. From the 983 reflections collected covering the indices, -8 ≤ h ≤ 
+8, -14 ≤ k ≤ 14, -8 ≤ l ≤ 8, 518 reflections were found that were symmetry independent and an R1 
+value of 0.0521 was obtained, indicating a good fit. R1 is calculated as: 
+!! =
+!!"# − !!"#!
+!!"#
+ 
+There is no detectable electron density between layers, indicating that Yb3+ does not intercalate 
+between layers in CrI3. 
+Variable-Temperature Photoluminescence (VTPL). Samples for VTPL measurements 
+were prepared by placing a single crystal between two quartz disks and loading into a closed-
+cycle helium cryostat. PL spectra were collected by exciting the sample with a continuous-wave 
+660 nm (1.88 eV) diode at 4 mW/cm2. Emission was collected and focused into a 
+monochromator with a spectral bandwidth of 0.627 nm and detected by a Hamamatsu 
+InGaAs/InP NIR photomultiplier tube, with signal recorded using a photon counter. Temperature 
+was varied from 4 to 300 K, starting at low temperature. All spectra were corrected for 
+instrument response. 
+Magnetic Circularly Polarized Luminescence (MCPL). Samples for MCPL measurements 
+were prepared as single crystals placed between two quartz disks and loaded into a 
+superconducting magneto-optical cryostat (Cryo-Industries SMC-1659 OVT) oriented in the 
+Faraday configuration. For full-spectrum measurements at static fields, samples were excited 
+with a 660 nm (1.88 eV) diode at approximately 40 mW/cm2. For field-sweep measurements, 
+samples were excited with a linearly polarized HeNe laser (632.8 nm/1.96 eV, 27 mW/cm2 for -6 
+to +6 T scans, 55 mW/cm2 for -0.4 to +0.4 T scans). No distinguishable difference was found in 
+the either the PL spectra or variable-field data between the two excitation sources. For field-
+sweep measurements, the monochromator was centered at 1.117 eV with a 6 nm spectral 
+bandwidth, and the signal was continuously monitored as the field was swept at a rate of 0.10 
+T/min and 0.45 T/min for the 0.4 T and 6 T scans, respectively. PL was collected along the 
+magnetic-field axis and passed through a liquid-crystal variable retardation plate set at λ/4, 
+followed by a linear polarizer to separate the left- and right-circularly polarized components. The 
+PL was then focused into a fiber-optic cable and fed into a monochromator with a spectral 
+bandwidth of 0.627 nm and detected by a Hamamatsu InGaAs/InP NIR photomultiplier, with 
+signals recorded using a photon counter. Polarization ratios are defined as	ρ = (σ- – σ+)/(σ- + σ+) 
+= (IL – IR)/(IL + IR) = ΔI/I, following the sign conventions outlined in Piepho and Schatz.11 
+Magnetic Measurements. Magnetic data on individual single-crystal flakes (Fig. 1) were 
+collected using a Quantum Design PPMS DynaCool vibrating sample magnetometer (VSM). A 
+flake was affixed to the end of a quartz paddle with varnish (VGE 7031). The paddle was then 
+snapped into the VSM brass sample holder with another quartz paddle placed symmetrically 
+above the sample. The weak background signal from the sample holder was removed in the data 
+analysis. The sample was probed with the external field aligned perpendicular to the face of the 
+crystal, and magnetization data were collected as a function of applied field and temperature. The 
+masses of individual flakes are below 0.1 mg and could not be accurately measured, so the 
+magnetic data are reported in units of emu. 
+
+	
+S-3 
+Ligand-field calculations within the Angular Overlap Model (AOM). Yb3+ 
+ligand(crystal)-field energies and g factors were calculated using the BonnMag package.12 
+Crystallographic data13 on CrI3 were used to create an [YbI6]3- unit with reduced symmetry 
+(point group C2). Crystallographic parameters were not adjusted for size differences between 
+Cr3+ and Yb3+. The electronic structure of Yb3+ was calculated using the spin-orbit coupling 
+parameter ζ as well as AOM parameters eσ and eπ to describe σ and π interactions with the I- 
+ligands, respectively. The value for eπ was taken to be isotropic. The Slater-Condon-Shortley 
+(SCS) parameters F2, F4, and F6 were taken to be 0, as is typically the case for Yb3+ (4f13 
+configuration). The Stevens orbital reduction factor k was taken to be equal to 1.0. Increasing 
+(decreasing) ζ while keeping all other parameters constant results in an increase (decrease) in all 
+transition energies while retaining peak splitting energies. Adjusting eσ or eπ alters the relative 
+energies of the peaks but maintains the barycenters.  
+ 
+ 
+ 
+	
+	
+Figure S1. Images of an individual Yb3+:CrI3 single-crystal flake under an optical microscope at 
+various magnification levels, viewing the flake's (a,b) edge, and (c) face. The flake thickness is 
+estimated to be 5.1 ± 0.3 µm.  
+ 
+Side 
+View!
+Yb3+:CrI3!
+Tape!
+Side 
+View!
+Top 
+View!
+a!
+b!
+c!
+
+100μm10 μm100 μm	
+S-4 
+	
+Figure S2. Analysis of XRD reflections collected using a powder diffractometer for 4.9% Yb3+-
+doped and undoped CrI3 single-crystal flakes (same data as shown in Fig. 1bc). Using the 
+method described by Jesche,14 the lattice parameter c for oriented single crystals with a 
+monoclinic space group can be extracted from XRD data from a powder diffractometer using the 
+following equation: 
+2c ∙ sinβ ∙ sin θ − S cosθ
+2
+= λℓ 
+Here, β is the obtuse angle in the monoclinic unit cell (108.507° for CrI3), λ is the x-ray 
+wavelength (Cu, 1.5406 Å), ℓ is the Miller index of each reflection in the XRD spectrum and 
+S
+!"#!
+!  is a correction factor related to the displacement of the x-ray focal plane relative to the 
+sample surface. Plotting 2θ values of the peak maxima vs λℓ, the data can be fit using the 
+equation above. For fitting, β and θ were taken in radians. By this method, the c lattice 
+parameters were found to be 6.996 ± 0.002 and 7.013 ± 0.002 Å for the undoped and doped 
+samples, respectively. From the lattice parameter c, the position of the (00ℓ) powder 
+diffractometer XRD peaks for a monoclinic single crystal can be calculated using the following 
+equation: 
+2θ = 2sin!!
+λ
+2sinβ
+ℓ
+c  
+The zero-shift in 2θ was determined by adding an offset to the experimental 2θ values and 
+adjusting the offset to minimize the difference between experimental and calculated peak 
+positions across all peaks in the XRD spectrum. This offset accounts for the measurement 
+discrepancy due to the thickness of the single crystals displacing the x-ray focal plane. For CrI3, 
+a zero-shift of -0.015° was found, contrasted to a zero-shift of +0.164° for Yb3+-doped CrI3. The 
+displacement-corrected XRD spectra are shown in Fig. 1c in the main text. 
+ 
+ 
+ 
+ 
+10
+8
+6
+4
+2
+0
+λℓ (Å)
+90
+80
+70
+60
+50
+40
+30
+20
+10
+2θ (deg.)
+CrI3
+Yb
+3+:CrI3
+
+	
+S-5 
+ 
+Table S1. Single-crystal X-ray diffraction data for 2.5% Yb3+:CrI3 measured at 263 K, 
+compared to literature data for CrI3. 
+ 
+ 
+Yb3+:CrI3 
+CrI3 (250 K, ref. 13) 
+Space group 
+C2/m 
+C2/m 
+ 
+ 
+ 
+a 
+6.86 Å 
+6.87 Å 
+b 
+11.89 Å 
+11.89 Å 
+c 
+6.99 Å 
+6.98 Å 
+α 
+90.0° 
+90.0° 
+β 
+108.7° 
+108.5° 
+γ 
+90.0° 
+90.0° 
+ 
+ 
+ 
+[(Yb/Cr) – Cr]avg 
+3.96 Å 
+3.96 Å 
+[(Yb/Cr) – I]avg 
+2.72 Å 
+2.72 Å 
+[(Yb/Cr) – I – (Yb/Cr)]avg 
+93.3° 
+93.6° 
+[I – (Yb/Cr) – I]avg 
+86.8° 
+86.9° 
+ 
+ 
+ 
+ 
+Figure S3. Visualization of the experimental room-temperature single-crystal XRD structure as 
+viewed along the a, b, and c principal axes (left to right). Yb3+ (cyan) is found to substitute for 
+Cr3+ (blue) in the edge-sharing octahedra formed by I- (purple) anions. No excess electron 
+density is observed between layers. Intralayer disorder is observed. The structure refines to the 
+expected high-temperature C2/m monoclinic symmetry. Some intralayer disorder was observed 
+(not shown). 
+
+aa	
+S-6 
+Figure S4. (a) Variable-temperature PL spectra of CrI3 measured from 4 to 200 K under 1.88 eV 
+CW excitation (from Fig. 2 of the main text). (b) Scatter plot depicting total integrated area of 
+the CrI3 PL from panel (a). The 200 K intensity is 7.5% that of the 4 K value. (c) Variable-
+temperature PL spectra of 4.9% Yb3+:CrI3 measured from 4 to 200 K under 1.88 eV CW 
+excitation (from Fig. 2 of the main text). (d) Scatter plot depicting total integrated area of the 
+Yb3+ PL from panel (c). The 200 K intensity is 0.8% that of the 4 K value. (e) Variable-
+temperature PL spectra of 5.0% Yb3+:CrI3 measured from 4 to 200 K under 1.88 eV CW 
+excitation (from Fig. 2 of the main text). (f) Scatter plot depicting total integrated area of the 
+Yb3+ PL from panel (e). The 200 K intensity is 7.5% that of the 4 K value. Note that a second, 
+broad "trap" PL band is observed at ~0.98 eV in samples made from Cr metal powder precursor 
+(99.94%, panel (c)) but not in samples made from Cr chip precursor (99.995%, panel (e)). Ni is 
+the primary impurity in the powder precursor (see Methods), and Ni is detected in this CrI3 
+sample at 0.4% cation mole fraction. Ni2+ 3A2g ! 3T2g absorption in NiI2 and Ni2+:CdI2 is 
+centered around 0.93 eV,15 and the broad "trap" PL band in panel (c) is thus tentatively attributed 
+to Ni2+ impurities in CrI3. 
+
+a
+b
+1.0
+Crl3 Undoped
+Integrated Area (norm.)
+1.0
+Crl3
+PL Intensity (norm.)
+4 - 200 K
+0.89 - 1.24 eV
+0.8
+Cr3+
+0.8
+0.6
+0.6
+0.4
+0.4 
+0.2
+0.2
+0.0
+0.0-
+1.2
+1.1
+1.0
+0.9
+0
+50
+100
+150
+200
+Energy (eV)
+Temperature (K)
+c
+d
+1.0
+Crl3
+Integrated Area (norm.)
+1.0
+(wuou) /
+0.8
+Yb 3+
+4 - 200 K
+1.07 - 1.2 eV
+0.8
+_ Intensity
+0.6.
+0.6-
+Trap
+0.4 -
+0.4
+0.2
+0.2-
+0.0
+0.0-
+1.2
+1.1
+1.0
+0.9
+T
+0
+50
+100
+150
+200
+Energy (eV)
+h
+e
+Temperature (K)
+1.0
+1.0
+Yb°
++ :Crl3
+Integrated Area (norm.)
+ Intensity (norm.)
+0.8
+4 - 200 K
+0.8
+1.07 - 1.2 eV
+3+
+h
+0.6
+0.6
+0.4.
+0.4
+0.2
+0.2
+0.0
+0.0.
+1.0
+0
+50
+100
+1.2
+1.1
+0.9
+150
+200
+Energy (eV)
+Temperature (K)	
+S-7 
+ 
+Figure S5. Comparison of the 5 K experimental data and calculated (AOM) f-f PL transition 
+energies for 4.9% Yb3+:CrI3. A best fit to the experimental PL data resulted in the following 
+values: ζ = 2665 cm-1 (330.4 meV), eσ = 176.5 cm-1 (21.9 meV), eπ = 122.5 cm-1 (15.2 meV). The 
+calculated transition energies using these parameters are shown as the vertical red lines in both 
+panels. (a) Comparison of calculated transition energies obtained by changing from ζ = 2665 cm-
+1 (red) to ζ = 2675 cm-1 (blue), with all other parameters constant to the best-fit (red). (b) 
+Comparison of calculated transition energies obtained by individually changing the values of eσ 
+and eπ. The gray traces show the effect of changing from eσ = 176.5 cm-1 (red) to eσ = 206.5 cm-1 
+with all other parameters constant to the best fit (red). The green traces show the effect of 
+changing from eπ = 122.5 cm-1 (red) to eπ = 152.5 cm-1 with all other parameters constant to the 
+best fit (red). From the best-fit parameters, g is anisotropic (g1 = 2.672, g2 = 2.686, g3 = 2.642) 
+and an average ground-state g value of ~2.7 is predicted.  
+ 
+ 
+ 
+ 
+ 
+
+a
+b
+1.0
+1.0
+{ = 2665 cm
+ = 2665 cm
+ Intensity (norm.)
+. Intensity (norm.)
+0.8
+0.8 -
+e。 = 176.5 cm
+e = 122.5 cm
+e = 122.5 cm
+-1
+ = 2665 cm
+0.6
+0.6 -
+-
+e. = 206.5 cm
+e= 122.5 cm
+ = 2675 cm
+0.4
+0.4 -
+{ = 2665 cm
+e。 = 176.5 cm
+-1
+PL
+0.2
+e = 122.5 cm
+P
+0.2
+=152.5cm
+0.0
+0.0
+9400
+9200
+9000
+8800
+9400
+9200
+9000
+8800
+Wavenumber (cm-")
+Wavenumber (cm"")	
+S-8 
+ 
+ 
+Figure S6. The Yb3+ valence energy level diagram described by the best-fit parameters of Fig. 
+S5. The energies of the crystal field states in eV are: 0.0000, (0.0179, 0.0182), 0.0496, (1.1667, 
+1.1668), 1.2013 eV.  
+ 
+ 
+ 
+ 
+
+Energy (cm"1)
+10000
+9689
+(9410, 9411)
+8000
+Energy (cm
+6000
+4000
+2000
+400
+-0
+(144, 147)
+0
+Free lon +
+Spin Orbit Coupling + Crystal Field	
+S-9 
+Table S2. Energies (cm-1) of the valence electronic states, 2F5/2 and 2F7/2 barycenter 
+energies,a and ΔE(Barycenter) for Yb3+ ions in several host crystals, and for the free ion. 
+These data were used to generate Fig. S7 (after converting to eV). Many of these entries are 
+compiled in ref. 16. 
+Host Lattice 
+0 
+1 
+2 
+3 
+2F7/2 
+Barycenter 
+0' 
+1' 
+2' 
+2F5/2 
+Barycenter 
+ 
+ΔE(Bary) 
+ref. 
+Ca2Ga2SiO7 (CGS) 
+0.0 
+300 
+490 
+970 
+440 
+10250 
+10570 
+11010 
+10610 
+10170 
+17 
+SrLaGa3O7 (SLG) 
+0.0 
+220 
+386 
+910 
+379 
+10190 
+10450 
+11025 
+10555 
+10176 
+17 
+Ca4GdO(BO3)3 
+(GdCOB) (site I, Gd) 
+0.0 
+423 
+668 
+1003 
+524 
+10246 
+10706 
+11089 
+10680 
+10157 
+18 
+GdCOB (site II, Ca) 
+0.0 
+437 
+694 
+1003 
+534 
+10261 
+10737 
+11150 
+10716 
+10183 
+18 
+GdCOB (site III, Ca) 
+0.0 
+417 
+688 
+1003 
+527 
+10240 
+10682 
+11026 
+10649 
+10122 
+18 
+Ca4YO(BO3)3 (YCOB) 
+0.0 
+427 
+556 
+1023 
+502 
+10242 
+10537 
+11109 
+10629 
+10128 
+19 
+Sc2O3 
+0.0 
+474 
+634 
+1076 
+546 
+10250 
+10640 
+11198 
+10696 
+10150 
+20 
+Ca5(PO4)3F (CFAP) 
+0.0 
+409 
+597 
+1099 
+526 
+10178 
+10496 
+11069 
+10581 
+10055 
+21 
+Sr5(PO4)3F (SFAP) 
+0.0 
+362 
+618 
+1190 
+543 
+10150 
+10512 
+11108 
+10590 
+10048 
+22 
+Sr5(VO4)3F (SVAP) 
+0.0 
+321 
+562 
+1078 
+490 
+10141 
+10740 
+11050 
+10644 
+10153 
+23 
+Y3Al5O12 (YAG) 
+0.0 
+584 
+635 
+783 
+501 
+10328 
+10752 
+10917 
+10666 
+10165 
+24 
+BaCaBO3F (BCBF) 
+0.0 
+303 
+533 
+902 
+435 
+10204 
+10570 
+11000 
+10591 
+10157 
+25 
+LiNbO3 
+0.0 
+352 
+448 
+788 
+397 
+10204 
+10471 
+11090 
+10588 
+10191 
+26 
+KGd(WO4)2 (KGW) 
+0.0 
+163 
+385 
+535 
+271 
+10188 
+10471 
+10682 
+10447 
+10176 
+27 
+KY(WO4)2 (KYW) 
+0.0 
+169 
+407 
+568 
+286 
+10187 
+10476 
+10695 
+10453 
+10167 
+27 
+CaWO4 
+0.0 
+220 
+366 
+492 
+270 
+10278 
+10366 
+10665 
+10436 
+10167 
+28 
+YAlO3 
+0.0 
+209 
+341 
+590 
+285 
+10220 
+10410 
+10730 
+10453 
+10168 
+28 
+LiYF4 
+0.0 
+216 
+371 
+479 
+267 
+10288 
+10409 
+10547 
+10415 
+10148 
+28 
+YAl3(BO3)4 (YAB) 
+0.0 
+94 
+185 
+581 
+215 
+10194 
+10277 
+10672 
+10381 
+10166 
+29 
+Cs2NaYbCl6 
+0 
+225 
+225 
+573 
+256 
+10243 
+10243 
+10708 
+10398 
+10142 
+30, 31 
+Cs3Yb2Br9 
+0.0 
+144 
+201 
+421 
+192 
+10277 
+10301 
+10578 
+10385 
+10194 
+32 
+CsCdBr3 
+0.0 
+114 
+140 
+441 
+174 
+10119 
+10146 
+10590 
+10285 
+10111 
+32 
+CuInS2 
+0.0 
+32 
+87 
+182 
+75 
+10033 
+10060 
+--- 
+10095a 
+10020 
+33 
+InP 
+0 
+35.5 
+35.5 
+97.5 
+42 
+10018 
+10064 
+10064 
+10049 
+10007 
+34 
+Free ion 
+--- 
+--- 
+--- 
+--- 
+0.0 
+--- 
+--- 
+--- 
+10213 
+10213 
+35 
+CrI3 
+0.0 
+146 
+146 
+400 
+173 
+9410 
+ 
+--- 
+9551a 
+9379 
+this 
+work 
+aFor the entire data set of complete entries, the ratio of 2F5/2:2F7/2 CF splitting energies, (E(2F5/2 Barycenter) - 
+E0')/(E(2F7/2 Barycenter)) is 0.82 ± 0.14. The 2F5/2 barycenter energies for Yb3+:CrI3 and Yb3+:CuInS2 were thus set 
+equal to the 2F7/2 barycenter energies for the same compounds. The resulting uncertainties in ΔE(Bary) are estimated 
+to be < ~1%, close to or smaller than the data points in Fig. S7. For comparison, the Yb3+:CrI3 AOM calculations 
+above yield: 2F7/2 barycenter = 173 cm-1 (21 meV), 2F5/2 barycenter = 9503 cm-1 (1.178 eV), ΔΕ(Bary) = 9330 cm-1 
+(1.157 eV), within this uncertainty range. 
+ 
+ 
+ 
+	
+ 
+ 
+			
+
+	
+S-10 
+ 
+Figure S7. Plot of the difference between experimental Yb3+ 2F5/2 and 2F7/2 barycenter energies 
+(ΔE(Bary)) for the compounds listed in Table S2, and for the free ion, vs the barycenter energy 
+for the 2F7/2 ground multiplet. The compounds associated with select data points are labeled. The 
+dashed blue line shows the value of the free ion. 
+ 
+ 
+Figure S8. (a) Power dependence of !- (red) and !+ (black) PL peak intensities and circular 
+polarization (ρ, blue) of the Γ8 ! Γ7 transition. The data were collected at 0.5 T and 5 K and the 
+sample was excited with linearly polarized light at 1.96 eV. The PL intensities show a linear 
+increase with power, resulting in a constant polarization ratio. The error bars represent 
+uncertainty estimated from the linear fit of the polarization intensities. (b) The !- (red) and !+ 
+(black) component of the Γ8 ! Γ7 transition normalized across all powers. The traces overlay 
+each other well, showing no detectable power dependence. 
+ 
+
+b
+a
+0.20
+, Signal Intensity (norm.)
+1.0
+1.0
+a-
+Polarization Ratio (
+Intensity (norm.)
+0.8
+0.8
+a+
+0.15
+a+
+0.6
+0.6
+.0.10
+0.4.
+0.4
+0.05
+0.2
+0.2
+0.0-
+-0.00
+0.0 -
+0
+100
+200
+300
+400
+1.120
+1.116
+1.112
+Energy (eV)
+Power Density (mW/cm3)KGW
+1.28
+CaWO4
+Free lon
+Cs3Yb2Brg
+LiYF4_ KYW
+1.26-
+QOYAIO3
+YAB
+CsCdBr3
+Cs2NaYbCl6
+Oxides
+00
+△E(Bary) (eV)
+1.24-
+CulnS2
+InP
+1.22-
+1.20 -
+1.18-
+1.16-
+0
+10
+20
+30
+40
+50
+60
+70
+2F7/2 Barycenter (meV)	
+S-11 
+ 
+Figure S9. Comparison of full MCPL spectra across two different samples, measured at 0.5 T, 5 
+K. (a) The sample used in Fig. 3b,c,e,f of the main text. (b) The sample used in Fig. 3d of the 
+main text. The two samples show very similar spectra, with slight differences in polarization 
+magnitude. 
+ 
+Figure S10. Magnetic data for a single-crystal flake of 5% Yb3+:CrI3, measured by VSM. The 
+sample was probed with the external field aligned perpendicular to the face of the crystal. (a) 
+Plots of magnetization vs external field measured at various temperatures. The data are similar to 
+those collected on undoped CrI3 bulk crystals (e.g., Fig S11). At 2 K, a coercive field of ~44 mT 
+was found. (b) Plot of magnetization vs temperature measured in the field-cooled and field-
+warmed directions. The inset shows the derivative of the field-cooled data as a function of 
+temperature, where the Curie temperature is found to be 60.4 K. These data show that Yb3+ 
+doping has no significant effect on the magnetism of CrI3 in these samples. 
+ 
+
+a
+b
+1.0
+1.0.
+0.5 T
+0.5 T
+ Intensity (norm.)
+5 K
+ Intensity (norm.)
+5 K
+0.8
+0.8 -
+a+
+a+
+0.6 -
+0.6 -
+0.4 -
+0.4 -
+0.2
+0.2 -
+P
+0.0 -
+0.0 -
+1.18
+1.16
+1.14
+1.12
+1.10
+1.08
+1.18
+1.16
+1.14
+1.12
+1.10
+1.08
+Energy (eV)
+Energy (eV)a
+b
+(emu)
+0.008
+Magnetic Moment (emu)
+0.02 T
+0.0008
+Magnetic Moment
+0.004
+0.0006
+dM/dT
+0.000
+Field Warmed
+2 K
+0.0004
+Tc = 60.4 K
+Field Cooled
+20 K
+0.004
+40 K
+60 K
+0.0002
+0
+50
+100
+80 K
+T(K)
+0.008
+100 K
+-3
+-2
+-1
+0
+1
+2
+3
+20
+40
+60
+80
+100
+Field (T)
+Temperature(K)	
+S-12 
+ 
+Figure S11. The same polarization data as featured in Fig. 3d of the main text, overlayed with 
+CrI3 magnetization data measured from -3 to +3 T with the field oriented parallel to the 
+crystallographic c axis (blue) by single-crystal vibrating sample magnetometry (VSM).36 For 
+comparison, the magnetization perpendicular to c (green) is also shown. The Yb3+ MCPL 
+polarization ρ is superimposable with the CrI3 magnetization measured in the same 
+configuration. 
+ 
+ 
+ 
+Figure S12. (a) Individual circularly polarized MCPL components measured during continuous 
+field sweeps from -6 to +6 T and back at 5 K. (b) The same data, displayed as the polarization 
+ratio (ρ, normalized). Panel (b) is shown as Fig. 3d of the main text. Data measured using 14 
+mW/cm2 excitation. 
+ 
+ 
+
+b
+a
+9000
+1.0 -
+5 K
+ Intensity (counts)
+8000
+(norm.)
+0.5
+g+
+7000
+0.0
+2
+6000-
+6
+-0.5
+p
+P
+5000
+-1.0
+T
+9-
+-4
+-2
+0
+2
+4
+6
+-6
+-4
+-2
+0
+2
+4
+6
+Field (T)
+Field (T)1
+Crls Single
+1.0-
+1.0
+Magnetization (norm.)
+Crystal VSM
+p 1.117 ev (norm.)
+0.5-
+0.5
+BIl c BIla,b
+0.0.
+0.0
+-0.5-
+-0.5
+-1.0-
+-1.0
+9-
+-2
+0
+2
+4
+6
+Field (T)	
+S-13 
+ 
+ 
+Figure S13. Comparison of field-dependent polarization ratios (ρ, normalized) measured with 
+(a) linearly polarized and (b) unpolarized excitation at 5 K. In panel (b), no data were collected 
+above 2 T. Panel (a) is shown as Fig. 3d of the main text. 
+ 
+ 
+ 
+Figure S14. (a,b) Effect of excitation power on the polarization ratio (ρ, normalized). Magnetic 
+hystereses measured under (a) low- and (b) higher-power excitation (14 vs 55 mW/cm2, 5 K) 
+show no difference. The black (red) trace corresponds to the sweep from negative (positive) to 
+positive (negative) fields. (c, d) The separate circularly polarized PL components from the same 
+(c) low- and (d) high-power measurements. 
+ 
+
+a
+b
+1.0
+5 K
+1.0
+5 K
+(wou)
+('wuou)
+0.5
+0.5
+1.117 ev
+0.0
+1.117 eV
+0.0
+-0.5
+-0.5
+p
+p
+-1.0
+-1.0-
+-6 
+-4 
+-2
+0
+2
+4
+6
+-2
+-1
+0
+1
+2
+Field (T)
+Field (T)a
+b
+1.0 -
+1.0 -
+55 mW/cm
+2
+14 mW/cm
+0.5
+0.5 -
+0.0
+0.0
+-0.5 -
+-0.5-
+Q
+-1.0-
+--
+-1.0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Field (T)
+Field (T)
+c
+d
+PL Intensity (counts)
+PL Intensity (counts)
+120 x10
+30 x10
+115-
+g+
+28
+9+
+55 mW/cm
+14 mW/cm
+110-
+105-
+26
+100 -
+24 -
+95
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Field (T)
+Field (T)	
+S-14 
+ 
+Figure S15. Temperature dependence of the Γ8 ! Γ7 PL feature of 4.9% Yb3+:CrI3 measured 
+from 4 to 200 K under no external magnetic field (from Fig. 4 of the main text, T = 4, 15, 30, 40, 
+50, 55, 58, 60, 62, 65, 70, 85, 100, 125, 150 K). A linear baseline was subtracted from each 
+spectrum here to facilitate viewing and determination of the peak's FWHM. 
+ 
+
+1.0-
+Baseline Subtracted
+PL Intensity (norm.)
+0.8 -
+0.6-
+0.4 -
+0.2
+0.0
+1.135
+1.130
+1.125
+1.120
+1.115
+1.110
+Energy (eV)	
+S-15 
+	
+ 
+Figure S16. (a,b) False-color plots of the Yb3+ PL intensities vs temperature measured for the 
+two samples shown in Fig. S4c,e, respectively, from 4 to 150 K at zero external magnetic field. 
+The horizontal dashed line indicates TC = 61 K. The two samples show the same temperature 
+dependence, but the features are slightly better resolved in panel (a). Panel (a) is shown as Fig. 
+4a of the main text. 
+ 
+ 
+ 
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+ 
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+Two-dimensional tile displacement
+can simulate cellular automata
+Erik Winfree1,2,3⋆ and Lulu Qian1,2,3⋆
+1Bioengineering, 2Computer Science, 3Computation and Neural Systems
+California Institute of Technology, Pasadena, CA 91125, USA
+⋆e-mail: [email protected], [email protected]
+Tile displacement is a newly-recognized mechanism in DNA nanotechnology that exploits principles anal-
+ogous to toehold-mediated strand displacement but within the context of self-assembled DNA origami tile
+arrays. Here, we formulate an abstract model of tile displacement for the simplest case: individual assemblies
+interacting with monomer tiles in solution. We give several constructions for programmable computation
+by tile displacement, from circuits to cellular automata, that vary in how they use energy (or not) to drive
+the system forward (or not), how much space and how many tile types they require, and whether their com-
+putational power is limited to PTIME or PSPACE with respect to the size of the system. In particular, we
+show that tile displacement systems are Turing universal and can simulate arbitrary two-dimensional syn-
+chronous block cellular automata, where each transition rule for updating the state of a 2× 2 neighborhood
+is implemented by just a single tile.
+Keywords: DNA origami, tile displacement, cellular automata, reversible computation
+1. INTRODUCTION
+A guiding principle in theoretical computer science has
+been “mechanism-to-model” exploration of connections
+between physical implementation and computational ca-
+pabilities. For example, what can be computed by sys-
+tems of AND gates and OR gates is strictly less than what
+can be computed by systems of NOR gates, which in turn
+is less than what can be computed by finite state machines
+coupled with an unbounded memory tape [1]. Likewise,
+molecular programming theory aims to understand how
+fundamental molecular mechanism can be used to build
+systems, and how the choice of mechanism determines
+the range of what can be built. An example would be the
+self-assembly of molecular structures by programmable
+cooperative binding, which can reliably grow structures
+that cannot grow reliably via non-cooperative binding [2].
+When a new molecular mechanism is discovered, it is of
+interest to understand the nature – the limitations and
+capabilities – of systems that exploit that mechanism.
+Doing so entails formulation of an abstract model that
+captures the essential features of the mechanism, which
+can then be rigorously analyzed.
+Since its invention two decades ago [4],
+toehold-
+mediated DNA strand displacement has been a central
+mechanism for programming dynamical function in DNA
+nanotechnology [5, 6]. As shown in figure 1A, a stable
+complex of two strands can be reconfigured such that an
+invading strand replaces the original partner via a branch
+migration process.
+The single-stranded portion of the
+original complex – known as the toehold because that
+is where the invading strand initiates contact – is critical
+for the displacement: an invader that does not match and
+bind to the toehold may be a million-fold slower to per-
+form the displacement, and thus neglected as “leak”. In
+abstract models that consider networks of more complex
+(but still non-pseudoknotted) DNA molecules that inter-
+act in solution using toehold-mediated strand displace-
+ment reactions – including both the irreversible mecha-
+nism shown here and a reversible variant known as “toe-
+hold exchange” – have been shown capable of simulating
+arbitrary formal chemical reaction network dynamics and
+even Turing-universal computation [7, 8, 9, 10, 11, 12, 13].
+However, a limitation of these results is that they are in-
+trinsically distributed computations, where state is en-
+coded within a collection of molecules in solution, and
+therefore a single test tube can perform only one com-
+putation at a time.
+More complex molecular mecha-
+nisms, such as the hypothetical polymer-modifying en-
+zymes envisioned by Bennett [14], are in principle capable
+of performing independent Turing-universal computation
+in parallel in the same test tube.
+The mechanism of “tile displacement”, shown in fig-
+ure 1B, was recently discovered during investigations into
+why the self-assembly of DNA origami tiles [15, 16, 17]
+failed to become kinetically trapped in undesired inter-
+mediates that the naive theory predicted [3]. There is a
+strong analogy to toehold-mediated strand displacement.
+Beyond using components that are two orders of magni-
+tude larger than the individual strands involved in strand
+displacement, the tile displacement mechanism has sev-
+eral distinct features. (1) Nucleotides are on multiple he-
+lices that are oriented orthogonally to the axis of branch
+1
+arXiv:2301.01929v1  [cs.ET]  5 Jan 2023
+
+A
+B
+C
+invader tile
+released tile
+branch migration
+game board
+player 1
+player 2
+
+ꓳ
+
+ꓳ
+
+
+ꓳ
+
+ꓳ
+
+ꓳ
+
+ꓳ
+
+
+ꓳ
+
+ꓳ ꓳ
+
+
+ꓳ
+
+ꓳ ��
+  
+
+(1,3)
+ꓳ
+(1,1)
+
+(3,1)
+ꓳ
+(2,2)
+
+(3,3)
+ꓳ
+(2,3)
+
+(3,2)
+day 0
+day 1
+day 2
+day 3
+day 4
+day 5
+day 6
+day 7
+invader strand
+released strand
+branch migration
+Figure 1:
+(A) Strand displacement mechanism. For scale, the DNA molecules are roughly 2 nm in diameter and 7 nm long.
+(B) Tile displacement mechanism. These hypothetical 10-helix DNA origami tiles are smaller than the 22-helix square tiles
+from [3]. (C) A tic-tac-toe game implemented using tile displacement (adapted from reference [3]). Scale bar for atomic force
+microscopy (AFM) images is 100 nm.
+migration, rather than being on a single helix that is par-
+allel to (identical to) the axis of branch migration. (2)
+Tile-tile binding may be due to multiple helix-end stack-
+ing bonds [18] in addition to (or instead of) being due to
+base-pair formation. (3) Toehold and branch migration
+domain specificity can be encoded both by tile geometry
+and by the sequences in multiple very short (1 or 2 nt)
+sticky ends, rather than being exclusively encoded by nu-
+cleotide sequences within a single helix. (4) The released
+tile will be less flexible than a single-stranded oligonu-
+cleotide. (5) Rather than having just one “side” and ini-
+tiating displacement via a single toehold, tiles may have
+many (e.g. four) sides and may initiate displacement via
+cooperative action of multiple toeholds, as highlighted by
+the replacement of the central tile of a 3 × 3 tic-tac-toe
+game board [3] shown in figure 1C. Despite these dif-
+ferences, it remains that tile displacement is highly sen-
+sitive to toehold and branch migration sequences, such
+that the kinetics of tile displacement without a matching
+toehold may be orders of magnitude slower and similarly
+negligible as “leak”. Indeed, systems of interacting tile
+monomers and tile assemblies were shown to be reconfig-
+urable by toehold-mediated tile displacement [3], and the
+same or similar constructs ought to be sufficient to im-
+plement more complex information-processing networks
+following, for example, the seesaw motif for circuits [19]
+or the two-domain scheme for formal chemical reaction
+network dynamics [10].
+Here we are interested in whether the tile displacement
+mechanism enabled new ways of programming dynamical
+behaviors, beyond simply replicating strand displacement
+on a larger scale. The ability to perform displacement
+within a two-dimensional array being an especially novel
+feature of tile displacement, we ask whether – unlike ex-
+isting strand displacement constructions – reconfiguration
+of a single tile assembly in a constant soup of monomer
+tiles might be sufficient for substantial computation, in
+which case parallel computation could be achieved with
+each tile assembly performing an independent computa-
+tion. We present three results. First, with a feedforward
+Boolean circuit laid out on the initial array, there is a tile
+set that, via displacement, propagates signals along wires
+and executes the specified logic. This system is powered
+by the energy of toehold formation; the final state is in an
+energy minimum and cannot be reused. Second, as a sim-
+plification and generalization of the first construction, any
+one-dimensional cellular automaton can be directly trans-
+lated into a set of tiles such that a wave of tile displace-
+ment converts an assembly, initially empty but for the
+input, into the space-time history of the cellular automa-
+ton. This system is powered by a concentration difference
+between the monomer tiles that are invading over those
+that are displaced. The above two constructions displace
+each tile in the original array at most once, using energy
+that is linear in the area used. The third construction ad-
+dresses whether iterated computation can be performed
+in-place, which requires replacing the tile at a given loca-
+tion an unbounded number of times. Remarkably, using
+locally reversible asynchronous tile displacement, we can
+simulate arbitrary synchronous block cellular automata
+that use the 2 × 2 Margolus neighborhood, including his
+globally reversible Billiard Ball Model that is known to
+be Turing universal by simulation of infinite or finite re-
+current Boolean circuits [20, 21, 22]. A key issue is how to
+bias the computation forward; we show that it is enough
+to include a large empty part of the array into which en-
+tropy is injected.
+This paper does not aim for novel advances in molecu-
+2
+
+全全lar programming that make technical applications closer
+to reality. We are sharing these observations mainly be-
+cause we find them to be beautiful and surprising. Tile
+displacement may indeed be useful for reconfiguration of
+adaptive molecular systems, but for most implementation
+goals that are merely computational, there are more di-
+rect and more reliable ways to achieve them using other
+mechanisms in DNA nanotechnology. However, it is re-
+markable that a molecular mechanism accidentally dis-
+covered in the laboratory gives rise to a theoretical model
+with such natural and direct connections to an esoteric
+but well-studied model of computation that arose in the
+study of the fundamental physics and ballistic motion.
+We hope you will see through our imperfect figures and
+clumsy explanations to see the poetry within the con-
+cepts [23].
+2. TILE DISPLACEMENT MODEL
+The abstract model developed in this work, which we call
+the Single-Assembly Tile Displacement (SATiDi) model,
+defines the behavior of a single tile assembly within a
+sea of monomer tiles. There are a number of assumptions
+that must hold in order for the model to be experimentally
+plausible, while also allowing its definition to be fairly
+clean.
+Singularity. The concentration of multi-tile assemblies
+is sufficiently low (e.g. there is exactly one) that they
+do not interact with each other.
+Monomers. Binding between two monomer tiles is suf-
+ficiently weak (at the given temperature and con-
+centrations) that any dimers are fleeting and their
+presence can be neglected.
+Stability. Tiles within an assembly (e.g.
+with four
+neighbors) are sufficiently strongly attached that
+they will not dissociate; tiles on the boundaries and
+corners, with only two or three neighbors, may have
+special binding interactions that make them as stable
+as the internal tiles.
+No growth. With a single-side attachment being unsta-
+ble for dimers, similarly new tiles may not attach by a
+single side to a facet of a multi-tile assembly. When
+the assembly is rectangular, as will be exclusively
+considered here, that means the number of tiles in
+an assembly will never change.
+Full toeholds. For consistent tile displacement kinetics,
+we require that the displacement process on each
+side has its own mediating toehold, so a tile that
+is bound to four neighbors will be displaced by a tile
+that forms a toehold on each of the four sides. While
+fewer toeholds may be sufficient for displacement, it
+is all too plausible that their kinetics would be ir-
+regular; our simulator will issue a warning whenever
+such a displacement possibility is encountered.
+Energetics. Tile displacement reactions must be either
+energetically neutral or energetically downhill, i.e.
+the number of toeholds formed is either the same
+as or more than the number of toeholds broken.
+Uniform design. Each side of every tile will consist
+of a first toehold domain, a branch migration do-
+main, and a second toehold domain.
+We will as-
+sume that the branch migration domains are distinct
+on the north, east, west, and south such that they
+force tile to maintain a specific orientation (although
+non-oriented versions of the model could be formu-
+lated when non-oriented tiles are desirable). Because
+branch migration domains cannot be changed by tile
+displacement, they will not be formally represented
+or accounted for in the model.
+The model is illustrated in figure 2A, where both a
+valid neutral tile displacement and a valid downhill tile
+displacement are shown.
+Neutral displacement can be
+though of as generalizing the “toehold exchange” mech-
+anism from strand displacement [24]: formation of me-
+diating toehold ensures fast kinetics, while dissociation
+of prior toeholds both ensures that the reaction is en-
+ergetically neutral and opens up those toeholds for use
+in subsequent steps, as shown in figure 2B. Toehold ex-
+change in tile displacement was demonstrated experimen-
+tally [3], although not in the exact geometric configura-
+tion required here; tuning of toehold strength (e.g. via
+temperature) would be required to ensure that the dis-
+sociation step (which may involve breaking four toeholds
+simultaneously) is sufficiently fast while still being effec-
+tive for mediating the reaction.
+Formally, a SATiDi system is defined by (1) a finite
+set of square tile types S, each of which specifies an or-
+dered pair of bond types (toeholds) for each of the four
+sides, (2) a bond strength function for each bond type b,
+Eb > 0, (3) a concentration for each tile type i, ci, and
+(4) a standard tile displacement rate constant k. The as-
+sociated set of assemblies A consists of finite arrays of tile
+types (or empty). Given a specific assembly, we say that
+a specific toehold on a specific tile is closed if the cor-
+responding toehold on the neighboring tile has the same
+(i.e. matching) bond type (i.e. they form a bond), while
+we say that it is open otherwise. The bond energy E(A)
+of an assembly A is the sum � −Eb over all closed toe-
+holds in the assembly, while the free energy G(A) of the
+assembly is its bond energy plus the sum � ln ci/c0 over
+all tiles in the assembly, where c0 is the reference con-
+centration (e.g. 1 M). Given these, we associate a formal
+chemical reaction network (CRN) with reactions
+A + ti
+k
+−→ A′ + tj
+where A is an assembly with tile tj at some position x, A′
+is the same assembly but with ti instead at that same po-
+sition x, and ti is a valid displacement: on all sides where
+tj has a neighbor, ti forms a matching bond with (at
+3
+
+A
+tiles that can displace the center tile:
+tiles that cannot displace the center tile:
+neutral
+downhill
+neutral 
+but no west toehold
+uphill
+downhill 
+but no north toehold
+warning
+a
+b
+c
+d
+x
+w
+y
+z
+a
+b
+c
+d
+x
+w
+y
+z
+a
+b
+c
+d
+C
+D
+B
+irreversible 
+strand displacement
+reversible 
+strand displacement
+Figure 2:
+(A) Abstract tile displacement model. (B) Irreversible and reversible strand displacement. (C) DNA origami
+tile implementation. (D) Single-stranded tile implementation.
+least one) open toehold, and the total number of match-
+ing bonds increases or stays the same (i.e. the assembly’s
+bond energy decreases or stays the same). When all reac-
+tions are reversible, which implies that the bond energy
+of the assembly never changes, the CRN satisfies detailed
+balance with respect to the assembly bond energy, with
+monomer tiles having zero energy.
+We consider standard stochastic kinetics according
+to
+Gillespie
+simulation
+with
+chemostatted
+constant
+monomer tile concentration [25, 26]. For an initial state
+containing a single assembly, this results in a finite
+continuous-time Markov chain (CTMC) where the set of
+states are all assemblies reachable via tile displacement
+reactions, and the transition A → A′ involving invading
+tile ti, as above, will have rate k × ci. If all reactions
+are reversible, this CTMC will satisfy detailed balance
+with respect to the assembly free energy, such that the
+equilibrium probability of assembly A is
+p(A) = 1
+Z e−G(A)
+with
+Z =
+�
+A′
+e−G(A′)
+where the partition function sum Z is taken with respect
+to all assemblies reachable by tile displacement.
+A tile displacement system simulation is considered un-
+reliable if at any time there is an energetically neutral or
+downhill tile replacement that does not form at least one
+new toehold with each neighboring tile. In this case, the
+simulation issues a warning, as illustrated in figure 2A.
+We briefly consider possible experimental implementa-
+tions of single-assembly tile displacement systems. Fig-
+ure 2C shows the motivating DNA origami tile scheme,
+using a geometrically-symmetrical tile design modeled af-
+ter those used in several prior experimental works [27, 17,
+3, 28, 29]. More speculatively, in figure 2D we envision
+an implementation that makes use of topologically two-
+dimensional arrays of single-stranded tiles [30, 31], which
+have been shown to tolerate a wide variety of structural
+variations (including single-stranded regions as we would
+require for toeholds) and permitting strand displacement
+reactions that remove tiles from the array [32, 33]. How-
+ever, single-strand tile reactions analogous to the four-
+toehold reversible tile displacement reactions required
+here have not been experimentally demonstrated.
+Re-
+gardless of whether considering DNA origami tiles or
+single-stranded tiles – or something else – a major ob-
+stacle to any experimental implementation would be the
+creation of the initial array with a desired initial pattern.
+One possible avenue – still difficult – would be to initially
+assemble a uniquely-addressed DNA origami array [17]
+or single-stranded tile array [31], use that array to geo-
+metrically organize the desired pattern of non-uniquely-
+addressed tiles needed for tile displacement behaviors,
+and then via photocleavable bonds or other mechanisms,
+remove and dispose of the uniquely-addressed array. But
+for now, we will assume that arbitrary initial assemblies
+can be synthesized.
+3. WIRES, GATES, AND CIRCUITS
+To get a feel for how tile displacement systems can be
+programmed, we begin with the most basic task: signal
+transmission. As shown in figure 3A, this can be accom-
+plished using a single tile type (“wire”) that is used in
+the initial assembly to indicate where the wire is, plus a
+single tile type (“signal”) that carries the signal x. Two
+additional tile types (“top” and “bottom”) are used to
+provide neighboring tiles for the wire, as in general the
+4
+
+A
+B
+C
+signal
+wire
+bottom
+top
+reversible transmission of signal x ⇌ x:
+neutral
+irreversible transmission of signal x → x:
+downhill
+signal
+wire
+bottom
+top
+neutral
+reversible transmission of signals w + x ⇌ y + z:
+signal w
+wire (S-N)
+wire (W-E)
+signal x
+wire cross
+signal y
+signal z
+gate
+downhill
+neutral 
+but no south toehold
+warning
+irreversible transmission of signals w + x → y + z:
+D
+warning
+neutral 
+but no south toehold
+❷ position (3,3)
+❸ position (4,3)
+❶ position (4,3) 
+signal w
+signal x
+wire cross
+signal y
+signal z
+gate
+wire (S-N)
+wire (W-E)
+Figure 3:
+(A) Reversible wire. All toeholds are strength 1 except for toehold “−”, which is inert, i.e. strength 0. The
+top and bottom tiles have toeholds that are not shown, such that in the assembly the unlabeled sides are bound to each
+other via matching closed toeholds. In assemblies, closed toeholds are shown with light grey labels and a solid dark grey bar
+indicates their bond. For open toeholds, black or light grey is used to highlight relevant locations for the tile displacement
+reaction of interest, but have no formal meaning. In the monomer tile that is a reactant of the indicated reaction, solid dark
+grey bars indicate where new toehold bonds will be formed. Here and in later figures, only the forward reaction is shown
+for any reversible reactions (i.e. the assembly and monomer tile that are the products of the indicated reaction do not have
+their relevant toeholds highlighted for the backward reaction). (B) Irreversible wire. (C) Reversible wire cross. The initial
+assembly shown here illustrates the moment when both reversible signals arrive at the wire cross location. At this time, a
+reversible bond-energy neutral reaction can occur that inserts the gate tile in the central location, enabling reversible signals
+y and z to propagate on the output wires. (D) An unreliable irreversible wire cross that has two possible types of warnings.
+5
+
+wire will be embedded within a larger assembly. Each
+tile displacement reaction is neutral with respect to the
+bond energy, so when both the wire and signal tiles are
+at the same concentration, every tile displacement occurs
+at the same rate, and the signal transmission performs an
+unbiased random walk. Thus the expected time for signal
+transmission along a wire of length N is O(N 2).
+Faster signal transmission is possible if each tile dis-
+placement step is irreversible, which can be accomplished
+if new toehold bonds are formed such that the bond en-
+ergy change is downhill. Shown in figure 3B, the wire is
+as before, but now the signal tile has an additional toe-
+hold. Thus, tile displacement reactions are energetically
+downhill, forming one net additional bond with each re-
+action step, and the expected time for signal transmission
+is now O(N).
+When a horizontal and a vertical wire meet, we can
+perform a computational step. Figure 3C shows two re-
+versible wires, one carrying signal x and the other carry-
+ing signal w, meeting at a “wire cross” tile in the center.
+At this location, reversible tile displacement by a “gate
+tile” can effect the w + x ⇀
+↽ y + z reaction.
+Because
+the initial wire cross tile has four closed toeholds, tile dis-
+placement by the gate tile must form all four new toehold
+bonds, and thus tile displacement here prior to arrival of
+both the x and w signals would be energetically unfa-
+vorable and would not occur. This gate design is robust
+and flexible: it is straightforward to design more powerful
+variants. For example, the horizontal wire can carry one
+of two signals, 0 or 1, the vertical wire also can carry 0
+or 1, and there are now four gate tiles, one for each input
+combination, with output signals that effectively compute
+the logic function of interest.
+Specifically, to compute
+NAND and output using the same signal varieties, we
+would use four gate tiles that replace (w, x, y, z) respec-
+tively by (0, 0, 1, 1), (0, 1, 1, 1), (1, 0, 1, 1), and (1, 1, 0, 0).
+Can we similarly perform logic gate operations using
+irreversible wires, thus making computation faster? Un-
+fortunately, the above schemes no longer work in this case,
+as illustrated in figure 3D. The problem is that now, prior
+to arrival of the second signal, an energetically neutral tile
+displacement is possible at the gate position that simply
+ignores the missing input wire.
+Indeed, if the vertical
+wire is meant to be capable of carrying two signals (here
+w or y), then an energetically neutral tile displacement
+could analogously flip the signal content. Thus, this tile
+set and gate design is deemed unreliable and would issue
+warnings in our simulator. The lesson is that irreversible
+reactions will form an additional toehold when operating
+as intended, and this presents the possibility that a sim-
+ilar context that differs by just one open toehold, where
+the reaction is not intended to occur, will be energetically
+neutral and lead to an error.
+So does this mean that linear-time binary signal trans-
+mission and circuit computation is impossible with tile
+displacement systems? Thankfully, no. The trick is that
+while the leading wavefront of signal propagation and
+computation still must be reversible, in order to reliably
+discriminate single-toehold differences, it can be safe to
+irreversibly latch a decision in a context where all neigh-
+boring tile contain the same information, so differences
+between a 0 signal and a 1 signal by necessity involve
+two toeholds. Now, if the irreversible tile displacement
+involves the formation of one extra toehold in the correct
+context, in the incorrect context it would have to ignore
+two toeholds and thus would be uphill.
+This principle
+is illustrated in the design shown in figure 4A, where the
+latch tile can irreversibly insert itself into a three-tile-long
+segment of signal-carrying wire. Exactly where the latch
+tile inserts does not matter; the signal ratchets forward
+either way.
+We are now ready to take these designs for wires and
+gates, and combine them to construct feedforward logic
+circuits that compute in time linear with the depth of the
+circuit (as laid out in an array). There is, however, one
+more problem to solve if we want to build circuits that
+utilize multiple types of logic gates (e.g.
+XOR, AND,
+OR, NAND, NOR, WIRECROSS, and others). When an
+invading gate tile (e.g. “XOR 10” shown in figure 4B)
+displaces the initial gate tile (e.g.
+“XOR”), it makes
+bonds with toeholds on the neighboring four tiles but
+not with the displaced tile itself – therefore, information
+about which function should be computed must be con-
+tained in the neighboring tiles, and not just the gate tiles.
+We achieve this goal by using a gate-specific toehold in
+the initial gate tile, which directs the incorporation of a
+translator tile in the final position of each wire, as shown
+in figure 4B. Now the translator tile contains information
+about which logic function should be computed. Thus, an
+arbitrary number of gate types may coexist in the same
+system.
+Simulations of two feedforward circuit computations
+are shown in figure 5. Using just a systolic array of XOR
+gates, a collection of parity outputs (involving different
+subsets of the inputs) are produced, incidentally creating
+a Sierpinski triangle pattern within the completed wires
+and gates.
+The second circuit makes use of four gate
+types: a NOR gate that produces the same signal on both
+output wires, a NAND/XOR gate that produces NAND
+to the north and XOR to the east, a WIRECROSS that
+sends it south input to the north and its west input to the
+east, and a WIREPASS that sends its south input to the
+east and its west input to the north. The positions and
+identities of the gates are laid out in the initial tile array.
+Computation of an N ×N circuit will take expected time
+O(N).
+The tile system with eastward and northward latching
+binary wires and the five types of gate functions discussed
+above consists of 66 tile types altogether. For a circuit
+that can be laid out effectively in this format, an area of
+O(N 2) tiles can support N 2 gates. Arbitrary feedforward
+circuits with N gates can be implemented in O(N 2) area
+using a standard crossbar array architecture (for exam-
+ple see [34]) and a new FANOUT gate that copies one
+6
+
+A
+B
+wire
+bottom
+top
+signal 0
+latch 0
+signal 1
+latch 1
+irreversible transmission of signals 0 and 1:
+translator 
+0 ⇌ 0⊕ (W-E)
+translator
+1 ⇌ 1⊕ (W-E)
+translator
+0 ⇌ 0⊕ (S-N)
+translator
+1 ⇌ 1⊕ (S-N)
+XOR
+XOR 00
+XOR 01
+XOR 10
+XOR 11
+❶
+❷
+❸
+❹
+Figure 4:
+(A) Irreversible wire with no warnings. The next effect of two separate, independent tile displacement steps is
+shown for each wire. Importantly, no unreliable tile displacement reactions are possible. (B) An XOR gate. The gate itself
+is entirely reversible; latch steps in the input and output wires are sufficient for ensuring net progress within a circuit.
+input and ignores the other (using 75 tiles types if the
+new gate is just added, or 57 tile types if the redundant
+NAND/XOR and XOR gates are removed).
+Can we do better than just feedforward circuits? It is
+clear from inspection that our latching binary-signal wires
+can transmit information in either direction, depending
+on where the signal first arrives, and it is straightforward
+to implement gates that receive inputs from any two sides
+and produce outputs on the two other sides, so we can ar-
+range for signals to go around in cycles. Furthermore, the
+7
+
+A
+B
+T = 0
+T = 1000
+T = 2000
+T = 3000
+T = 0
+T = 1000
+T = 2000
+T = 3000
+Figure 5:
+(A) A 9 by 9 array of XOR gates. Black tiles are “caps” that terminate the output wires. (B) A 18-input 18-output
+logic circuit composed of XOR, NAND, and NOR gates. Two types of wire routing are implemented with a WIRECROSS
+tile that sends it south input to the north and its west input to the east, and a WIREPASS tile that sends its south input to
+the east and its west input to the north. The simulation time T measures the number of tile displacement reactions that have
+occurred, rather than the real time in the Gillespie simulation.
+tile displacement model in principle allows displacement
+to occur an arbitrary number of times in a given location.
+As a trivial example, the reversible wire of figure 3A can
+endlessly perform a random walk, back and forth forever.
+This raises the prospect of a tile displacement system of
+size N simulating a recurrent (iterated, feedback) circuit
+of size O(N), which can perform computations of length
+2O(N) – exponentially more than what a feedforward cir-
+cuit of the same size can do. This is to say, with respect
+to the size of the initial array and tile set, our existing
+construction can solve PTIME problems, while a reach
+goal would be to solve PSPACE problems like recur-
+rent circuits can. Unfortunately, this is not compatible
+with the use of latching wires to ensure linear-time signal
+propagation: an area-N tile array initially has at most
+O(N) open toeholds, and thus at most O(N) irreversible
+8
+
+tile displacement steps can take place before the system
+comes to a standstill – or more precisely, until it must
+henceforth rely exclusively on reversible steps.
+4. 1D CA SPACE-TIME HISTORIES
+Boiling down what we learned about circuits to its re-
+versible essence, we can re-implement the above compu-
+tations using fewer tile types, more compact layouts with
+just one tile per logic gate, and power for driving the com-
+putation forward coming from concentration differences
+rather than from irreversible toehold formation.
+We start by providing generalized construction for sim-
+ulating the space-time history of one-dimensional block
+cellular automata (1D BCA) that is very similar to
+their simulation by algorithmic self-assembly of DNA
+tiles [35, 2]. The instantaneous state of a 1D BCA is just
+a one-dimensional array of symbols from a given alphabet
+A, and in each time step the entire array is synchronously
+updated by applying a rule (x, y) → (f(x, y), g(x, y)) to
+a partitioning of the array into pairs, where f and g are
+functions that define the BCA and the parity of the par-
+tition alternates on each time step. The size of the ar-
+ray may be infinite, finite, or expanding, with given ini-
+tial state and boundary conditions (typically a finite core
+then periodic). Our tile displacement system construc-
+tion, shown in figure 6A, makes use of 2 + 2N + N 2 tile
+types for a 1D BCA with an alphabet of size N. The
+initial array uses 1 tile in the lower-left corner, N tiles to
+define input boundary conditions to be fed in at each time
+step from the left, N tiles to define the input boundary
+conditions to be fed in at each time step from the bottom,
+and 1 tile type filling in the remaining “blank” uncom-
+puted region of the array. The remaining N 2 tile types
+encode every input/output case for the update rule. For
+example, a binary alphabet (N = 2) will result in 10 tile
+types (figure 6B). The nth synchronous update of the 1D
+BCA will be encoded in the nth diagonal of the tile ar-
+ray. Similar to the gate tiles in the circuit construction,
+displacement must match all four open toehold positions,
+else it will be energetically unfavorable. This can only
+happen when both the tile to the left and the tile below
+have already updated, thus ensuring that the computed
+information is based on the correct information from the
+preceding diagonal.
+Because our model insists that any tile that can be
+displaced in a simulation must have a non-zero concen-
+tration as a monomer in solution, every reaction will
+be reversible. However, by chemostatting the blank tile
+at a lower concentration than the rule tiles, each dis-
+placement reaction can be biased forward by some factor
+r = crule/cblank. From detailed balance of the CRN and
+CTMC, this ensures that the equilibrium probability of
+the rule-tile containing assembly is r times higher than
+that of the blank-tile containing assembly. Although the
+system will never get irreversibly locked into a final out-
+put assembly state, the complete assembly with all rule
+tiles in place will be rm times more likely than an assem-
+bly with m blank tiles still present, which we consider
+“good enough”. Note that if a final irreversible step is
+desired to lock in place the completed computation, this
+is also possible by adapting the techniques used in the
+circuit construction, just at the upper right corner.
+Comparing the circuit construction of figure 5A to the
+cellular automaton space-time history construction in fig-
+ure 6BC, both of which compute parallel systolic arrays
+of XOR gates, we see that for the same size array, the
+cellular automaton approach computes roughly 9 times
+more gates. It also uses just 10 tile types, compared to
+30 for the circuit construction (if the tiles used for logic
+gates other than XOR are omitted).
+However, our cellular automaton construction, by its
+very nature as a cellular automaton, receives information
+only in the initial 1D boundary conditions, and thus an
+assembly cannot specify a two-dimensional layout for the
+circuit that will be computed by tile displacement. A sim-
+ple modification of the ideas resolves this apparent limita-
+tion: we generalize the construction to cellular automaton
+transformers whose cell update now depends both on the
+current state (x, y) ∈ A1 and a time-and-space-dependent
+input pattern (p, q) ∈ A0, as shown in figure 6D. Instead
+of an initial array containing uniform blank tiles, the ini-
+tial array will contain a layout of “pattern” tiles that each
+encode the information p that the gate below it will need
+to read, as well as the information q that the tile to its
+left will need to read.
+If A1 is size N and A0 is size
+M, then there are 2N input tiles, M 2 pattern tiles, and
+N 2M 2 rule tiles. Each reversible tile displacement reac-
+tion now must match four variable pieces of information,
+in two pattern toeholds and two state toeholds. As shown
+in figure 6EF, laying out exactly the same circuit as in
+figure 5B now requires 9 times less space, uses just 39 tile
+types (N = 2 state bit values plus a terminator, M = 5
+logic functions, but not all combinations are needed) in-
+stead of 57, and, with concentration bias again, computes
+significantly faster.
+Both these constructions exhibit strong similarities to
+computation via algorithmic growth during self-assembly
+of tiles – in the first case, 2D tiles growing a 2D structure
+from a 1D boundary [35, 2], and in the second case, 3D
+tiles growing an additional layer on top of a patterned
+2D initial assembly [36]. A significant difference is that
+rather than growing in size, the tile displacement system
+always remains the same size; rather than each tile attach-
+ment requiring new bond energy to counteract the lost en-
+tropy due to localization of the tile, the tile displacement
+system remains neutral with respect to bond energy be-
+cause each incoming tile is balanced by an outgoing tile.
+Thus, rather than finding suitable operating conditions
+by balancing temperature (controlling the bond energies)
+against tile concentrations (which simultaneously affect
+the kinetics), in tile displacement we balance concentra-
+tion against concentration (which permits similar bias
+at different speeds and temperatures).
+These benefits
+reflect similar observations about the increased robust-
+9
+
+B
+C
+input 0 (N)
+input 1 (N)
+input 0 (E)
+input 1 (E)
+blank
+XOR00
+XOR10
+XOR01
+XOR11
+E
+T = 0
+T = 1000
+T = 2000
+T = 3000
+NOR00
+NOR10
+NOR01
+NOR11
+XOR-NAND00
+XOR-NAND10
+XOR-NAND01
+XOR-NAND11
+XOR01
+XOR01
+XOR10
+diagonal-wire01
+XOR-NAND11
+XOR-NAND10
+A
+1D block cellular automaton update rule:
+(������������, ������������) → (������������ ������������, ������������ , ������������(������������, ������������))
+corner
+input (N)
+input (E)
+blank
+rule
+1D block cellular automaton transformer update rule:
+state (������������, ������������) pattern (������������, ������������) → state (������������, ������������)
+corner
+input (N)
+input (E)
+pattern
+rule
+D
+F
+T = 0
+T = 30
+T = 60
+T = 90
+corner
+Figure 6:
+(A) General case implementation of 1D block cellular automaton. Here a and b, written in roman font, denote
+specific toeholds. In contrast, x, y, f and g are variables and thus shown in italics. There will be a separate rule tile for each
+possible pair x, y ∈ A, with f and g being dependent on x and y, and similarly for the input tiles. (B) An example 1D block
+cellular automaton that computes the same function as the circuit shown in Fig. 5A. (C) Simulation snapshots. (D) General
+case implementation of 1D block cellular automaton transformer. (E) An example 1D block cellular automaton transformer
+that computes the same function as the circuit shown in Fig. 5B. (F) Simulation snapshots.
+10
+
+ness of strand displacement and toehold exchange com-
+pared to direct hybridization of complementary oligonu-
+cleotides [37, 38]. Seen more generally, tile displacement
+systems involve reconfiguration of a constant-sized assem-
+bly via local propagation of information, which is remi-
+niscent of the distinction between crystal growth from
+monomers in dilute solution (the case generally assumed
+in algorithmic self-assembly of DNA tiles) versus crys-
+tallization from the melt (wherein the initial state is a
+disorganized constant-density liquid of monomers, within
+which crystalline order locally propagates during crystal
+growth).
+Have we identified new concepts for tile displacement
+systems that allow us to perform more computation in a
+limited space? Powering computation forward via con-
+centration bias in reversible reactions has given rise to
+compact constructions that naturally avoid the unrelia-
+bility warnings that plagued our initial wire and circuit
+constructions, but the computational power still remains
+PTIME. One way of looking at this is that the free en-
+ergy of the assembly, G(A) decreases every time a higher-
+concentration tile replaces a lower-concentration tile, yet
+the minimum (most favorable) free energy occurs if all
+tiles in the array are highest-concentration tiles. That is
+to say, the free energy is bounded below, and if each for-
+ward computational step is biased by a minimum amount,
+there are a bounded number of such steps that can occur
+before the computation is done. The situation is not so
+different from the limitation we encountered when power-
+ing computation by new toehold formation in irreversible
+displacement steps. Is this limitation to PTIME a feature
+of tile displacement systems in general, or is it particular
+to the lack of imagination in the constructions we have
+presented so far?
+5. 2D CA IN-PLACE EXECUTION
+We can get some ideas from the notion of a cellular au-
+tomaton transformer, which reads a 2D pattern as a wave
+of activity passes over it, leaving a new pattern in its
+wake. Suppose that the new pattern can be read by a sec-
+ond wave, corresponding to a second cellular automaton
+transformer using a new set of rule tiles. For example,
+the initial pattern might use toehold alphabet A0, the
+first cellular automaton transformer uses states in alpha-
+bet A1 and writes a new pattern using alphabet A2 by
+utilizing the two locations that, in figure 6D, have useless
+inert “−” toeholds. Then, the second cellular automaton
+transformer can read A2, store its transient state in A3,
+and write a third pattern using A4. To drive the com-
+putation forward, the first transformer’s rule tiles should
+have a higher concentration than the pattern tiles, and
+the second transformer’s rule tiles should have a higher
+concentration than the first transformer’s rule tiles. This
+idea could be extended to K waves, each with its own
+set of rule tiles. This would improve upon the previous
+constructions, in which each location in the array expe-
+riences just net one forward tile displacement step – at
+that location, either one has the initial tile, or the final
+tile. Whereas, in an implementation of a multiple-wave
+cellular automaton transformer, each location would go
+through a sequence of changes, one for each wave.
+In
+a sense, we achieve K-fold more computation within the
+same assembly area. This is somewhat analogous to freez-
+ing cellular automata, which are restricted to change a
+cell’s state a limited number of times [39].
+There are two problems here, as you have probably
+already noticed.
+First, if the concentration ratio from
+wave to wave is r, then a K-wave computation requires
+a ratio of rK between the lowest-concentration tiles and
+the highest-concentration tiles. That quickly becomes im-
+practical, and theoretically unappealing.
+Second, each
+wave requires a new set of tiles – yet for PSPACE com-
+putations we would require an exponential number of tile
+updates and thus a comparable number of waves. So this
+idea doesn’t get us where we want to go.
+To keep a constant number of tile types while allowing
+an unbounded number of tile displacement steps per site,
+perhaps we could have a small number K of waves, but
+have wave K output its new pattern using alphabet A0 so
+that the tiles of wave 1 can read it – thus allowing iterated
+computation, such as binary counters and perhaps univer-
+sal space-bounded algorithms. This is indeed the essence
+of the construction we’ll arrive at, but it comes at a cost:
+for wave 1 tiles to displace wave K tiles, they cannot be
+at a lower concentration, which basically implies that all
+rules tiles must be at the same concentration, and we have
+no concentration bias pushing the computation forward.
+(This conclusion is not specific to periodic waves of cel-
+lular automata transformers; it follows in general that if
+we want to implement a computation that may update a
+given site an unknown and unbounded number of times,
+then every tile type may at some point be an incoming
+tile and at other times be the outgoing tile, so the con-
+centrations of all rule tiles must be equal.) If we have
+already accepted that our designs should exclusively use
+bond-energy neutral tile displacement, then in fact the
+bond energy and free energy of our assembly will remain
+constant over time – we are truly dealing with reversible
+computation.
+Thankfully, reversible computation is by
+no means impossible [14, 40].
+Our approach will be to exhibit a surprisingly natural
+correspondence between certain tile displacement systems
+and the well-studied class of two-dimensional block cel-
+lular automata (2D BCA) that arose in the study of re-
+versible computation by discrete models of ballistic phys-
+ical dynamics [20, 21, 22]. The 2D BCA model is a natu-
+ral generalization of the 1D BCA discussed above: rather
+than partitioning a 1D array into pairs of cells that get
+synchronously rewritten with alternating partition par-
+ity on alternate time steps, we now partition a 2D array
+into 2 × 2 blocks of cells that get synchronously rewrit-
+ten with alternating partition parity on alternating time
+steps (compare figure 7A with figure 7E). The formalism
+11
+
+A
+( ⃗������������, ⃖������������) → (������������ ⊕ ������������, ������������ ⊕ ������������)
+������������ = 4
+0
+0
+������������ = 3
+1
+1
+1
+1
+������������ = 2
+1
+1
+0
+0
+1
+1
+������������ = 1
+0
+0
+1
+1
+1
+1
+0
+0
+������������ = 0
+0
+0
+0
+1
+1
+0
+0
+0
+B
+(������������, ������������) → (������������ ⊕ ������������, ������������ ⊕ ������������)
+10 steps
+0
+1
+1
+0
+0
+1
+1
+0
+9 steps
+0
+1
+1
+1
+1
+1
+1
+0
+8 steps
+0
+1
+1
+0
+1
+1
+1
+0
+⋮
+2 steps
+0
+0
+1
+1
+1
+0
+0
+0
+1 steps
+0
+0
+0
+1
+1
+0
+0
+0
+0 steps
+0
+0
+0
+1
+1
+0
+0
+0
+time sheet 0
+time sheet 8
+C
+������������ = 25
+������������ = 0
+������������ = 500
+������������ = 0
+D
+E
+→
+F
+������������
+������������
+↘ ↙
+↗ ↖
+������������
+������������
+↖
+↗
+������������ ������������
+������������ ������������
+↙
+↘
+bottom 
+view
+top 
+view
+side 
+view
+=
+21 tile attachment steps
+time sheet 21
+������������ = 0
+������������ = 1
+������������ = 1
+������������ = 2
+Figure 7:
+(A) Execution of a synchronous 1D block cellular automaton. (B) Asynchronous 2D tile self-assembly that
+simulates the computation in (A). (C) and (D) Simulations of two example 2D block cellular automata: Billiard Ball Model (C)
+and Critters (D). (E) Execution of a synchronous 2D block cellular automaton. (F) Asynchronous 3D tile self-assembly that
+simulates the computation in (E).
+allows the rewrite rules to be arbitrary functions
+f
+��
+a
+b
+d
+c
+��
+=
+�
+w
+x
+z
+y
+�
+but if the rewrite function is a bijection, then the 2D BCA
+is logically reversible in the sense that iterating with f −1
+instead of with f will bring the simulation backwards in
+time. The most famous 2D BCA rule, the Billiard Ball
+Model (BBM), is logically reversible, rotationally and
+mirror symmetric, conserves the total number of 1s, can
+directly simulate reversible circuits, and with an infinite
+periodic initial state can simulate universal Turing ma-
+chines [20]. Example simulations of two binary-state 2D
+BCA, the BBM and “Critters”, are shown in figure 7CD.
+With larger alphabets, 2D BCA can simulate arbitrary
+classical cellular automata and Turing machines, either
+of the irreversible or reversible variety. (Generalizations
+to using blocks larger than 2 × 2 is also natural, but will
+not be considered here.)
+There are three obstacles to implementing arbitrary 2D
+BCA as tile displacement systems, and we will solve them
+all. The first is that tile displacement reactions are asyn-
+chronous (occurring at random locations and in random
+orders) while 2D BCA require synchronous updates of the
+entire array (and fail utterly if the same update function
+is applied asynchronously with no other modifications).
+The second is that the mechanics of tile displacement
+must be designed to avoid irreversible steps that close
+too many toeholds at once.
+And the third obstacle is
+that with exclusively reversible reactions and no concen-
+tration bias, there must be some other way to drive the
+system forward if we don’t want to wait forever.
+For the first challenge, we adapt prior methods for im-
+buing asynchronous cellular automata with locally syn-
+chronizing mechanisms [41, 42, 43]. The specific approach
+used here generalizes the approach used for simulation of
+1D cellular block automata space-time histories in the
+previous section.
+Figure 7A gives an example of a 1D
+BCA, with boxes highlighting the partitioning into pairs
+with alternate parity on each synchronous time step. Fig-
+ure 7B shows the same computation interpreted as 2D
+tile self-assembly where, starting from the 5 tiles at the
+bottom that encode the 8 input bits as well as their par-
+titioning, rule tiles attach whenever they can match two
+sides of existing tiles in the assembly, thus asynchronously
+growing the space-time history. We have augmented the
+tiles with arrows that point to where incoming tiles could
+attach; thus, in the initial assembly of 5 tiles, the sites
+12
+
+where tile can attach are exactly those locations where ar-
+rows are pointing inward toward the incoming tile. A cut
+through the assembly’s space-time diagram corresponds a
+particular moment during the asynchronous self-assembly
+process – we show a cut after 0 tile additions (yellow)
+and another after 8 tile additions (orange). We call these
+“time sheets” because at different horizontal (x) posi-
+tions, they are at different heights (t), and thus reading
+out the binary (black/white) states along a time sheet
+path correspond to states at different time steps of the
+underlying synchronous cellular automaton. Nonetheless,
+the time-sheet state information, augmented with the rel-
+evant arrows, is all that is needed to correctly complete
+the computation using an asynchronous update rule that
+executes only when arrows point toward each other, oth-
+erwise leaving the cells untouched. This process exactly
+mimics the self-assembly of the deterministic space-time
+history, despite its non-deterministic order of execution.
+There is an exactly analogous arrow-augmented asyn-
+chronous update rule for 2D BCA. Rather than square
+tiles, we now have truncated octahedra as “tiles”, but
+the self-assembling structure is again a space-time history
+of the correct synchronous cellular automaton computa-
+tion. Tiles may attach when they match four hexagonal
+faces of existing tiles in the assembly. (The small square
+faces are inert.) Again, if we imagine arrows orthogonal
+to the hexagonal faces of tiles, pointing out of the tile,
+then valid sites for attachment of a new tile correspond
+exactly to situations where all four arrows on the match-
+ing faces are pointing toward each other.
+The growth
+front for a give stage of assembly again corresponds to
+a (now two dimensional) time sheet, and we can write
+out the states of each exposed hexagonal face in a two
+dimensional array along with the orientation of its corre-
+sponding arrow. It is now a simple observation that the
+asynchronous addition of a tile corresponds exactly to an
+asynchronous update of a 2 × 2 block with four inward-
+pointing arrows, resulting in updates of the four cells and
+reversing all four arrows. Another way of thinking of it
+is that after a block asynchronously updates, it will not
+be able to update again until all four overlapping 2 × 2
+blocks have first updated and flipped the arrows back.
+Thus, the arrow-augmented asynchronous updating cor-
+responds exactly to synchronous parallel updating with
+alternating-parity partitioning into blocks.
+The second challenge is to implement this type of asyn-
+chronous block cellular automaton updating rule using
+tile displacement. Our construction, shown in figure 8,
+introduces additional complexities due to the fact that
+all tile displacement reactions are physically reversible,
+even if the 2D BCA logic update rules are irreversible,
+combined with the need to ensure that when one toe-
+hold is closed, the neighboring toehold must be opened –
+thus we must be able to guarantee a mismatch. Tripling
+the cell state alphabet by adding α, β, γ markers solves
+both problems. For each 2D BCA update case, we make
+three tiles, one inputting α-symbols and outputting β-
+symbols, another inputting β-symbols and outputting
+γ-symbols, and the third inputting γ-symbols and out-
+putting α-symbols. When an α → β tile inserts into the
+array, that simulates a forward-time asynchronous up-
+date. The swapping of which toehold is open and which
+is closed reflects the flipping orientation of arrows in the
+asynchronous cellular automaton; we can read the arrows
+from a tile array by looking at the open toeholds and
+drawing the arrow from α to β, from β to γ, or from γ
+to α. Boundary conditions for finite arrays must also be
+handled, using the same principles.
+Each side of a tile encodes the state of a specific cell
+in the 2D BCA (at a particular time mod 3, as per α-
+β-γ of the open toehold), and thus the grid of simulated
+BCA cells is oriented at a 45◦ angle relative to the array
+of tiles.
+State being encoded on the sides of tiles also
+facilitates that each tile displacement step corresponds to
+an update of a whole 2 × 2 block, and the fact that two
+tiles share the same side location reflects that each cell in
+a 2D BCA can be updated either by an odd-parity block
+or an even-parity block.
+The final challenge concerns how to drive the compu-
+tation forward. Let us first consider reversible 2D BCA
+rules. In this case, after any forward tile displacement
+step, there is exactly one monomer tile type that can re-
+verse the reaction: the tile that was just displaced. What
+this means is that the full state space of the tile displace-
+ment system’s CTMC is essentially linear; though fat and
+fuzzy, it has the same thickness both arbitrarily far into
+the future and arbitrarily far into the past. The thickness
+has to do with all the possible contours of the time sheet
+for a given average time. Thus we can say that the state
+space of the tile displacement system consists exclusively
+of correct reachable states of the computation; for a re-
+versible 2D BCA simulating a compact recurrent circuit
+for solving a PSPACE problem, the tile displacement sys-
+tems’s state space will also be exponentially long and will
+reach the same correct conclusion. Stochastic Gillespie
+simulation of the tile displacement CRN will result in an
+unbiased random walk back and forth along this fuzzy-
+linear state space. (Every assembly in this reachable state
+space has the same energy.) However, unlike a standard
+reversible Turing machine with Brownian dynamics [14],
+whose state spaces is strictly a linear graph so the ex-
+pected random walk hitting time for reaching the end of
+an T step computation is O(T 2), the time sheet diffuses
+much more slowly. As a rough estimate for an N × N
+tile array that requires N 2 forward updates to move the
+time sheet 1 net synchronous update step into the future
+under ideal circumstances, the same N 2 updates if half
+forward and half backward will be expected to net move
+the time sheet N steps either forward or backward, which
+corresponds to just 1/N equivalent synchronous update
+steps. This being just a polynomial inefficiency, perhaps
+we should not be too concerned.
+More interesting is what happens if the 2D BCA rules
+are irreversible. This means there are multiple cases for
+13
+
+A
+B
+❶
+❷
+❸
+❹
+2D block cellular automaton update rule:
+(������������, ������������, ������������, ������������) → (������������, ������������, ������������, ������������)
+rule (time sheet γ → α)
+rule (time sheet β → γ)
+rule (time sheet α → β)
+Figure 8:
+(A) General case implementation of 2D block cellular automaton. (B) Example updates in the Billiard Ball
+Model.
+the 2×2 block input that map to the same output. There-
+fore the state space for the tile displacement system will
+be exponentially branched in the backwards-in-time di-
+rection (as pictured by Bennett in figure 10 of his re-
+view paper [14], but thicker and fuzzier). Consequently
+Brownian dynamics will tend to be entropically biased
+toward where there are more states, and the system will
+run backwards. Can this entropic driving force be used to
+encourage a system to perform a desired computation by
+designing a system whose reverse dynamics are what we
+want? Attempting to do so would be risky, and probably
+futile, because the 2D block update rule being irreversible
+means that there are some states that have no local pre-
+decessor, and backward progress will get stuck as such
+14
+
+local configurations are encountered.
+A better way to exploit an entropic driving force is to
+have nondeterministic, stochastic forward update rules
+added to an otherwise-reversible system.
+For exam-
+ple, consider a tile displacement simulation of the BBM
+model, with boundary tiles designed to implement a re-
+flecting boundary (as they must for the system to remain
+reversible). If we design a special boundary tile that can
+either reflect a ball or (in the forward direction) produce
+a new ball out of nothing, then we obtain a new system
+that is still entirely reversible in the sense that there ex-
+ists (at least one) possible applicable block update in all
+circumstances, so the system cannot get stuck either in
+the forward direction or the reverse direction. With uni-
+form tile concentrations, all assembly states will still be
+isoenergetic. But started with an empty N ×N array, for-
+ward updates of the special tile will about half the time
+produce a new ball, which will entropically drive the sys-
+tem to a density such that forward production of balls is
+balanced by the reverse reaction, the absorption of balls
+into the special tile. At this point, which will be O(N 2)
+synchronous time steps in the future, the time sheet will
+stop advancing on average. Another way of looking at
+it is that with all reactions being neutral, equilibrium
+will reflect equipartition among all reachable states, and
+the combinatorially greatest number of states will have a
+number of ball near the optimal density – so, that’s what
+we are likely to observe. And the only way to get there is
+to run the time sheet forward enough to emit that number
+of balls.
+A gas-filled BBM simulation is not of great use by itself,
+but we can make use of it by also placing a circuit in the
+array, and drawing a 2-cell-thick wall around it. In the
+BBM model, balls bounce off walls, and walls are stable.
+Thus, despite random stochastic gas entering the areas of
+the array outside the box, the circuit will remain perfectly
+isolated from the gas. But due to the time-sheet coupling
+enforced by the asynchronous arrow rules, the time sheet
+that is being driven forward by the expansion of the gas
+will simultaneously drive the circuit forward.
+Unfortunately, for an array of area N 2, we will only
+drive the computation forward by O(N 2) steps – this is
+no better than the PTIME computational power of the
+original circuit construction. Essentially, in a small con-
+fined space, our circuit “heats up” and stops working.
+To run it for a long time, we need a larger space into
+which we can release the simulated heat. For example,
+if we are willing to entertain a half-infinite-plane array
+for tile displacement, we can draw a BBM wall down the
+middle, release gas on one side, and let the other side
+simulate an interesting recurrent circuit. Now, although
+the array is infinite (or very large) in direct proportion
+to how much computation we want to do, we can say
+that we have confined the interesting part of the compu-
+tation – the circuit itself – to a very small area relative
+to the potentially exponentially long computation. This
+isn’t PSPACE computation in terms of the size of the
+array, but rather in terms of the size of the part of the
+array that we care about. Similar constructions can be
+used to drive forward computation not just for other re-
+versible 2D BCA rules, but even for irreversible rules: the
+cellular automaton alphabet can be expanded to encode
+an inert “wall”, time sheets within the walled region and
+outside of it remain coupled, and sufficient entropy must
+be generated by stochastic rules outside the wall, to be
+dissipated into a sufficiently larger area.
+6. DISCUSSION
+Tile displacement within arrays of square DNA origami
+tiles was discovered accidentally [3]. While some aspects
+of the formal model, such as the four-sided generaliza-
+tion of toehold exchange, were invented for mathemat-
+ical elegance rather than detailed realism, they are not
+too far flung from what has been experimentally demon-
+strated and characterized. So it is quite delightful that
+within the design space for tile displacement systems, we
+find natural implementations for feedforward circuits and
+one-dimensional cellular automata that compute in lin-
+ear time, powered by irreversible toehold formation or
+by concentration gradients. Even more delightful is that
+attempts to squeeze out more computational power per
+area seemed almost inevitably to lead us consider physical
+constraints such as energy, reversibility, and asynchrony
+– which in turn lead to classical two-dimensional cellular
+automata models that arose in early studies of the physics
+of computation [14, 20].
+Our strongest result (despite
+weak time efficiency) is that a tile displacement array of
+size N can reversibly simulate a recurrent reversible cir-
+cuit (via the Billiard Ball Model cellular automaton) for
+an arbitrary number of steps. In other words, the reach-
+ability question for tile displacement is PSPACE com-
+plete – a result strongly reminiscent of Thachuk & Con-
+don’s beautiful PSPACE-hardness result for CRNs and
+DSDs [44].
+We believe that there remains a lot undiscovered within
+the tile displacement design space. For example, while
+our constructions showed that the asynchronous tile dis-
+placement model can simulate synchronous cellular au-
+tomata, the needed flipping-arrow mechanism for local
+synchronization seems almost built-in to the tile displace-
+ment model in the form of open and closed toeholds for
+toehold exchange, and it’s not obvious how to directly
+simulate asynchronous cellular automaton models such as
+reversible surface CRNs [45, 34, 43]. We might also ask
+whether using information within the branch migration
+domains rather than just in toeholds – or whether having
+even more toeholds and branch migration domains on a
+tile’s sides – could have advantages either theoretically
+or experimentally. Further, the most interesting systems
+demonstrated experimentally in the initial work on tile
+displacement [3] involved systems of interacting multi-tile
+arrays, rather than a single array and a monomer. How do
+our single-assembly results fit into that larger picture? Fi-
+nally, might tile displacement systems be combined with
+15
+
+other molecular mechanisms to solve our problems driving
+the computation forward – for example, an oscillator [46]
+that periodically activates and deactivates the α, β, and
+γ monomer tiles in sequence.
+When first discovered, the tile displacement mecha-
+nism seemed most closely related to strand displace-
+ment mechanisms, only two dimensional. However, as we
+investigated the capabilities of single-assembly tile dis-
+placement, many parallels to passive tile assembly [2]
+became prominent.
+Tile displacement systems appear
+to combine the principles of DNA strand displacement
+and self-assembly in different ways than hairpin-based
+programmable self-assembly [47], signal-passing tile self-
+assembly [48, 49], CRN-controlled tile assembly [50, 51],
+and other models we are aware of. Comparing the ben-
+efits, drawbacks, and relationships between these models
+may help uncover a more unified way of thinking about
+programmable molecular systems.
+And even if tile displacement systems, as explored the-
+oretically here, never become useful experimentally, we
+hope that it was interesting and perhaps inspiring to look
+long and deep at a simple mechanism until intricate pat-
+terns emerge.
+ACKNOWLEDGEMENTS
+The authors thank William Poole and Ho-Lin Chen for
+useful discussions. This work was partially supported by
+NSF awards 2008589 and 1813550.
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+

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+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+1 
+ 
+An efficient hybrid classification approach for 
+COVID-19 based on Harris Hawks Optimiza-
+tion and Salp Swarm Optimization 
+ 
+Abubakr Issa, University of Technology, Baghdad, iraq 
+Yossra Ali, University of Technology, Baghdad, Iraq 
+Tarik Rashid, University of Kurdistan Hewler, KRG, Iraq 
+ 
+Abstract— Feature selection can be defined as one of the pre-processing 
+steps that decrease the dimensionality of a dataset by identifying the most signif-
+icant attributes while also boosting the accuracy of classification. For solv-
+ing feature selection problems, this study presents a hybrid binary version of the 
+Harris Hawks Optimization algorithm (HHO) and Salp Swarm Optimization 
+(SSA) (HHOSSA) for Covid-19 classification. The proposed (HHOSSA) pre-
+sents a strategy for improving the basic HHO's performance using the Salp algo-
+rithm's power to select the best fitness values. The HHOSSA was tested against 
+two well-known optimization algorithms, the Whale Optimization Algorithm 
+(WOA) and the Grey wolf optimizer (GWO), utilizing a total of 800 chest X-ray 
+images. A total of four performance metrics (Accuracy, Recall, Precision, 
+F1) were employed in the studies using three classifiers (Support vector machines 
+(SVMs), k-Nearest Neighbor (KNN), and Extreme Gradient Boosting 
+(XGBoost)). The proposed algorithm (HHOSSA) achieved 96% accuracy with 
+the SVM classifier, and 98% accuracy with two classifiers, XGboost and KNN. 
+ 
+ 
+Keywords—— Feature selection, Hybrid Swarm intelligence, classification, 
+Covid-19, medical image 
+1 
+Introduction 
+ 
+Medical image processing can be defined as one of the most significant areas in 
+medical science, and it has a substantial effect on visualization applications. Also, med-
+ical image processing has a broad range of applications in medical diagnoses (treating 
+and investigating diseases) and medical sciences (such as physiological and anatomi-
+cal studies). Medical physics, medical engineering, biology, and optics are some of the 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+2 
+ 
+fields of science that make up this medical science. With the discovery of X-rays, Wil-
+liam Roentgen initiated the first efforts at contemporary medical imaging. Coronavirus 
+(COVID-19), also known as SARS-Corona Virus-2, is a virus that results in causing 
+severe acute respiratory syndrome (SARS-CoV2), is a viral infection that first occurred 
+in Wuhan at the end of 2019. Due to such an outbreak, COVID-19 became a pandemic, 
+threatening human lives and wreaking havoc on the economy. Therefore, many stud-
+ies have been launched in an attempt to identify a way to restrict mortality and spread. 
+Those researches include the suggested treatment strategy, the screening method for 
+early-stage patients, and the evaluation of different phases and recovery of treated pa-
+tients. In hospitals, imaging techniques like chest X-rays are commonly utilized for 
+detecting the severity and existence of COVID-19 pneumonia [1][2]. For improving 
+the suggested system's training, X-ray images are often maintained in a medical data-
+base for subsequent investigation by multiple research organizations. Low contrast, 
+noise, blurs, and faded colors are frequent problems, and images should be pre-pro-
+cessed to enhance quality by reducing noise.  
+The second stage is image segmentation, which depends on some attributes includ-
+ing color, texture, and depth measurements. The type of image and characteristics of 
+the problem (disease) are chosen to determine which segmentation technique is used. 
+The identification and extraction of features is the third stage. As the number of features 
+that have been extracted from the image grows, the accuracy of classification decreases. 
+In the classification vision, we can call it the curse of dimensionality. Feature optimi-
+zation is a viable option for dealing with this issue.[3] 
+The 4th stage is the feature selection that has been obtained from the known proper-
+ties using robust Optimization algorithms for better disease identifications from the 
+medical images. The image was classified using one of the classifiers. Feature selection 
+is a step in the preprocessing process that tries to increase the relevancy of obtained 
+data by deleting irrelevant characteristics and choosing just relevant or useful variables 
+[5]. Feature selection comprises reviewing feature subsets, employing certain search 
+approaches to locate the best feature subset, assessing the chosen features, stopping cri-
+teria, and subset validation in general.[6] 
+There are three types of feature selection classifiers: wrapper schemes, filer 
+schemes, and embedding schemes. The filter method, in contrast to the wrapper 
+scheme, which is characterized by good classification accuracy and low speed, is rapid 
+but inaccurate. The embedded system is preferred in the case when handling a certain 
+model [7]. Filter techniques use the qualities of training data to assess the quality of 
+features. Those approaches do not employ machine learning algorithms. Before choos-
+ing features with the highest score, filter methods usually take into account the score of 
+all features. At the same time, other filtering approaches favor features with the greatest 
+score per iteration [8]. Other well-known methods, like the correlation-based feature 
+selection approach in [9] as well as dimensionality reduction methods and NNs in [10], 
+can greatly decrease computational load and system complexity. Filter approaches 
+overlook the performance regarding the chosen characteristics despite their speed and 
+low computational cost [11]. 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+3 
+ 
+Wrapper approaches utilize an evaluation algorithm to assess the specified features’ 
+quality. SVMs, Decision trees (DTs), KNN, Naïve Bayesian (NB), linear discriminant 
+analysis (LDA), local neighborhood structure preserving embedding (LNSPE), artifi-
+cial neural networks (ANNs), and local geometrical structure Fisher analysis 
+(LGSFA) are some of the major wrapper’s methods utilized for feature selection. In 
+almost all cases, wrapper approaches outperform filter ones. Meta-heuristic algorithms 
+are more advanced search algorithms that result from the evolution and expansion of 
+feature selection. For instance, ongoing research to increase the performance regard-
+ing evolutionary algorithms (EA) like Genetic Algorithms (GAs) and Swarm Intelli-
+gence (SI) like Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), and 
+Ant Colony Optimization (ACO) are underway. Grasshopper Optimization Algorithm 
+(GOA), Grey Wolf Optimizer (GWO), Butterfly Optimization Algorithm (BOA), Har-
+ris Hawks Optimization (HHO) Whale Optimization Algorithm (WOA), and Ant Lion 
+Optimization (ALO) are examples of recent algorithms. Metaheuristic algorithms are 
+classified according to their exploration and exploitation phases into single solution 
+based (i.e., Tabu Search (TS) and Simulated annealing (SA)) or population size based 
+(in other words, GA, ACO, and PSO). The key contributions of this research are listed 
+below:  
+ 
+• Suggest an effective hybrid classification method for COVID-19 with the use of 
+the hybrid swarm algorithms (HHO, SSA). 
+This novel hybrid algorithm must improve resource consumption and performance, 
+as well as storage capacity, reducing processing time. 
+• With the use of multiple classifiers (KNN, SVM, XGboost), test the sug-
+gested (HHOSSA) algorithm on datasets containing some positive negative COVID-19 
+chest X-ray scan images. 
+• Individual, hybridized predictor models and state-of-the-art techniques (WOA, 
+GWO) are compared in terms of performance. 
+ 
+The sections of this paper are organized as follows: Section 2 provides a concise 
+summary of some of the most related works. Section 3 discusses methodology. In sec-
+tion 4 we described in detail our proposed approach. Tools are illustrated in section 5. 
+Performance evaluation is described in section 6. Results and discussion are included 
+in section 7. Finally, the conclusions and future works are stated in section 8. 
+2 
+Related Works 
+Many studies have employed hybrid algorithms to handle a range of challenges re-
+cently. Hybrid algorithms have received a lot of attention lately, notably in feature se-
+lection optimization. Low-level hybrid algorithms and high-level hybrid algorithms are 
+the 2 forms of hybrid algorithms. There are 2 types of hybridization schemes in high-
+level hybrid algorithms: high-level teamwork hybridization (HTH) and high-level rely-
+on hybridization (HRH). The self-contained meta-heuristics have been carried out in 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+4 
+ 
+order in HRH, whereas in the HTH, one algorithm assists the other by supplying infor-
+mation via cooperative search. Low-level hybridization has been separated into two 
+types: low-level teamwork hybrid (LTH) and low-level rely-on hybrid (LRH), both of 
+which contain one meta-heuristic algorithm [12]. In the feature selection field, it has 
+been observed that hybrid algorithms surpass native algorithms concerning perfor-
+mance. In the year 2004, the search process has been controlled by merging local search 
+approaches with a GA algorithm, which was the first time a hybrid metaheuristics ap-
+proach was utilized in feature selection. A combination with the EGA filter has 
+been provided in a wrapper technique for text categorization [13]. A hybrid ap-
+proach for feature selection has lately been created in various metaheuristic algorithms. 
+In [13], the Binary Grey Wolf algorithm was combined with the Harris Hawks algo-
+rithm to create an excellent balance between exploitation and exploration to prevent 
+local optimum solutions and increase solution precision. Harris Hawks was hybridized 
+in [14] using Bitwise operations and Simulated Annealing for supporting the HHO al-
+gorithm's exploitation capacity and getting out of local optima. In [15], the Salp swarm 
+algorithm was used to modify teaching–learning based optimization. This integration 
+gives TLBO more flexibility in the exploration of population and achieving variety 
+while also allowing it to swiftly attain the optimal value. They combined the Salp 
+swarm algorithm with the Particle swarm algorithm in [16], in which the SSA was uti-
+lized for updating the salps positions and the PSO was utilized otherwise. This hybrid-
+ization was utilized for the improvement of the exploration and exploitation of the Salp 
+swarm algorithm. 
+3 
+Methodology 
+3.1 Harris Hawks optimization algorithm 
+HHO can be defined as one of the swarm metaheuristic algorithms inspired by Har-
+ris Hawks' hunting behavior of "seven kills" or "surprise pounce." Based on the prey's 
+fleeing behavior nature, hunting duration can range from some seconds to many hours. 
+The modeling algorithm of HHO is split into 2 parts (exploitation and exploration). 
+Harris' hawks have been employed as candidate solutions in the HHO algorithm, with 
+the best candidate solution reflecting the desired or optimum prey in each stage [17]. 
+The first phase pertains to the process of perching and detection of the prey. The algo-
+rithm simulates Harris' hawks' perching methods in 2 separate scenarios. Harris' hawks 
+are assumed to perch in various locations during their group home range in the first 
+scenario. In Eq (1), q=0.50 models that condition. 
+ 
+X1
+⃗⃗⃗⃗  (t+1)= {
+𝑋𝑟𝑎𝑛𝑑(𝑡) − 𝑟1|𝑋𝑟𝑎𝑛𝑑(𝑡) − 2𝑟2𝑋(𝑡)|, 𝑞 ≥ 0.50
+(𝑋𝑟𝑎𝑏𝑏𝑖𝑡(𝑡) − 𝑋𝑚(𝑡)) − 𝑟3(𝐿𝐵 + 𝑟4(𝑈𝐵 − 𝐿𝐵)), 𝑞 < 0.50( 1 ) 
+ 
+While the other likelihood is that Harris' hawks would perch on positions near other 
+swarm members and prey. This condition has been introduced in Eq1 for q < 0.50: 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+5 
+ 
+where  X1
+⃗⃗⃗⃗  (t+1)  is Hawks' position vector, t represents the following iteration, 
+𝑋𝑟𝑎𝑛𝑑(𝑡) is a hawk that has been chosen at random from the current population, 
+𝑋(𝑡) represents the position vector of hawks, r1, r2, r3, r4, and q represent random 
+numbers in the range of (0,1), Xrabbit(t) represents rabbit position, Xm denotes the aver-
+age position of the current population of the hawks, lower and upper bounds for gener-
+ating random locations inside the Hawks' stadium are Lb and Ub, respectively. 
+ 
+While in the phase of exploitation, the Harris' hawks attack prey which has been 
+identified in the preceding step. The algorithm has 4 different possibilities for modeling 
+various attacking styles that have been utilized by Harris' hawks. 
+While r denotes the probability of prey escaping, successful escape has been donated 
+by r < 0.50, whereas r ≥ 0.50 denotes failure to escape. Depending upon the prey's 
+chances of escaping (r), hawks will use either soft or hard besieges to catch prey. The 
+algorithm's parameter E has been utilized for the determination of the type of attacking 
+besieges. If the prey is unable to escape when r ≥ 0.50 hard besiege happens when |E| 
+< 0.50 and soft besiege takes the place in the case where |E|≥ 0.50 The mathematical 
+Modelling of soft besiege has been represented by Eqs (2) through (3), and hard besiege 
+has been shown by Eq (4): 
+ 
+𝑿(𝒕 + 𝟏)=∆X(t) –E|JxXrabbit (t) –X(t)| ( 2 ) 
+∆(t) =Xrabbit (t) –X(t) ( 3 ) 
+X( t+1) =Xrabbit(t) –E|∆X(t)| ( 4 ) 
+ 
+In the case of successful escaping of the prey (r<0.50), soft besiege with a progres-
+sive rapid dive take is applied in the case where |E|≥ 0.50 as shown in Eq (5), Eq (7), 
+Eq(8) while Hard besiege with the progressive fast dive occurs in a case where |E|≥ 
+0.50 as shown in Eqs (6), (7), and (8):  
+ 
+𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿(𝒕)| ( 5 ) 
+𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿𝒎(𝒕)| , 𝑿𝒎(𝒕) =
+𝟏
+𝑵 ∗ ∑
+𝑿𝒊
+𝑵
+𝒊=𝟏
+(𝒕) ( 6 ) 
+𝒁 = 𝒀 + 𝑺 × 𝑳𝑭(𝑫) ( 7 )  
+ 
+𝑿(𝒕 + 𝟏) = {𝒀, 𝒊𝒇 𝒇(𝒀) < 𝑭(𝑿(𝒕))
+𝒁, 𝒊𝒇 𝒇(𝒁) < 𝑭(𝑿(𝒕)) ( 8 ) 
+ 
+D represents the problem dimension and S represents the random vector by 1xD size 
+and LF represents the function of levy flight, estimated with the use of Eq. (9): 
+ 
+𝑳𝑭(𝒙) = 𝟎. 𝟎𝟏 ×
+𝒖 ×𝛔
+|𝒗|
+𝟏
+𝜷
+ , 𝛔 = (
+(𝜞(𝟏+𝜷) ×𝒔𝒊𝒏 (𝝅𝜷/𝟐)
+𝜞(𝟏+𝜷
+𝟐 )×𝜷×𝟐(𝜷−𝟏
+𝟐 ) ) ( 9 ) 
+ 
+The energy of a rabbit is modeled as 𝑬 = 𝟐𝑬𝟎 (𝟏 −
+𝒕
+𝑻) ( 10 ) 
+Where E represents the prey’s escaping energy, T represents the maximal number of 
+iterations, and Eo represents its initial energy state. 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+6 
+ 
+ 
+3.2 Salp swarm optimization algorithm 
+ 
+SSA can be defined as a swarm metaheuristic algorithm [18] that was created for 
+solving various optimization problems. It was inspired by the activity of Salps in na-
+ture; salps are a type of jellyfish with tissues comparable to jellyfish and a high water 
+percentage in their moving behavior and weights [19]. They move by contracting their 
+bodies and shifting positions by pumping water through them. The salp chain describes 
+the swarming behavior of salps in the ocean. By allowing for faster and more harmonic 
+changes, this behavior could benefit salps in foraging and better movement. [18] Salp 
+chains were theoretically modeled and after that tested in optimization problems as a 
+result of this characteristic [16]. The algorithm starts its work  by dividing the generated 
+population into 2 parts (which are: leader and followers   ( where the leader leads the salp 
+chain and the remaining salps play the role of followers. A salp uses the food source as 
+a target in an n-dimensional search space. The following equation has been used to 
+update the leader's position: 
+ 
+𝑿𝒋
+𝟏 = {
+𝑭𝒋 + 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 ≥ 𝟎 
+𝑭𝒋 − 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 < 𝟎
+} ( 11 ) 
+ 
+Where 𝑋𝑗
+1 represent the position of leader in the jth dimension and Fj is food's loca-
+tion. The upper is represented by 𝑉𝑚𝑎𝑥𝑗 and the lower bounds that have been denoted 
+by 𝑉𝑚𝑖𝑛𝑗. The search space is maintained using the 2 random variables 𝑟2 & 𝑟3 in the 
+range [0, 1].  
+ 
+The parameter 𝑟1 is also an important control parameter in the process of exploration 
+and exploitation and it is calculated by using Eq (12). 
+ 
+𝒓𝟏 = 𝟐𝒆(−𝟒𝒕
+𝑵 )𝟐 ( 12 )  
+ 
+Where t represents the current iteration and N denotes the maximum amount of iter-
+ations. In a case where the position of the leader has been changed, Eq (13) is used to 
+change the followers' position: 
+ 
+𝑿𝒋
+𝒊 =
+𝟏
+𝟐 (𝑿𝒋
+𝟏 − 𝑿𝒋
+𝒊−𝟏) ( 13 ) 
+ 
+Where 𝐗𝐣
+𝐢 denotes the ith follower's position in the jth dimension, and the value 
+of I must be > 1. 
+ 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+7 
+ 
+4 
+The proposed approach 
+Despite its simple structure and fast convergence rate, the HHO algorithm is not 
+without flaws. However, in the domain of feature selection optimization, the algorithm 
+may encounter a balancing problem between the exploration and exploitation phases, 
+resulting in a local optimum. Problems can arise during the feature selection process 
+when dealing with the high-dimensional feature set. In general, the HHO algorithm 
+optimization power depends on the best optimal solution selected based on the best 
+fitness value. In this paper, we present a strategy for improving the basic HHO's per-
+formance using the Salp algorithm's power to select the best solution.  
+4.1 The structure of HHOSSA 
+The proposed hybrid algorithm HHOSSA contains many stages: Initialization and 
+binarization function, Best fitness selection, and Evaluation. 
+4.2 Initialization and binarization function 
+In this phase, the HHO algorithm generates a random initial population X that con-
+tains k Hawks which is every k represents a new solution this vector of d dimension of 
+features and using binary representations of (0 and 1) to represent the selected features 
+where every feature that selected will represent by 1 and every refused feature will 
+represent by 0 by using of the following binarization function: 
+ 
+𝑿
+𝒃𝒊𝒏𝒂𝒓𝒚={𝟏 𝒊𝒇 𝒙>𝒕𝒉𝒓𝒆_𝒗𝒂𝒍
+𝟎 𝒊𝒇 𝒙<𝒕𝒉𝒓𝒆_𝒗𝒂𝒍  𝒘𝒉𝒆𝒓𝒆 𝒕𝒉𝒓𝒆_𝒗𝒂𝒍=𝟎.𝟓   ( 14 ) 
+4.3 Best fitness selection 
+In basic HHO the position vectors Xrand and Xrabbit are responsible for the explo-
+ration step that has been characterized by Eq1, which is critical for balancing the ex-
+ploitation and exploration phases. Position vectors with higher significance speed up 
+global exploration, while those with lower significance speed up exploitation. As a re-
+sult, an appropriate Xrand and Xrabbit selection should be made to achieve a stable 
+balance between local exploitation and global exploration [20]. In this phase, the SSA 
+algorithm will be used to find a better solution where the SSA algorithm finds the new 
+fitness and if the new one is better than the one that has been found by the HHO algo-
+rithm so the new one will be replaced and the Xrabbit will be changed also otherwise, 
+the HHO solution remains unchanged. 
+The goal of feature selection is to reduce the number of features and classification 
+error rate, i.e., through the removal of the redundant and irrelevant features and keeping 
+the relevant ones only, classification accuracy is improved. The KNN classifier was 
+used in this study because it is simple to evaluate the fitness function Eq (15), which 
+was used, expresses the fitness function that was used. 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+8 
+ 
+𝑭𝒊𝒕𝒏𝒆𝒔𝒔 = 𝒂 ∗ 𝒄𝒍𝒂𝒔𝒔𝒆𝒓𝒓 + 𝒃 ∗ (
+𝒇𝒔𝒆𝒍
+𝒇𝒎𝒂𝒙) (15) 
+Where a =0.9 is constant for controlling the accuracy, b=[0.1, a] random number 
+enhances the accuracy,classerr is the rate of classification error and 𝒇𝒔𝒆𝒍 represents the 
+number of the selected feature and 𝒇𝒎𝒂𝒙 represents the total amount of features. 
+ 
+Algorithm1 Pseudo-Code of HHOSSA Algorithm 
+Input: H population size, T iteration number, ub=1, lb=0, thre_val=0.5, 
+levy_beta=1.5 
+Output: Best selected features vector 
+ 
+Randomly initialize of population H random hawks xi (i=1,2,3,….., H) 
+Compute the fitness value of every one of the hawks Fhho 
+Xrabbit = best solution found  
+ 
+While (the stop condition isn’t met) do 
+Compute the fitness values of the hawks 
+Set Xrabbit as rabbit location (i.e. optimal location) 
+For (each hawk (Xi)) do 
+ Update (Eo , J)  
+ if (|E| ≥ 1) then  
+ Update location vector according to Eq1 
+ if (|E| < 1) then  
+ if (r ≥0.50 & |E| ≥ 0.50 ) then  
+ Update location vector through utilizing Eq. (2) 
+ else if (r ≥0.50 & |E| < 0.50 ) then  
+ Update location vector through utilizing Eq. (4) 
+ else if (r <0.50 & |E| < 0.50 ) then  
+ Update location vector through utilizing Eq. (8) 
+ else if (r <0.50 & |E| < 0.50 ) then  
+ Update location vector through utilizing Eq. (8) 
+Apply the SSA algorithm to find the best fitness Fssa using Eq. (15) 
+If (Fssa < Fhho ) 
+Update (Xrabbit, Xrand )  
+End if  
+ 
+End While  
+ 
+ 
+ 
+ 
+ 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+9 
+ 
+ 
+ 
+ Figure 1: Structure of the proposed  HHOSSA algorithm  
+Start  
+Randomly initialize of 
+population H 
+Calculate Fhho and Xrabbit 
+Initialize of E0 and 
+update E1 
+Stopping_condition 
+met ? 
+Yes 
+No 
+Apply SSA algorithm 
+to found Fssa 
+Fssa< Fhho 
+Update Xrabbit and 
+Xrand 
+Keep Xrabbit and 
+Xrand 
+Evalute the selected feature with 
+FS wrapper method by using 
+KNN classifier  
+Yes 
+No 
+Features Extraction  
+Split the dataset into 
+training and testing   
+Preprocessing   
+Stop 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+10 
+ 
+5 
+Tools  
+5.1 Dataset 
+We are working with a dataset of 800 chest X-ray images obtained from [21-25]. 
+This dataset consists of 400 chest X-ray images with confirmed COVID-19 infection, 
+and 400 chest X-ray images of normal condition. This dataset images come with PNG 
+file format and grey level scale and all images are resized to 200 �� 200 pixels. 
+5.2 Classifiers 
+The main goal of classification is to categorize new samples that haven't been labeled 
+for a particular class. However, we must first train the classifier for it to recognize the 
+characteristics of the data, as well as the relationship between attribute values and the 
+class label. Three classifiers are used in the methodology presented in this paper. The 
+first one K‑nearest neighbor classifier and it’s used for the reasons of its straightforward 
+implementation, with only one parameter K denoting the number of neighbors, which 
+makes it more useful for identifying the best subset of attributes [26]. The second one 
+is the SVM classifier which is a well-known constructive learning technique that is 
+formalized by a separating hyperplane. Making a nonlinear transformation of the orig-
+inal input set to the high-dimensional set of features, where the optimum separating 
+hyperplane may be found, can lead to a solution [27]. The third classifier is Extreme 
+Gradient Boosting (XGBoost) which is a machine learning method that has been used 
+for solving supervised learning problems. It has excellent scalability and a fast running 
+speed, making it a popular machine-learning method [28]. 
+ 
+6 
+Performance evaluation 
+The metrics of evaluation  that are used to measure classification performance in this 
+study are accuracy, precision, recall, and F1 as defined below: 
+ 
+𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 =
+𝑻𝑷+𝑻𝑵
+𝑻𝑷+𝑻𝑵+𝑭𝑷+𝑭𝑵 ( 16 ) 
+ 
+𝒑𝒓𝒆𝒄𝒊𝒔𝒊𝒐𝒏 = 
+𝑻𝑷
+𝑻𝑷+𝑭𝑷 ( 17 ) 
+ 
+𝒓𝒆𝒄𝒂𝒍𝒍 =
+𝑻𝑷
+𝑻𝑷+𝑭𝑵 ( 18 ) 
+ 
+𝑭𝟏 = 𝟐 ×
+𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 × 𝑹𝒆𝒄𝒂𝒍𝒍
+𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚+ 𝑹𝒆𝒄𝒂𝒍𝒍 (19), 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 =
+𝑻𝑵
+𝑻𝑵+𝑭𝑷 (20) 
+ 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+11 
+ 
+In which "TP" (true positives) denotes positive COVID-19 images which the classi-
+fier accurately labeled, and "TN" (i.e. true negatives) corresponds to the nega-
+tives COVID-19 images that have been successfully labeled by the classifier. False 
+positives (FP) are positive COVID-19 images mislabeled as negative, whereas false 
+negatives (FN) are negative COVID-19 images that have been incorrectly identified as 
+positive COVID-19 images [29]. 
+7 
+Results and discussion 
+A total of 800 X-ray images (400 covid-19 and 400 normal) have been collected 
+from the digital database and utilized for testing the efficacy of the suggested hybrid 
+approach, which utilized two state-of-art algorithms (SSA, HHO) for feature selection 
+to improve the classification of the covid-19 infection with the use of automatic AI 
+techniques and showed a high level of classification accuracy following testing and 
+training. The dataset was divided into two sections: 20% for validation and testing and 
+80% for training. Table 2 demonstrates that the suggested hybrid method has a high 
+accuracy percentage based on the classifiers utilized. The parameter setting for the sug-
+gested methodology has been listed in Table 1. 
+ 
+Table1: Parameter values for used methods  
+Methods 
+Parameter values  
+ 
+HHOSSA algorithm 
+ 
+Feature size: 126 
+Population size: 30 
+Number of iterations for HHO:100 
+Number of iterations for SSA:20 
+Ub:1 
+Lb:0 
+Thre_val:0.5 
+Beta:1.5 
+Random variables a and b: 0.9, [0.1, a ] 
+ 
+KNN classifier  
+ 
+K=5 
+Classes count:2 
+No.of training set:224 
+ 
+SVM classifier  
+ 
+Classes count:2 
+No.of training set:224 
+ 
+XGboost classifier 
+ 
+Classes count:2 
+No.of training set:224 
+ 
+Table 2: Performance of HHOSSA over three classifiers KNN, SVM, and XGboost. 
+Classifier  
+Accuracy  
+Precision 
+Recall 
+F1 
+KNN 
+98.21428571428571 
+0.97 
+0.99 
+0.98 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+12 
+ 
+SVM 
+96.42857142857143 
+0.96 
+0.96 
+0.96 
+XGboost  
+98.21428571428571 
+0.99 
+0.96 
+0.98 
+ 
+7.1 Comparative study 
+The suggested system's performance was assessed utilizing a variety of modern op-
+timization methods (GWO, WOA). Table (3) shows the performance of the HHO algo-
+rithm used for feature selection and gets 94%,89%, and 94% over three classifiers 
+KNN, SVM, and XGboost, while Table(4) shows the performance of the SSA algo-
+rithm used for feature selection and gets 96%,80%,94% over three classifiers KNN, 
+SVM, XGboost, Table (5) shows the performance of GWO algorithm used for feature 
+selection and gets 96%,82%,92% over three classifiers KNN, SVM, XGboost, While 
+Table (6) shows the performance of WOA algorithm used for feature selection and gets 
+96%,86%,96% over three classifiers KNN, SVM, XGboost. 
+ 
+Table 3: Performance of HHO over three classifiers KNN, SVM, and XGboost. 
+Classifier  
+Accuracy  
+Precision 
+Recall 
+F1 
+KNN 
+94.64285714285714 
+0.90 
+0.99 
+0.95 
+SVM 
+89.28571428571429 
+0.87 
+0.93 
+0.90 
+XGboost  
+94.64285714285714 
+0.93 
+0.96 
+0.95 
+ 
+Table 4: Performance of SSA over three classifiers KNN, SVM, and 
+XGboost. 
+Classifier  
+Accuracy  
+Precision 
+Recall 
+F1 
+KNN 
+96.64285714285714 
+0.93 
+0.96 
+0.95 
+SVM 
+80.35714285714286 
+0.81 
+0.79 
+0.80 
+XGboost  
+94.64285714285714 
+0.96 
+0.93 
+0.95 
+ 
+Table 5: Performance of GWO over three classifiers KNN, SVM, and XGboost. 
+Classifier  
+Accuracy  
+Precision 
+Recall 
+F1 
+KNN 
+96.42857142857143 
+0.93 
+0.99 
+0.97 
+SVM 
+82.14285714285714 
+0.74 
+0.99 
+0.85 
+XGboost  
+92.85714285714286 
+0.90 
+0.96 
+0.93 
+ 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+13 
+ 
+Table 6: Performance of WOA over three classifier KNN, SVM, and XGboost. 
+Classifier  
+Accuracy  
+Precision 
+Recall 
+F1 
+KNN 
+94.64285714285714 
+0.90 
+0.99 
+0.95 
+SVM 
+89.28571428571429 
+0.87 
+0.93 
+0.90 
+XGboost  
+96.42857142857143 
+0.99 
+0.93 
+0.96 
+ 
+ 
+Figure 2: The accuracy, precision, recall, and the F1 values for all algorithms over the KNN 
+classifier 
+ 
+0.84
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+0.98
+1
+HHOSSA
+HHO
+SSA
+WOA
+GWO
+Accuracy
+Precision
+Recall
+F1
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+14 
+ 
+ 
+Figure 3: The accuracy, precision, recall, and the F1 values for all algorithms over the SVM 
+classifier 
+ 
+Figure 4: The accuracy, precision, recall, and the F1 values for all algorithms over the 
+XGboost classifier 
+ 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+HHOSSA
+HHO
+SSA
+WOA
+GWO
+Accuracy
+Precision
+Recall
+F1
+0.84
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+0.98
+1
+HHOSSA
+HHO
+SSA
+WOA
+GWO
+Accuracy
+Precision
+Recall
+F1
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+15 
+ 
+7.2 Software and Hardware Requirements 
+The proposed system operates by using a personal computer Lenovo that has speci-
+fications such as Intel(R) Intel(R) Core(TM) i7-6500U @ 2.59 GHz for CPU, 8 GB 
+windows10 of RAM, and 64-bit Operating System. The proposed system is operated 
+by using python 10 languages with (Pycharm) IDE. Table (7) shows the processing 
+time of the proposed algorithm and stand-alone algorithms depending on the classifi-
+cation processing time of the testing dataset. 
+Table 7: Processing time of proposed (HHOSSA), HHO, SSA. 
+Algorithm 
+Total processing time (seconds) 
+HHOSSA 
+1.0661 
+HHO 
+0.9906 
+SSA 
+1.1425 
+ 
+It should be noted that the hybrid algorithm's processing time for completing the clas-
+sification process is less than the sum of the processing times for the Harris hawk and 
+Salp algorithms because the Salp algorithm's iterations are fewer than those of the 
+Harris hawk algorithm within the hybrid algorithm. However, this improved the clas-
+sification process and accelerated performance without degrading the hybrid algo-
+rithm's quality. 
+ 
+8 
+Conclusion and future works 
+The presented work presents a new hybrid swarm algorithm (referred to 
+as HHOSSA) that combines the SSA and HHO for selecting the best features subset to 
+improve the detection and classification of the COVID-19 virus with the use of chest 
+X-ray images. The novel method provided to improve the process of the feature section 
+and also for achieving the balance between exploitation and exploration of the HHO 
+algorithm with the use of the capability of SSA for finding the best features subset It is 
+noted that the processing time required to complete the classification process using the 
+hybrid algorithm is less than the sum of the processing time of the Harris hawk and 
+Salp algorithms because the number of iterations of the Salp algorithm is less than the 
+iterations of Harris hawk algorithm inside hybrid algorithm, However, this did not af-
+fect the quality of the hybrid algorithm, but rather it increased the speed of performance 
+and improved the classification process. A total of 800 (400 covid-19 and 400 normal) 
+X-ray images are taken from the digital database to assess the HHOSSA's performance. 
+XGboost and KNN classifiers get 98% accuracy, whereas SVM classifiers score 96%. 
+We want to adapt the suggested technique to more applications in the future, including 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+16 
+ 
+signal processing and cloud computing task scheduling. Furthermore, the HHO algo-
+rithm's searching power was used to construct a novel suggested algorithm in several 
+aspects. 
+9          Acknowledgment 
+The authors would like to thank the University of Technology, Baghdad, Iraq for their 
+continuous support for this research work. 
+10 
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+[14]  Abdel-Basset, Mohamed, Weiping Ding, and Doaa El-Shahat. "A hybrid Harris Hawks 
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+Lu. "Improved salp swarm algorithm based on particle swarm optimization for feature selection." 
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+optimization: Algorithm and applications,''Future Gener. Comput. Syst., vol. 97, pp. 849872, 
+Aug. 2019 
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+thinking the role of salps in the ocean." Trends in Ecology & Evolution 31, no. 9 (2016): 720-
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+Wang. "Opposition-based learning Harris hawks optimization with advanced transition rules: 
+Principles and analysis." Expert Systems with Applications158 (2020): 113510. 
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+[21] Radiological Society of North America (RSNA), “Radiological Society of North America (RSNA),” 
+Radiological Society of North America (RSNA), 2020. https://www.rsna.org/ 
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+[22] R. Dataset, “Radiopaedia dataset,” 2020. https://radiopaedia.org/articles/imaging-data-sets-artificial-in-
+telligence 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+18 
+ 
+ 
+[23] C. 
+G. Repository, 
+“Cohen’s 
+GitHub 
+repository,” 
+Cohen’s 
+GitHub 
+repository, 
+2021. 
+https://github.com/ieee8023 
+ 
+[24] Italian Society of Medical and Interventional Radiology (SIRM), “Italian Society of Medical and Inter-
+ventional Radiology (SIRM),” Italian Society of Medical and Interventional Radiology (SIRM), 2020. 
+https://sirm.org/la-radiologia-medica/ 
+ 
+[25] Kaggle, “Kaggle’s chest X-ray images (Pneumonia) dataset,” Kaggle’s chest X-ray images (Pneumonia) 
+dataset, 2020. https://www.kaggle.com/datasets/paultimothymooney/chest-xray-pneumonia 
+ 
+[26] Abdel-Basset, Mohamed, Weiping Ding, and Doaa El-Shahat. "A hybrid Harris Hawks optimization 
+algorithm with simulated annealing for feature selection." Artificial Intelligence Review54, no. 1 (2021): 
+593-637. 
+ 
+    [27] Fusilier, Donato Hernández, Manuel Montes-y-Gómez, Paolo Rosso, and Rafael Guzmán Cabrera. 
+"Detecting positive and negative deceptive opinions using PU-learning." Information processing & manage-
+ment 51, no. 4 (2015): 433-443. 
+ 
+[28] Chen, Tianqi, and Carlos Guestrin. "Xgboost: A scalable tree boosting system." In Proceedings of the 
+22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785-794. 2016. 
+ 
+[29] Yasir, M. A., & Ali, Y. H. (2021). Review on Real Time Background Extraction: Models, Applications, 
+Environments, Challenges, and Evaluation Approaches. International Journal of Online and Biomedical En-
+gineering (iJOE), 17(02), pp. 37–68. https://doi.org/10.3991/ijoe.v17i02.18013 
+ 
+11            Authors 
+Abubakr S. Issa received his bachelor’s degree in computer science department – 
+Artificial intelligence branch from the University of Technology (UOT) – Iraq 2014. 
+Since 2014, he is working as a programmer at the Information Technology Center, at 
+the University of Technology up till now. Meanwhile, he is an M.Sc candidate at the 
+University of Technology (UOT) – Iraq. 
+Assistant Professor Dr. Yossra Hussain Ali. She received her B.Sc, M.Sc, and Ph.D. 
+degrees in 1996, 2002, and 2006 respectively from Iraq, the University of Technology, 
+Department of Computer Sciences. She joined the University of Technology, Iraq in 
+1997. During her postgraduate studies, she worked on Computer Networks, Infor-
+mation systems, Agent Programming and Image Processing as well as some experience 
+in Artificial Intelligent and Computer Data Security. She is a reviewer at many confer-
+ences and journals and she supervised several undergraduate and postgraduate (PhD. 
+and MSc.) dissertations in Computer sciences. Yossra has many professional certifi-
+cates and she has published in well-regarded journals (e-mail: yossra.h.ali@uotechnol-
+ogy.edu.iq). 
+Tarik A. Rashid received his Ph.D. in Computer Science and Informatics degree from 
+the College of Engineering, Mathematical and Physical Sciences, University College 
+Dublin (UCD) in 2001–2006. He pursued his Post-Doctoral Fellow at the Computer 
+Science and Informatics School, College of Engineering, Mathematical and Physical 
+Sciences, University College Dublin (UCD) from 2006–2007. He joined the University 
+
+Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on 
+Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi-
+cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 
+ 
+19 
+ 
+of Kurdistan Hewlêr (UKH) in 2017. He has also been included in the prestigious Stan-
+ford University list of 2.% of the best world researchers for the years 2020 and 2022. 
+

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+The detection of an extraordinarily-luminous
+high-redshift optical/ultraviolet flare by Swift/UVOT
+Zhi-Ping Jin1,2,3†, Hao Zhou1,2,3†, Yun Wang1,3, Jin-Jun Geng1,
+Stefano Covino4, Xue-Feng Wu1,3, Xiang Li1,
+Yi-Zhong Fan1,2,3∗, Da-Ming Wei1,2,3, and Jian-Yan Wei5
+1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China
+2Key Laboratory of Dark Matter and Space Astronomy of Chinese Academy of Sciences,
+Nanjing 210023, China
+3School of Astronomy and Space Science, University of Science and Technology of China,
+Hefei 230026, China
+4INAF/Brera Astronomical Observatory, via Bianchi 46, I-23807 Merate (LC), Italy
+5National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100049, China
+∗To whom correspondence should be addressed; E-mail: [email protected].
+†These authors contributed equally to this work.
+Hyper-luminous optical/ultraviolet flares have been detected in Gamma-ray
+Bursts and the record was held by naked eye event GRB 080319B. Such flares
+are widely attributed to internal shock or external reverse shock radiation.
+With a new dedicated method developed to derive reliable photometry from
+saturated images of Swift/UVOT, here we carry out time-resolved analysis of
+the initial White band 150 s exposure of GRB 220101A, a burst at the red-
+shift of 4.618, and report a rapidly-evolving optical/ultraviolet flare with an
+unprecedented-high absolute AB magnitude ∼ −39.4. At variance with GRB
+080319B, the temporal behavior of this new flare does not trace the gamma-
+ray activities. Rather than either internal shocks or reverse shock, this opti-
+1
+arXiv:2301.02407v1  [astro-ph.HE]  6 Jan 2023
+
+cal/ultraviolet monster is most likely from the refreshed shocks induced by the
+catching-up of the late-ejected extremely-energetic material with the earlier-
+launched decelerating outflow. We detect the first ultraviolet/optical flare with
+an absolute AB magnitude brighter than −39 and reveal the efficient process
+to power such transients.
+Gamma-ray bursts are widely believed to originate from the internal energy dissipation of
+a highly relativistic and narrowly collimated outflow that was launched by a nascent stellar
+mass black hole or magnetized neutron star. Shortly after the onset of prompt emission of
+GRBs, there could come very bright optical/ultraviolet flashes arising from either the internal
+shocks in specific conditions or the external reverse shock radiation (1). An apparent ∼ 9th
+mag optical radiation was detected in GRB 990123 at a redshift of z = 1.62 (2, 3). Its rapid
+rise and the quick decline are consistent with the reverse shock radiation model (4–6), and
+the late more-detailed afterglow modeling revealed that the reverse shock region should be
+significantly more magnetized than the forward shock region (7,8). A long-holding record was
+set by GRB 080319B, a burst at a redshift of z = 0.937. Its peak visual magnitude reaches 5.3
+(corresponding to an absolute AB magnitude of −38.7), which is so bright that an observer in
+a dark location could have seen it with the naked eyes (9)! The correlated temporal behaviors
+of the prompt gamma-ray emission and the optical radiation are in favor of the internal shock
+process (10,11). In the past decade, no similar or even comparable events have been reported.
+GRB 220101A was discovered simultaneously by Swift Burst Alert Telescope (BAT) (12),
+the Fermi satellite (13) and the AGILE satellite (14). Before the so-called finding chart exposure
+ranging from 90 to 240 seconds with the White filter (12), UVOT observed the target in V band
+for 9 seconds. The estimated average magnitude in the White band for an exposure of ∼ 150 s
+is ∼ 14.7th Vega mag (12,15). The redshift was measured to be z = 4.618 and in the spectrum
+a broad absorption feature, which results from the Lyman alpha absorption (16, 17), is evident
+2
+
+centered at ∼ 6820 ˚A. The corresponding isotropic equivalent gamma-ray energy is ∼ 4 × 1054
+erg and the peak luminosity is ∼ 9 × 1053 erg s−1, both are in the rank of the brightest ones
+among current GRBs (18,19). After the redshift correction, the observed optical photons were
+intrinsically in the ultraviolet bands. Therefore, all the emission detected by Swift suffered from
+serious absorption (in the observer’s frame, the V band absorption is about 2 mag stronger than
+that in the I-band (16)) and thus the intrinsic emission would be much brighter. This is in
+particular the case for the White filter because of its large effective area in the blue part (i.e.,
+U, UVW1, UVM2 and UVW2) and the Lyman alpha/intergalactic medium (IGM) absorption
+would be very strong. We concentrate on the first ∼ 150 s exposure with White filter in the
+event mode (i.e. photon counting mode) that can be efficiently divided into short bins according
+to the signal-to-noise ratio (SNR). Our time-resolved analysis reveals that the measurements
+in the time range of ∼ 106 − 150 s after the BAT trigger suffered from strong saturation, as
+shown in Fig. 1, Table S5 and Fig. S1. The absence of clear signal of the read-out streaks in the
+raw data, indicating a moderate saturation, however hampers a correction following procedures
+proposed in the literature (20,21). Therefore we propose a new method to correct the saturation
+effect. The basic idea is that though the pile up at the source site is so serious that can not be
+reliably corrected, the surrounding but relatively “separated” pixels are possibly unaffected by
+saturation and therefore the enhancement of the counts should be correlated with the intrinsic
+count rate of the source. To clarify whether it is the case, we need some data with known
+magnitudes as well as the count rates in external annuli. For the unsaturated data with relatively
+low ring count rate, we simply take UVOT/White measurements of GRB 220101A at 150−240
+seconds after the burst trigger. For the moderate saturation that is of our great interest, we
+take the UVOT measurements of GRB 130427A in the time interval of 500 − 2000 seconds.
+Though the moderately saturated White band emission of GRB 130427A can not be directly
+measured, we infer them with the UVOT emission in other bands since the spectrum can be
+3
+
+well fitted by a single power-law, see Fig. S2. With these two sets of data, we do find a tight
+correlation between photon count rate in 5′′ aperture ( ˙Naper, directly measured if unsaturated,
+or inferred from the “intrinsic” count rate ˙Nint measured in other ways) and in the 15′′ − 25′′
+ring ( ˙Nring, directly measured in UVOT images), which reads ˙Naper = (22.22 ± 0.84) ˙Nring
+for ˙Nring ≤ 80 s−1 (see Fig. S3). The correlation efficient for such an empirical relation is
+0.99 (22). The other essential correction is on the absorption of the ultraviolet photons at high
+redshift. In the analysis we correct such a factor, i.e., 4.78 ± 0.10 mag in the White band, with
+the wide band energy spectrum and further check it with the other two GRBs at rather similar
+redshifts (see Fig. S5).
+In Fig. 1 we show the lightcurves of the prompt gamma-ray emission and the very early
+optical emission. The first White exposure with a duration of 150 s was in the events mode.
+In our approach, a bin size of 4s is adopted. In principle, a narrower bin size is helpful in
+revealing the peak or structure of the flash, but a reasonably wide bin is necessary for a high
+SNR. The optical/ultraviolet flash lightcurve is relatively smooth and there is no evidence for
+tracing the temporal behavior of prompt gamma-rays. This is very different from the case of
+GRB 080319B, where the naked-eye optical flash shows strong variabilities and the optical
+lightcurve resembles that of the gamma-rays (see the insert of Fig. 1), indicating a new origin.
+We have also constructed the “prompt” optical, X-ray and gamma-ray SED. In Fig. 2 we show
+three representative time intervals of the first UVOT White band exposure, including the very
+beginning, the peak, and the final shallow decline phase. In the rise and the quick decline
+phases, the extrapolation of the high energy radiation spectrum into the optical is well below
+the White band measurements, which again points towards different physical origins of the
+optical and high energy radiation. While in the t−2.3±0.3 shallow decline phase, the optical
+to X-ray emission are consistent with being a single power-law, which may be dominated by
+the external reverse shock radiation. In Fig. 3 we present the absolute AB magnitudes of the
+4
+
+very early optical emission of GRB 220101A and the other three remarkable events, including
+GRB 990123 (3), GRB 050904 (23) and GRB 080319B (9), distinguished by the extremely
+bright optical emission. After the proper saturation, absorption and cosmological corrections, it
+turns out that GRB 220101A sets a new record. The prompt ultraviolet to X-ray spectrum at the
+optical/ultraviolet emission peak time is softer than ν−1.3 (see Fig.2). If this soft spectrum could
+extend to the optical band in the frame of the burst, GRB 220101A would be so far the unique
+source with an absolute AB magnitude brighter than −40 in the visible band (22). Note that the
+peak optical emission of GRB 220101A could be even stronger than presented here since our
+current fluxes are the average of the radiation in each 4s bin.
+As already mentioned before, for GRB 080319B, the internal shock model is favored by
+the similar temporal behaviors of the prompt gamma-ray and optical radiation. While for GRB
+990123, the external reverse shock model has been widely accepted to account for the optical
+flash. The optical/ultraviolet flare detected in GRB 220101A, however, should have a differ-
+ent physical origin for the following facts: (i) In contrast to GRB 080319B, the optical flash
+lightcurve of GRB 220101A does not trace the variability of the prompt gamma-rays (see Fig.
+1), requiring different radiation processes/sites of these two components; (ii) The t−2.3-like de-
+cline of the optical/ultraviolet flare of GRB 220101A may be due to the reverse shock emission,
+but the ∼ t20 increase is much quicker than that of GRB 990123 and hence strongly in tension
+with the standard external reverse/forward shock emission model (5, 6). Here we present a re-
+freshed shock model for the brightest optical/ultraviolet spike of GRB 220101A. Looking at the
+gamma-ray lightcurve, the main burst phase consisting of two giant gamma-ray spikes appears
+at ∼ 90 s after the BAT trigger, and the earlier emission was much weaker (i.e., the time-
+averaged luminosity is ∼ 1052 erg s−1). As indicated by the bulk Lorentz factor−luminosity
+correlation (24,25) of Γ ∝ L0.3
+γ , the weak/slow GRB outflow component launched at the early
+times is expected to have a Γ ∼ 102 and the surrounding interstellar medium further decelerates
+5
+
+the outflow to a Lorentz factor of ΓW, while the outflow component yielding the most luminous
+part of GRB 220101A likely has a Lorentz factor of ΓM ∼ 103. The first giant spike comes
+from the energy release of the main outflow, either through the internal shocks or the magnetic
+re-connection within it. Soon the main outflow would catch up with the decelerating weak part
+at a time of ∼ Γ2
+WδtWM/Γ2
+M ∼ O(10) s, which explains the second gamma-ray spike and the
+delayed onset of the optical/ultraviolet flare, where δtWM ∼ 100 s is the delay of the onset of the
+main part with respect to that of the weak part (started at ∼ 60 s before the trigger, see Fig. 1).
+The collision of the late/fast material shell(s) with the early/decelerating material will generate
+strong refreshed shocks and then produce energetic emissions. Following the treatments pre-
+sented in Sec. 2.1 of the Ref. (26), it is straightforward to show that for the internal shocks taking
+place at ∼ 2Γ2
+WcδtWM/(1 + z) ∼ 1016 cm (ΓW/102)2(δtWM/102 s), the typical synchrotron
+radiation frequency is indeed within the optical/ultraviolet bands. The bulk Lorentz factor of
+the merged shells can be approximated to be ¯Γ ≈
+�
+[MWΓW + MMΓM]/[MW/ΓW + MM/ΓM]
+and the Lorentz factor of the internal shocks can be estimated as Γsh ≈ ΓM/¯Γ + ¯Γ/ΓM, where
+MW and MM are the rest masses of the ejecta powering earlier weak gamma-ray emission and
+the main outburst, respectively (27). Indeed, for GRB 220101A-like burst, we have the outflow
+luminosity of Lm ∼ 1053 − 1054 erg s−1, with the fractions of the shock energy given to the
+magnetic fields (electrons) ϵB ∼ 0.1 (ϵe ∼ 0.3), ¯Γ ∼ several × 100 and Γsh ∼ a few, it is natural
+to have an optical/ultraviolet flux (26) of ∼ 1 Jy even for a redshift as high as ∼ 5 (a numerical
+example is presented in (22) and Fig.S6).
+Note that the very energetic prompt emission appearing at ∼ T0 +90 s, which partly overlap
+with the optical/ultraviolet flash, after the BAT trigger should also effectively cool the electrons
+accelerated in the collision discussed above. Such a process would produce GeV emission,
+which is expected to last longer than the overlapping phase of the prompt MeV emission and
+ultraviolet/optical flare. Indeed, at t ∼ 100 − 150 s after the BAT trigger, GeV emission was
+6
+
+detected from GRB 220101A (28).
+Though the hyper-luminous very early optical/ultraviolet emission are not common, we
+suggest that the bursts with prompt emission resembling GRB 220101A (i.e., the much more
+energetic outflow is well separated from the early ejecta) are good candidates for hosting the
+extraordinarily bright flares. The problem is how to catch such signals promptly. Small tele-
+scopes with a large field of view should be very helpful and the I/R-band observation of these
+telescopes can catch the monsters in a wide range of redshifts. Anyhow, such observations are
+limited by the weather, the time (day or night) and the burst site. The space telescopes like
+Swift/UVOT and SVOM/VT (29) may play an important role for the high redshift events. Since
+the optical/ultraviolet flash of GRB 220101A was observed by Swift/UVOT, below we focus on
+the upcoming 0.4m SVOM/VT with two channels, including the blue (400 − 650 nm) and the
+red (650 − 1000 nm) bands. For the shortest exposure time of 1s, the saturation limit is about
+9th magnitude. Given its higher sensitivity in comparison to Swift/UVOT V filter, the seriously
+absorbed “ultraviolet” emission of GRB 220101A/GRB 080319B-like extra-luminous events,
+even taking place at the even higher redshift (say, z ∼ 6), can still be caught by the blue channel
+of SVOM/VT though the red channel might be saturated (22). Dedicated observation strategies
+are needed to optimize the potential of the discoveries.
+7
+
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+-50
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+0
+200
+400
+600
+800
+1000
+1200
+BAT count rate (count/s/det)
+UVOT count rate (count/s)
+Time since trigger (s)
+BAT
+V
+White
+ 
+ 0
+ 1
+ 2
+ 3
+ 4
+ 0
+ 20
+ 40
+ 60
+ 800.0E0
+5.0E4
+1.0E5
+1.5E5
+GRB 080319B V
+Figure 1:
+Photon count rates of the prompt gamma-ray (Swift/BAT) and optical
+(Swift/UVOT V and White band) emission of GRB 220101A. The prompt gamma-ray
+lightcurve is highly variable, while the prompt optical emission lightcurve is relatively smooth
+and does not trace that of gamma-rays. The red filled circles are from the aperture measurement
+while the open circles are obtained with the new method developed in this work. The energetic
+optical/ultraviolet flash just overlaps with the late part of the giant outburst phase of the prompt
+gamma-rays. The prompt gamma-ray and optical lightcurves of GRB 080319 are inserted for
+comparison.
+8
+
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+Energy (keV)
+10
+2
+10
+1
+100
+101
+102
+103
+104
+Flux density (keV/cm2/s/keV)
+Unabsorbed CPL model (91.96 - 93.62 s)
+Unabsorbed CPL model (113.64 - 117.62 s)
+Unabsorbed PL model (173.63 - 239.56 s)
+Mape AB = 17.75 ± 0.24 (91.96 - 93.62 s)
+Mring AB = 13.56 ± 0.19 (113.64 - 117.62 s)
+Mape AB = 15.42 ± 0.04 (173.63 - 239.56 s)
+Figure 2: The “prompt” optical to γ-ray SEDs of GRB 220101A. The data in blue (red)
+are collected in the very beginning (peak) of the UVOT/White band emission. The optical
+emission in both cases are well above the extrapolation of the high energy spectrum, suggesting
+an origin different from the prompt X-rays and gamma-rays. While in the time interval of
+t ∼ 173.6 − 239.6 seconds, the extrapolation of the X-ray and gamma-ray spectrum into the
+optical is in agreement with the UVOT data.
+9
+
+-40
+-39
+-38
+-37
+-36
+-35
+-34
+-33
+-32
+-31
+-30
+-29
+-28
+101
+102
+103
+Absolute Magnitude (AB)
+t′ (second)
+GRB 220101A
+GRB 080319B
+GRB 050904
+GRB 990123
+-39
+-38
+-37
+-36
+1015 1016
+ 
+ν′ (Hz)
+Figure 3: The ultraviolet/optical flare of GRB 220101A (red) in comparison to that of
+GRB 990123 (green) (3), GRB 050904 (pink) (23) and GRB 080319B (blue) (9) in rest
+frame. The White band emission of GRB 220101A has been corrected for total extinction
+of Aλ = 4.78 ± 0.1 mag, including the tiny softening of E(B − V ) = 0.0483 mag in the
+Milky Way. The absolute AB magnitude of GRB 220101A exceeds that of GRB 080319B, the
+so-called naked burst, rendering it the most energetic optical/ultraviolet flare recorded so far.
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+28. M. Arimoto, L. Scotton, F. Longo, Fermi-LAT Collaboration, GRB Coordinates Network
+31350, 1 (2022).
+29. S.-J. Yu, F. Gonzalez, J.-Y. Wei, S.-N. Zhang, B. Cordier, Chin. Astron. Astrophys. 44, 269
+(2020).
+Acknowledgments
+Funding:
+This work was supported in part by NSFC under grants of No. 12225305, 11921003
+and 11933010, the China Manned Space Project (NO.CMS-CSST-2021-A13), Major Science
+and Technology Project of Qinghai Province (2019-ZJ-A10), Key Research Program of Frontier
+Sciences (No. QYZDJ-SSW-SYS024). SC has been supported by ASI grant I/004/11/0.
+Author Contributions:
+Y.Z.F and Z.P.J launched the project. Z.P.J, H.Z., Y.W, X.L, S.C and
+J.Y.W carried out the data analysis. Y.Z.F, J.J.G., X.F.W, D.M.W and Z.P.J interpreted the data.
+Z.P.J, H.Z. and Y.Z.F prepared the paper and all authors joined the discussion. Z.P.J and H.Z
+contributed equally.
+12
+
+Competing Interests:
+The authors declare that they have no competing financial interests.
+Author Information:
+Correspondence and requests for materials should be addressed to Y.Z.F
+([email protected]).
+Code availability:
+The codes used in this analysis are standard in the community, as intro-
+duced in the supplementary materials.
+Data availability:
+The Swift observation data analysed/used in this work are all publicly avail-
+able.
+13
+
+Supplementary materials
+Materials and Methods
+Tables S1 to S6
+Figs. S1 to S7
+References (30-52)
+Materials and Methods
+1
+A new method to measure the saturated sources in Swift
+UVOT images
+UVOT is a photon counting detector and typical read-out rate is once every ∼ 11 ms. If the
+source is bright enough (> 10 counts s−1), coincidence losses start to be significant and a
+correction is necessary. When the incident photon counts rate beyond the read-out rate ∼ 86 s−1,
+the source is fully saturated and proper coincidence loss correction is impractical (30). However
+for extremely saturated sources with read-out streaks, a calibration method has been developed
+based on the measurement of read-out streak line strength (20). Anyhow, the read-out streak
+lines are only present in the extremely saturated sources or those with very long time exposure.
+For the moderate saturation with relatively short exposure, it cannot be applied and our main
+goal is to provide a new way. Below we focus on the White band, but our method can be applied
+to other UVOT filters as well (indeed, as a validation, we also show in the end of this subsection
+that a rather similar empirical correction function holds for the V band).
+The saturated pattern of an UVOT image can be divided into three parts. The first is a
+point source like structure at the center of saturated pattern, which represents the location of the
+saturated source. The second part is a dark square structure caused by coincidence loss and the
+half length of its diagonal line is about 14 arcsec. A more detailed explanation is that UVOT has
+14
+
+actually a 256×256 CCD which records the flash pattern produced by the incident photon after
+several amplifiers and there is a centroid algorithm to calculate positions of incident photons
+whose accuracy could reach 0.125 pixel. As a result, each physical pixel could be subsampled
+to 8×8 virtual pixels with a resolution of 0.5 arcsec/pixel. The side length of the dark square is
+about 20 arcsec, that is 40 virtual pixels, corresponding to an area of 5×5 pixels region on real
+physical CCD which is the affected region of coincidence loss. The third part is the halo ring,
+which is distinct for saturated sources and some unsaturated sources but with low background.
+Fig. S1 shows such a saturated pattern. We attribute the halo rings to the wing of the Point
+Spread Function (PSF) of UVOT detector. To test this conjecture, we will examine whether
+the “intrinsic” photon counts rates of saturated sources is proportional to photon counts rates of
+halo rings.
+To avoid the influence of the coincidence loss, the best measurement region to get the highest
+S/N ratio is the area between a circle with a radius of 25 arcsec and a square, with the same
+center and with a side length of 20 arcsec, like a Chinese copper cash. However, if Swift rotated
+during observations, the dark region of final stacked science image are not necessarily a square
+due to that the coincidence loss square is aligned to the edge of CCD. Hence, we used an
+annulus of an inner radius of 15 arcsec and an outer radius of 25 arcsec (i.e., the outer edge
+of halo rings) to measure photon count rate in the ring ( ˙Nring), where the background should
+be removed and the coincidence loss has been corrected. The crucial step is to reliably derive
+the corresponding photon count rate of the saturated source within the standard aperture with
+a radius of 5 arcsec( ˙Naper). As mentioned above, if the incident photon counts rate is beyond
+the CCD readout rate, the source is saturated. Fortunately, the UVOT White band is much
+wider than other 6 bands (hence, we will call them the narrow bands), which means although
+a source is saturated in White band, it could be unsaturated in narrow bands. It is therefore
+plausible to measure the spectrum with other filters of UVOT and then convolve it with the
+15
+
+White filter to get the corresponding “intrinsic” emission. This can be done for the power-law
+like afterglow spectrum of GRBs and the very early time optical flash of GRB 130427A is a
+nice sample. The earliest UVOT measurements of this burst were highly saturated and some of
+them can be analyzed with the readout streak method (20). Moreover, as shown in Maselli et
+al. (31) and the left panel of Fig. S2, when the White filter was still saturated, there were usable
+measurements in other bands. In the right panel of Fig. S2, we show the ultraviolet/optical
+SED of GRB 130427A with the UVOT observations. Note that these data were re-measured
+in this work and they are consistent with that reported in the literature (31). We performed
+the early time photometry of GRB 130424A with HEASoft and the results are summarized in
+Table S1. The first exposure in B band and the first 2 exposures in U band were saturated,
+hence we took the values from Maselli et al (31). Light curves of 6 narrow bands were fitted
+to found their magnitudes simultaneous with White band exposures, the results are listed in
+Table S2. We then carry out the power-law spectral fit to the SED and estimate the White band
+magnitudes, as summarized in the last column of Table S2, which are further used in Table S3
+to yield the ˙Nint (in another word, the inferred ˙Naper). It is worth noting that in epoch 1 there
+was an optical/ultraviolet flare and hence it is not suitable to evaluate the White band emission
+with this method. Moreover, the White band measurement in the first, second and third epochs
+were significantly saturated with readout streaks, for which the fluxes were reported before.
+As show in Fig. S3, in epoch 2 our calculated flux is consistent with that reported in Maselli
+et al. (31), validating the method proposed in this work. Our downloaded image of the epoch
+3 mentioned in Maselli et al. (31) is distorted and we have hence focused on the subsequent
+observation data with an exposure of 20 s. Our estimated flux is still well consistent with that
+reported in Maselli et al. (31), which is expected because these two measurements were almost
+simultaneous. Anyhow, in the plot the data point reported in Maselli et al. (31) is not shown
+because we can not measure its ring count rate because of distortion. For epoch 4 to epoch
+16
+
+8, there were no readout streaks and the method developed by Page et al. (20) does not work
+any longer. Our method mentioned above applies to these data and yield reasonable results.
+As for GRB 220101A, shortly after its peak, the ultraviolet/optical flash is not saturated any
+more. For these observations we can reasonably measure its White band emission. HEASoft
+UVOT pipeline was used to make photometry of barely saturated images of GRB 220101A with
+a circle aperture with a radius of 5 arcsec. However, a reliable measurement of the ring count
+rate requires a somewhat long exposure. Therefore, we just divide the “tail” part of the flash
+into two time intervals. We also notice 3 bright stars in the field and then measure them for
+independent check. These five data points are summarized in Table S3. The White band fluxes
+measured (indirectly and directly, respectively) in the above events and field stars are used to
+clarify whether there is a tight correlation between the ring counts and the intrinsic source
+emission. For such a purpose, these three data sets have been fitted with a linear function of a
+model of y = ax and a least square cost function was applied, χ2 = �
+i
+(yi−axi)2
+y2
+err,i+(axerr,i)2, where
+yi and xi represent extracted White-band photon counts rates and halo ring photon counts rates,
+respectively, and yerr,i and xerr,i are the corresponding uncertainties. The Pearson correlation
+coefficient is 0.99, which reveals a very strong linear correlation, and the χ2/d.o.f value is
+∼ 0.90, which implies a reasonable fit, where d.o.f denotes the degree of the freedom. Hence,
+we conclude that ˙Naper = 22.22 ± 0.84 ˙Nring can yield a reasonable estimation of “true” photon
+counts rates of saturated sources in White band. Fig. S3 presents our best fitting result which
+confirms our early speculation and suggests that the outer part of the PSFs of such sources are
+nearly unmodified.
+The ground-based telescopes can well measure the V-band emission of the sources, which
+can thus provide an economical way to calibrate the saturated V-band observations of Swift/UVOT.
+Interestingly, GRB 080319B is a nice example. For the UVOT V-band observations, in total we
+have 22 sub event files, which were later converted to images with HEASoft for measurements.
+17
+
+The first 4 exposure duration are 30s, 40s, 50s and 55s, which are same as the time bins in Page
+et al. (20). These exposures display readout streaks and have been analyzed with the method
+of Page et al. (20), which are shown in the right panel of Fig. S3 (see the light green empty
+squares). We measured the counts rate in the halo rings, which is defined above, with HEASoft,
+but made coincidence loss correction manually. Another 18 images are unsaturated, the intrin-
+sic emission were directly measured, and they are marked with dark green empty triangles in
+the right panel of Fig. S3. These measurements are summarized in Table S4. In addition, the
+optical emission of GRB 080319B was well measured by the ground based telescopes (32), and
+the accurately measured V-band emission from RAPTOR-T can be taken as the intrinsic ones
+(i.e., we have the ˙Nint, in another word, ˙Naper defined in this paper). The difference between the
+V filter of UVOT and that of RAPTOR-T is small and the magnitude difference can be ignored,
+as demonstrated by the overlapping data points in the left lower corner of the right panel of Fig.
+S3. Since the very early UVOT/V band observations were in event mode, we can re-bin them
+into the time intervals the same as that of RAPTOR-T and then get the ˙Nring. Time bins of our
+measurements are listed in Table S4. Therefore, we apply the linear fit to the data sets and find
+an empirical function of ˙Naper = 20.6 ± 0.4 ˙Nring with a high correlation coefficient of 0.998.
+Such a correlation is nicely consistent with that for the UVOT/White band. It is worth noting
+that for GRB 220101A, the photons collected in the White band are dominated by those passing
+the V filter because of the serious absorption in the bluer region. Indeed we find rather sim-
+ilar count rates for the (almost) simultaneous White and V-band measurements (see Fig. S4).
+Therefore, the rather similar correction function for UVOT/V filter strongly suggests that our
+White band analysis of GRB 220101A is robust.
+18
+
+2
+Data analysis
+2.1
+Swift UVOT data analysis
+Swift/UVOT observed GRB 220101A in V, B, U, W1, M2, W2 and White bands for several
+epochs. For data in image mode, we started from the level 2 UVOT products and used standard
+aperture photometry, background was measured in a nearby region without sources in stacked
+images. Reliable detections were only obtained in V and White bands, and the photon count
+rates were measured in 3 or 5 arcsec apertures, depending on SNR. Coincidences loss correction
+and aperture correction were applied. For images without detection, upper limits were assuming
+count rates would have reached the SNR of S/N = 3. Finally zeropoints including long-term
+sensitivity correction were used for absolute calibrations. The results are shown in Table S5.
+The first white-band exposure under event mode (incident positions and time of every pho-
+ton are recorded) began at about 90 seconds after the trigger time, which lasted about 150
+seconds. Due to the fact that the luminosity of GRB 220101A changed rapidly at early epochs,
+although the transient seems to be unsaturated on the image for the total 150s exposure, it could
+be saturated in its peak phase. Hence, we screened the calibrated event data into slices whose
+exposure time is ∼4s to check whether the situation mentioned above had happened. Follow-
+ing the guidance of UVOT data process, event slices were transformed into images and image
+calibrations (flat field and mod 8 corrections) were applied. Since the transient is bright and iso-
+lated on reduced images, standard aperture photometry method was applied. From 90s to 100s,
+the transient was brightening rapidly and then became saturated for about 50 seconds. After ∼
+150s since the trigger time, it became unsaturated, again. We found that there are halo rings
+around the transient on barely saturated and saturated images, which we think are the ’wings’
+of point spread functions, hence, we analyzed the data with our calibration method described
+before. The results are summarized in Table S5.
+19
+
+2.2
+Swift-BAT/XRT and Fermi-GBM data analysis
+We processed Swift-BAT data according to standard procedures, using the software HEASoft
+(ver. 6.29) and calibration database (CALDB), which are available at
+https://www.swift.ac.uk/analysis/bat/setup.php. The mask weighting file used in extracting the
+light curve is generated by batgrbproduct (a complete GRB processing script in HEASoft). We
+extract event data at time intervals between -60 to 340 seconds related to the trigger time, the
+energy range is 15-350 keV, and the time bin size is 1 second. Our BAT analysis results are
+plotted with our Swift UVOT analysis results in Fig. 1.
+We also present a spectral analysis in a broad gamma-ray band (0.3 - 40000 keV) from Swift-
+BAT/XRT and Fermi-GBM data. The files used include the source and background spectrum
+files, as well as the corresponding response functions. For BAT file extraction and correction,
+we used standard procedures as in the manual (33). XRT files were created by online analysis
+tools provided by Swift official website (34, 35). The Fermi-GBM data have been processed
+with GBM Data Tools (36). There are different statistics used for each dataset (cstat for Swift-
+XRT, χ2 for Swift-BAT and pgstat for Fermi-GBM data). We use Bilby (37) in the framework
+of PyXspec for model parameter estimation. The results are shown in Fig. 2.
+3
+Intrinsic optical/ultraviolet emission of GRB 220101A
+To estimate a reliable un-absorbed optical/ultraviolet emission, we need an intrinsic spectrum
+to evaluate the absorption in different observation bands. For such a purpose, in addition to the
+UVOT V and White band observations, we adopt the g, r, i, z-band data from Liverpool tele-
+scope measured at t ∼ 0.625 day after the burst (38) and the simultaneous XRT data. Such
+a set of ground-based telescope observation data are chosen because they are almost simulta-
+neous with one UVOT White measurement and at late times the White band emission was not
+20
+
+detectable any longer (see Table S5 and Fig. S4). The SED from i to g declines very rapidly,
+requires a spectral index β ∼ 8 (see Fig. S5). Similar rapid declines, due to the serious Lyman
+forest absorption, have been observed in GRB 000131 (39) and 100219A (40) at redshifts of
+z = 4.500 and 4.667, respectively. Since the i and z observations do not suffer from strong
+absorption and there is no evidence for the presence of a flare at that time, we adopt them to
+construct the intrinsic optical (z band) to X-ray SED to be fν ∝ ν−0.70±0.05, with which we can
+obtain the absorption correction in r, g as well as UVOT White and V bands. In the direction
+of GRB 220101A, the Galactic extinction is E(B − V ) = 0.0483 (41). Basing on the intrinsic
+spectrum of and assuming no extinction from the GRB host galaxy, we find an absorption in
+White band as high as Aλ = 4.78 ± 0.10 mag, including Lyman absorption and the Galactic
+extinction, see the right panel of Fig. S5. Note that here the central frequency of the White band
+observation has been taken as the same as that of the V band because of the serious absorption
+of the bluer photons, as demonstrated in Fig. S4.
+In this work we adopt a cosmology with with H0 = 67.4 km s−1 Mpc −1, ΩM = 0.315 and
+ΩΛ = 0.685 (42), a redshift z =4.618 leads to a distance modules DM= 48.19. The absolute
+peak magnitude is calculated via Mpeak,abs = Mpeak −DM−Aλ +2.5(1−βi) log(1+z), where
+the last term is the k-correction and βi is the intrinsic spectral slope. The pity is that none of the
+extremely luminous flashes in GRB 990123, GRB 050904, GRB 080319B and GRB 220101A
+have a measured optical/ultraviolet spectrum. For GRB 220101A, the UVOT and XRT data
+suggest an “overall” optical to soft X-ray spectrum softer than ν−1.3. If this holds in the optical
+band (i.e., βi ≥ 1.3) in the rest frame, then we would have Mpeak,abs ≤ −40 mag in the visible
+band. It is so far the unique event to be brighter than the absolute AB magnitude of −39 mag,
+see Table S6 for a comparison of the brightest flare in history. If there are spectral information
+of optical flares in the future, these bursts would be able to directly compared in the same band.
+21
+
+4
+The numerical interpretation of the optical emission as well
+as the X-ray afterglow emission
+Here we call the X-ray emission after ∼ 170 s after the Swift trigger as the afterglow since the
+earlier emission are most likely the low energy part of the prompt radiation arising from the
+internal energy dissipation.
+4.1
+Refreshed shock emission for the peak of the optical/ultraviolet flare
+In the prompt γ−ray emission lightcurve, there are several weak gamma-ray spikes from earlier
+outflow before the main pulse starting at ∼ T0 +65 s. The front half part (between ∼ T0 +65 to
+102 s) of the giant gamma-ray pulse should come from the energy release of the main outflow,
+either dissipated through internal shocks or magnetic re-connections within it. For the later part
+(> T0 + 102 s) of the giant pulse, it overlaps with an energetic optical/ultraviolet flash, which
+indicates the rise of an additional dissipation process. As the preceding weak outflow gets
+decelerated to a bulk Lorentz factor of Γ1, a later launched but faster shell (with a bulk Lorentz
+factor of Γ4) will catch up with it at a radius of R0, so that a collision between two shells would
+occur. Note that Γ1 and Γ4 correspond to ΓW and ΓM mentioned in the main text, which is used
+here for the convenience of the discussion below. If the fast shell is not extremely magnetized,
+the collision would produce a refreshed forward shock (FS) propagating into materials of the
+preceding shell, and a refreshed reverse shock (RS) propagating into the fast shell. As a result,
+an optical/ultraviolet flash is expected from the radiation in the downstream of the refreshed
+RS, which has been initially proposed and works well for optical flash in the early afterglow
+stage (5). Here we show that this scenario could account for the prompt optical emission of
+GRB 220101A with a detailed numerical approach.
+Two refreshed shocks separate the system into four regions: (1) the unshocked slow shell,
+(2) the shocked slow shell, (3) the shocked fast shell, and (4) the unshocked fast shell. Here-
+22
+
+after, Xi denotes the value of the quantity X in Region “i” in its own rest frame. Unlike the
+preceding shell that exhausts the magnetic energy in the early stage (σ1 = 0), the later fast shell
+may keep the magnetic fields advected from the central engine, which could be parameterized
+by the magnetization of σ4 = B2
+4/4πn4mpc2, where n4 is the particle density in the comoving
+frame of Region 4 and mp is the proton mass. Let’s introduce an equivalent “luminosity” of
+the kinetic, internal and the magnetic energy for the two shells measured in the lab frame, Li,
+the corresponding particle density is then ni = Li/4πR2βiΓ2
+i mpc3(1 + σi), where i = 1, 4,
+βi = 1/
+�
+(1 − 1/Γ2
+i ) and R is the radius from the central engine. Due to the highly vari-
+able nature of the outflow from the central engine, the luminosity of the later fast ejecta could
+be further described by L4 = Lf(R/Rpeak)qr for R ≤ Rpeak and L4 = Lf(R/Rpeak)qd for
+R > Rpeak, where Rpeak is the radius that the RS reaches its peak luminosity, and qr (qd) is
+the rising (decaying) index of the luminosity before(after) Rpeak. We assume that Region 2 and
+Region 3 share a common bulk Lorentz factor (Γ2 = Γ3). After applying the hydrodynami-
+cal/magnetohydrodynamical jump conditions (43,44) to the FS/RS respectively and the energy
+conservation law to the FS-RS system (45), the evolution of Γ2 and relevant quantities within
+these regions could be solved numerically given the total isotropic energy of each shell (Ef and
+Es).
+The kinetic particle-in-cell simulations reveal that particle acceleration is less efficient in
+strongly magnetized shock than that of weakly magnetized shock (46). The shock is considered
+to be moderately magnetized, and it is reasonable to assume that the distribution of electrons in-
+jected downstream is Maxwellian both for the FS/RS (47), i.e., Qi(γe, t) = Ci (γe/γc,i)2 exp−γe/γc,i,
+where γc,i = 1
+3ϵe,i
+ei
+ρic2
+mp
+me is the typical Lorentz factor of the thermal distribution, ϵe,i is the frac-
+tion of post-shock energy that goes into electrons for each region, ei and ρi is the energy and den-
+sity of protons. The normalization constant Ci is obtained from the relevant mass conservation.
+The instantaneous electron spectrum can be obtained by solving the continuity equation of elec-
+23
+
+trons in energy space (48). Integrating the synchrotron radiation power from the electron spec-
+trum in Regions 2 and 3 and considering the effect of synchrotron self-absorption and the equal-
+arrival-time surface, the radiation spectra and the light curves are then derived. With a starting
+radius of R0 = 1015 cm for the collision and a set of parameters of L1 = 5.6 × 1052 erg s−1,
+Lf = 4.5 × 1053 erg s−1, Γ1 = 100, Γ4 = 1000, qr = 1.3, qd = −0.5, σ4 = 0.1, ϵB,3 = 0.08,
+ϵe,2 = 0.1, ϵe,3 = 0.07, Es = 5.8 × 1053 erg, Ef = 6.0 × 1054 erg. We get numerical optical
+lightcurves in good agreement with the observed data.
+4.2
+The external forward and reverse shock afterglow emission
+In our modeling, it turns out that the shallow-declining part of the optical flare is hard to be
+accounted for (see Fig. S6). A possibility is the emergence of the reverse shock, as observed in
+for instance GRB 990123 (3, 5). Indeed, a reverse and forward shock scenario can reasonably
+reproduce the optical and X-ray data. The magnetic field in the reverse shock region should
+be stronger than that in the forward shock region by a factor of quite a few ×10 otherwise the
+induced optical flash can not be brighter than the forward shock peak optical emission by a
+factor of ∼ 1000 (7, 8). The following physical parameters are found to be able to reasonably
+reproduce the afterglow data: the isotropic energy is Eiso = 1.0 × 1055 erg with a half open
+jet angle θj = 0.025, the initial Lorentz factor is Γ = 800, the fraction of forward and reverse
+shock energy given to the electrons is ϵe = 0.4, the fraction of the forward (reverse) shock
+energy given to the magnetic field is ϵb,fs = 2.5 × 10−5 (ϵb,rs = 0.3), the number density of the
+interstellar medium is n = 0.05 cm−3 and the power-law index for shock-accelerated electrons
+is p = 2.26. Such a p is well consistent with that needed in reproducing the optical to X-ray
+spectrum and lightcurves shown in Fig. S5 and Fig. S6, including Swift data analyzed in this
+work and Liverpool telescope data from GCN (38,49).
+24
+
+5
+The prospect of detecting ultra-luminous optical/ultraviolet
+flares at high redshifts with SVOM/VT
+Optical/ultraviolet flares at high redshift will surfer from serious absorption. Following Moller
+& Jakobsen (50), we estimate the absorption correction to be AB ∼ 5 mag (the received photons
+are mainly caused by red leak of blue filter) and AR ∼ 1 mag for the sources at z ∼ 6, based
+on the responses of SVOM/VT blue and red channels (i.e., B and R). For flares as luminous as
+GRB 080319B or GRB 220101A, if taken place at z ∼ 6, then we would have MR ∼ 10.5 mag
+and MB ∼ 15 mag. With the shortest exposure of 1s, SVOM/VT has a dynamic range of 9 − 18
+mag, which is sufficiently sensitive to catch the signals mentioned above. However, usually the
+exposure time of SVOM/VT should be 10-100 seconds, for which the R filter may get saturated
+but the B filter is not. We therefore conclude that SVOM/VT is a suitable equipment to detect
+the extremely bright optical flares of GRBs at z ∼ 6.
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+47. Giannios, D. & Spitkovsky, A. Signatures of a Maxwellian component in shock-accelerated
+electrons in GRBs. Mon. Not. Roy. Astron. Soc. 400, 330–336 (2009). 0905.1970.
+48. Geng, J.-J., Huang, Y.-F., Wu, X.-F., Zhang, B. & Zong, H.-S. Low-energy Spectra of
+Gamma-Ray Bursts from Cooling Electrons. Astrophys. J. Supp. 234, 3 (2018). 1709.
+05899.
+49. Perley, D. A. GRB 220101A: Additional Liverpool telescope photometry. GRB Coordi-
+nates Network 31425, 1 (2022).
+50. Møller, P. & Jakobsen, P. The Lyman continuum opacity at high redshifts - Through the
+Lyman forest and beyond the Lyman valley. Astron. Astrophys. 228, 299–309 (1990).
+51. Li, W. et al.
+The Calibration of the Swift UVOT Optical Observations: A Recipe for
+Photometry. Publ. Astron. Soc. Pac. 118, 37–61 (2006). astro-ph/0505504.
+52. Kuin, N. P. M. & Rosen, S. R. The measurement errors in the Swift-UVOT and XMM-OM.
+Mon. Not. Roy. Astron. Soc. 383, 383–386 (2008). 0709.1208.
+27
+
+Supplementary Tables
+T-T0
+Exp
+V
+B
+U
+W1
+M2
+W2
+(s)
+(s)
+(AB)
+(AB)
+(AB)
+(AB)
+(AB)
+(AB)
+367.38
+19.46
+...
+...
+...
+...
+...
+12.67±0.04
+391.76
+19.45
+12.01±0.04
+...
+...
+...
+...
+...
+416.18
+19.45
+...
+...
+...
+...
+12.65±0.04
+...
+440.84
+19.44
+...
+...
+...
+12.60±0.04
+...
+...
+465.10
+19.44
+...
+...
+12.09±0.38a
+...
+...
+...
+490.09
+19.45
+...
+11.28±0.40a
+...
+...
+...
+...
+540.86
+19.44
+...
+...
+...
+...
+...
+13.12±0.04
+565.28
+19.40
+12.41±0.04
+...
+...
+...
+...
+...
+589.61
+19.46
+...
+...
+...
+...
+13.04±0.04
+...
+614.82
+19.44
+...
+...
+...
+13.06±0.04
+...
+...
+639.13
+19.46
+...
+...
+12.90±0.07a
+...
+...
+...
+663.96
+19.46
+...
+12.69±0.04
+...
+...
+...
+...
+713.68
+19.45
+...
+...
+...
+...
+...
+13.50±0.04
+737.97
+19.44
+12.65±0.04
+...
+...
+...
+...
+...
+762.19
+19.44
+...
+...
+...
+...
+13.30±0.04
+...
+786.88
+19.44
+...
+...
+...
+13.31±0.04
+...
+...
+811.16
+19.44
+...
+...
+13.06±0.04
+...
+...
+...
+835.91
+19.45
+...
+12.98±0.04
+...
+...
+...
+...
+1136.89
+19.45
+...
+...
+13.50±0.04
+...
+...
+...
+1161.73
+19.46
+...
+13.39±0.04
+...
+...
+...
+...
+1213.17
+19.44
+...
+...
+...
+...
+...
+14.10±0.04
+1237.51
+19.44
+13.34±0.05
+...
+...
+...
+...
+...
+1261.91
+19.43
+...
+...
+...
+...
+13.92±0.05
+...
+1286.75
+19.44
+...
+...
+...
+13.90±0.04
+...
+...
+1311.01
+19.45
+...
+...
+13.73±0.04
+...
+...
+...
+1335.68
+19.44
+...
+13.60±0.04
+...
+...
+...
+...
+1385.28
+19.40
+...
+...
+...
+...
+...
+14.20±0.04
+1409.66
+19.43
+13.53±0.05
+...
+...
+...
+...
+...
+1433.98
+19.45
+...
+...
+...
+...
+14.09±0.05
+...
+1458.64
+19.44
+...
+...
+...
+14.08±0.04
+...
+...
+1482.87
+19.44
+...
+...
+13.82±0.04
+...
+...
+...
+1508.07
+19.45
+...
+13.77±0.04
+...
+...
+...
+...
+1557.68
+19.44
+...
+...
+...
+...
+...
+14.39±0.04
+1581.95
+19.45
+13.69±0.05
+...
+...
+...
+...
+...
+1606.20
+19.44
+...
+...
+...
+...
+14.25±0.05
+...
+1630.88
+19.45
+...
+...
+...
+14.24±0.04
+...
+...
+1655.08
+19.41
+...
+...
+13.98±0.04
+...
+...
+...
+1679.93
+19.44
+...
+13.87±0.04
+...
+...
+...
+...
+1729.85
+19.44
+...
+...
+...
+...
+...
+14.42±0.04
+1754.32
+19.46
+13.76±0.05
+...
+...
+...
+...
+...
+1779.91
+19.55
+...
+...
+...
+...
+14.28±0.05
+...
+1804.55
+19.45
+...
+...
+...
+14.30±0.04
+...
+...
+1828.75
+19.45
+...
+...
+14.08±0.04
+...
+...
+...
+1853.48
+19.45
+...
+13.90±0.04
+...
+...
+...
+...
+1903.00
+19.44
+...
+...
+...
+...
+...
+14.52±0.05
+1927.21
+19.45
+13.81±0.05
+...
+...
+...
+...
+...
+1951.62
+19.45
+...
+...
+...
+...
+14.38±0.05
+...
+a. Taken from Maselli et al. (31).
+Table S1: Early observations of GRB 130427A by Swift-UVOT. Galactic extinction AV =
+0.055, AB = 0.071, AU = 0.087, AW1 = 0.118, AM2 = 0.163 and AW2 = 0.156 have been
+applied. These data points have been plotted in the left panel of Fig. S2.
+28
+
+Epoch
+T-T0
+Exp
+V
+B
+U
+W1
+M2
+W2
+Whitea
+(s)
+(s)
+(AB)
+(AB)
+(AB)
+(AB)
+(AB)
+(AB)
+(AB)
+1
+515.57
+19.44
+12.32±0.04
+11.51±0.35b
+12.31±0.30b
+12.82±0.04
+12.89±0.04
+13.05±0.04
+12.62±0.44
+2
+688.49
+19.45
+12.59±0.04
+12.79±0.04
+12.93±0.06
+13.18±0.04
+13.20±0.04
+13.45±0.04
+12.95±0.06
+3
+860.19
+19.45
+12.82±0.04
+13.00±0.04
+13.11±0.04
+13.41±0.04
+13.44±0.04
+13.75±0.04
+13.19±0.09
+4
+1187.86
+19.44
+13.28±0.05
+13.42±0.04
+13.58±0.04
+13.80±0.04
+13.85±0.04
+14.08±0.04
+13.60±0.06
+5
+1359.98
+19.45
+13.48±0.05
+13.63±0.04
+13.77±0.04
+13.98±0.04
+14.02±0.05
+14.18±0.04
+13.77±0.05
+6
+1532.32
+19.44
+13.65±0.05
+13.78±0.04
+13.86±0.04
+14.15±0.04
+14.19±0.05
+14.37±0.04
+13.93±0.05
+7
+1704.19
+19.44
+13.75±0.05
+13.88±0.04
+14.02±0.04
+14.28±0.04
+14.27±0.05
+14.41±0.04
+14.03±0.05
+8
+1877.72
+19.44
+13.80±0.05
+13.90±0.04c
+14.08±0.04c
+14.29±0.04c
+14.32±0.05
+14.50±0.05
+14.07±0.05
+a. Interpolated White band AB magnitude of GRB 130427A. To derive the intrinsic count rate in a 5 arcsec aperture, galactic extinction
+AWH = 0.0875 have been accounted. Fitting uncertainties and standard deviation of fitting residuals contribute to uncertainties have been
+considered.
+b. At early phase, there is an additional radiation component, hence these 2 data points are excluded from SED fitting algorithm.
+c. These data points are results of extrapolation, hence they are excluded from SED fitting algorithm as well.
+Table S2: White band emission interpolated by Swift-UVOT narrow bands. These data
+(except for the last column) have been plotted in the right panel of Fig. S2.
+
+T-T0
+Exposure
+˙Ntot,raw
+ring
+˙Nbkg,raw
+ring
+COItot(bkg)
+LSS
+˙Nring
+˙Naper
+(s)
+(s)
+(count/s)
+(count/s)
+(count/s)
+(count/s)
+GRB 220101A
+165.95
+27.56
+83.08±1.86
+74.07±1.18
+1.033(1.029)
+0.998
+11.23±2.75
+257.17±22.69
+209.76
+58.67
+79.73±1.25
+73.84±0.81
+1.031(1.029)
+0.998
+7.28±1.85
+150.41±5.44
+GRB 130427Aa
+515.57
+19.44
+139.77±3.05
+72.21±1.21
+1.056(1.028)
+0.997
+80.66±4.00
+2030.98±831.79b
+688.49
+19.45
+133.49±2.98
+72.09±1.21
+1.054(1.028)
+0.996
+73.20±3.90
+1493.50±86.74
+860.19
+19.45
+120.21±2.83
+70.98±1.21
+1.048(1.028)
+0.996
+58.19±3.69
+1204.35±101.10
+1187.86
+19.44
+102.65±2.30
+71.17±1.06
+1.041(1.028)
+0.996
+36.99±3.00
+826.44±43.24
+1359.98
+19.45
+92.88±2.19
+71.45±1.06
+1.037(1.028)
+0.996
+25.09±2.86
+703.53±29.41
+1532.32
+19.44
+96.50±2.23
+71.19±1.04
+1.038(1.028)
+0.997
+29.68±2.90
+608.85±30.53
+1704.19
+19.44
+92.26±2.18
+71.03±1.05
+1.037(1.028)
+0.997
+24.86±2.84
+553.88±26.34
+1877.72
+19.44
+89.60±2.15
+71.26±1.05
+1.036(1.028)
+0.998
+21.47±2.81
+531.31±26.60
+RA
+DEC
+˙Ntot,raw
+ring
+˙Nbkg,raw
+ring
+COItot(bkg)
+LSS
+˙Nring
+˙Naper
+(J2000)
+(J2000)
+(count/s)
+(count/s)
+(count/s)
+(count/s)
+stars in GRB 220101A fieldc.
+00:05:43.983
++31:47:20.11
+80.83±0.76
+74.00±0.49
+1.032(1.029)
+1.006
+8.58±1.14
+168.01±4.10
+00:05:33.844
++31:42:10.45
+81.85±0.75
+74.00±0.48
+1.032(1.029)
+1.014
+9.94±1.12
+208.39±6.07
+00:05:26.211
++31:48:43.76
+83.42±1.06
+73.99±0.67
+1.033(1.029)
+0.996
+11.73±1.56
+230.06±7.61
+a. ˙Naper is derived from SED.
+b. This data is not fitted since U-band exposures were saturated around this exposure, hence it could be unreliable(see Fig. S2).
+c. These data are measured with the first 150 second White band exposure in window timing mode.
+Table S3: Photon count rates measured in aperture and halo ring methods in White band.
+Sensitivity correction factors are 1.175 and 1.102 for GRB 220101A field and GRB 130427A
+field, respectively. The factor from count rate to flux is 0.01327 mJy/(count/s) for white band.
+These data points have been plotted in the left panel of Fig. S3.
+
+T-T0
+Exp
+˙Ntot,raw
+ring
+˙Nbkg,raw
+ring
+COItot(bkg)
+˙Nring
+Magaper
+a
+˙Nb
+aper
+(s)
+(s)
+(count/s)
+(count/s)
+(count/s)
+(AB)
+(count/s)
+080319B V band measurements with Wo´znika et al. as reference
+180.60
+9.84
+74.86±1.19
+15.11±0.20
+1.030(1.006)
+65.43±1.35
+10.06±0.02
+1349.02±24.85
+193.52
+9.84
+65.04±1.39
+14.95±0.20
+1.026(1.006)
+54.63±1.56
+10.23±0.02
+1152.44±21.23
+206.25
+9.85
+54.36±1.50
+14.77±0.19
+1.021(1.006)
+43.01±1.67
+10.44±0.02
+952.40±17.54
+218.97
+9.85
+52.19±1.52
+14.94±0.20
+1.020(1.006)
+40.43±1.68
+10.55±0.02
+861.43±15.87
+231.70
+9.84
+47.80±1.53
+14.35±0.19
+1.019(1.006)
+36.23±1.69
+10.72±0.02
+731.85±13.48
+244.43
+9.84
+42.70±1.53
+14.32±0.19
+1.017(1.006)
+30.68±1.68
+10.88±0.02
+635.65±11.71
+257.45
+9.84
+39.72±1.52
+14.38±0.19
+1.015(1.006)
+27.36±1.67
+11.00±0.02
+567.05±10.45
+270.38
+9.84
+38.63±1.51
+14.77±0.19
+1.015(1.006)
+25.75±1.66
+11.12±0.02
+509.59±9.39
+283.30
+9.84
+36.73±1.50
+14.34±0.19
+1.014(1.006)
+24.15±1.64
+11.23±0.02
+457.95±8.44
+296.23
+9.84
+35.39±1.49
+14.10±0.19
+1.014(1.005)
+22.94±1.63
+11.36±0.02
+408.53±7.53
+309.15
+9.84
+31.18±1.45
+14.37±0.19
+1.012(1.006)
+18.09±1.58
+11.52±0.02
+351.58±6.48
+322.08
+9.84
+28.19±1.41
+14.05±0.19
+1.011(1.005)
+15.20±1.54
+11.61±0.02
+323.91±5.97
+334.80
+9.84
+28.53±1.41
+14.00±0.19
+1.011(1.005)
+15.62±1.54
+11.72±0.02
+291.08±5.36
+360.05
+29.52
+24.58±0.78
+14.14±0.11
+1.010(1.005)
+11.21±0.85
+11.91±0.02
+244.35±4.50
+395.50
+29.52
+23.49±0.77
+13.98±0.11
+1.009(1.005)
+10.21±0.84
+12.20±0.02
+187.25±3.45
+431.04
+29.53
+22.41±0.76
+14.19±0.11
+1.009(1.005)
+8.82±0.82
+12.40±0.02
+156.18±2.88
+466.29
+29.52
+21.29±0.74
+14.14±0.11
+1.008(1.005)
+7.67±0.81
+12.54±0.02
+137.03±2.52
+502.44
+29.52
+20.02±0.73
+13.90±0.11
+1.008(1.005)
+6.55±0.79
+12.74±0.02
+137.03±2.52
+537.68
+29.52
+19.13±0.72
+13.96±0.11
+1.007(1.005)
+5.53±0.78
+12.80±0.02
+114.29±2.11
+080319B V band measurements with Page et al. as reference
+189.92
+29.49
+67.15±0.79
+14.97±0.11
+1.026(1.006)
+56.97±0.88
+10.07±0.26
+1335.42±319.79
+224.89
+39.36
+49.06±0.76
+14.58±0.10
+1.019(1.006)
+37.38±0.84
+10.44±0.29
+949.77±253.68
+269.89
+49.21
+38.26±0.67
+14.48±0.09
+1.015(1.006)
+25.67±0.74
+10.89±0.38
+627.51±219.62
+322.39
+54.13
+29.98±0.61
+14.06±0.08
+1.012(1.005)
+17.13±0.67
+11.60±0.74
+326.30±222.40
+080319B V band measurements with HEASoft
+322.39
+54.13
+29.98±0.61
+14.06±0.08
+1.012(1.005)
+17.13±0.67
+11.75±0.02
+284.20±5.24c
+357.39
+14.77
+23.47±1.09
+14.15±0.16
+1.009(1.005)
+10.00±1.18
+11.91±0.04
+245.26±9.04
+372.39
+14.76
+24.23±1.10
+14.37±0.16
+1.009(1.006)
+10.58±1.20
+12.10±0.04
+205.88±7.58
+387.39
+14.77
+24.79±1.11
+14.16±0.16
+1.010(1.005)
+11.40±1.21
+12.14±0.04
+198.44±7.31
+402.40
+14.77
+22.70±1.08
+13.84±0.15
+1.009(1.005)
+9.51±1.17
+12.21±0.04
+186.05±6.85
+417.40
+14.76
+22.44±1.07
+14.46±0.16
+1.009(1.006)
+8.56±1.17
+12.35±0.04
+163.54±6.02
+432.40
+14.77
+22.51±1.07
+14.14±0.16
+1.009(1.005)
+8.97±1.17
+12.44±0.04
+150.53±5.55
+447.40
+14.76
+20.96±1.05
+13.89±0.15
+1.008(1.005)
+7.58±1.14
+12.49±0.04
+143.75±5.30
+462.40
+14.77
+21.64±1.06
+14.21±0.16
+1.008(1.005)
+7.96±1.15
+12.51±0.04
+141.13±5.20
+477.40
+14.76
+21.09±1.05
+14.22±0.16
+1.008(1.005)
+7.36±1.14
+12.60±0.04
+129.90±4.79
+492.39
+14.77
+21.44±1.06
+13.95±0.15
+1.008(1.005)
+8.03±1.15
+12.62±0.04
+127.53±4.70
+507.39
+14.76
+19.91±1.03
+13.98±0.15
+1.008(1.005)
+6.35±1.12
+12.77±0.04
+111.08±4.09
+522.39
+14.77
+19.06±1.01
+13.82±0.15
+1.007(1.005)
+5.60±1.10
+12.79±0.04
+109.05±4.02
+537.40
+14.77
+19.94±1.03
+13.76±0.15
+1.008(1.005)
+6.62±1.12
+12.88±0.04
+100.37±3.70
+552.40
+14.76
+19.08±1.01
+14.00±0.16
+1.007(1.005)
+5.43±1.10
+12.89±0.04
+99.45±3.66
+567.30
+14.57
+18.62±1.01
+14.19±0.16
+1.007(1.005)
+4.75±1.10
+12.94±0.04
+94.98±3.50d
+719.60
+19.47
+16.60±0.84
+13.88±0.13
+1.006(1.005)
+2.90±0.91
+13.35±0.04
+65.11±2.40d
+1073.88
+196.67
+15.55±0.26
+14.04±0.04
+1.006(1.005)
+1.61±0.28
+14.13±0.02
+31.74±0.58d
+1273.77
+196.77
+15.34±0.25
+14.16±0.04
+1.006(1.005)
+1.26±0.28
+14.52±0.02
+22.16±0.41d
+a. Magnitudes are taken from Wo´znika et al. (32) and Page et al. (20). It is not necessary to take account for the very small(∼ 0.01) difference
+between Vega magnitude and AB magnitude in V band.
+b. Only values in the last sub table 080319B V band measurements with HEASoft are directly measured, others are all inferred values(i.e.
+˙Nint).
+c. This exposure is close to saturation, and Page et al. (20) derived a photometry result with readout streak method, which is consistent with
+the aperture photometry result given by HEASoft.
+d. These points are not plotted in Fig 3 and not used in fitting algorithm as well.
+Table S4: Photon count rates measured in aperture and halo ring methods in v band. The
+large scale structure correction factor and the sensitivity correction factor are 1.001 and 1.056,
+respectively. The factor from count rate to flux is 0.25491 mJy/(count/s) for V band. These data
+points used have been plotted in the right panel of Fig. 3.
+
+Filter
+Tstart
+Tend
+Texp
+Signala
+Sky
+Magb
+second
+second
+second
+count/s
+count/s/pixel
+(AB)
+v
+70.94
+80.61
+9.52
+6.137 ± 1.074
+0.0313
+16.12 ± 0.19
+white
+91.96
+93.62
+1.64
+21.74 ± 4.74
+0.0145
+17.75 ± 0.24
+white
+93.64
+97.62
+3.93
+50.37 ± 4.79
+0.0148
+16.83 ± 0.10
+white
+97.63
+101.62
+3.94
+123.7 ± 8.9
+0.0150
+15.86 ± 0.08
+white
+101.63
+105.63
+3.94
+360.4 ± 30.3
+0.0147
+14.70 ± 0.09
+white
+105.64
+109.62
+3.93
+(634.4 ± 171.8)c
+0.0146
+(14.08 ± 0.29)
+white
+109.63
+113.63
+3.94
+(765.0 ± 175.6)c
+0.0147
+(13.88 ± 0.25)
+white
+113.64
+117.62
+3.93
+(1033 ± 184.4)c
+0.0150
+(13.56 ± 0.19)
+white
+117.63
+121.63
+3.94
+(544.8 ± 168.7)c
+0.0146
+(14.25 ± 0.34)
+white
+121.64
+125.62
+3.93
+(570.2 ± 170.7)c
+0.0147
+(14.20 ± 0.33)
+white
+125.63
+129.62
+3.94
+(374.0 ± 165.0)c
+0.0148
+(14.66 ± 0.48)
+white
+129.64
+133.62
+3.93
+(411.7 ± 167.3)c
+0.0149
+(14.55 ± 0.44)
+white
+133.63
+137.62
+3.94
+(383.1 ± 165.9)c
+0.0149
+(14.63 ± 0.47)
+white
+137.63
+141.62
+3.93
+(333.5 ± 163.1)c
+0.0146
+(14.78 ± 0.53)
+white
+141.63
+145.62
+3.94
+(381.1 ± 163.3)c
+0.0145
+(14.64 ± 0.47)
+white
+145.63
+149.63
+3.94
+(346.4 ± 163.6)c
+0.0147
+(14.74 ± 0.51)
+white
+149.64
+153.62
+3.93
+330.2 ± 26.4
+0.0148
+14.79 ± 0.09
+white
+153.63
+157.63
+3.94
+328.3 ± 26.1
+0.0147
+14.80 ± 0.09
+white
+157.64
+161.62
+3.93
+303.1 ± 23.3
+0.0149
+14.89 ± 0.08
+white
+161.63
+165.63
+3.94
+291.6 ± 22.0
+0.0145
+14.93 ± 0.08
+white
+165.64
+169.62
+3.93
+257.8 ± 18.7
+0.0147
+15.06 ± 0.08
+white
+169.63
+173.62
+3.94
+224.0 ± 15.8
+0.0149
+15.21 ± 0.08
+white
+173.63
+177.62
+3.93
+215.8 ± 15.2
+0.0146
+15.26 ± 0.08
+white
+177.63
+181.62
+3.94
+202.9 ± 14.2
+0.0148
+15.32 ± 0.08
+white
+181.63
+185.62
+3.93
+208.6 ± 14.6
+0.0149
+15.29 ± 0.08
+white
+185.63
+189.62
+3.94
+186.0 ± 12.9
+0.0147
+15.42 ± 0.08
+white
+189.63
+193.63
+3.94
+192.9 ± 13.4
+0.0146
+15.38 ± 0.08
+white
+193.64
+197.62
+3.93
+179.6 ± 12.5
+0.0147
+15.45 ± 0.08
+white
+197.63
+201.62
+3.94
+164.6 ± 11.5
+0.0147
+15.55 ± 0.08
+white
+201.64
+205.62
+3.93
+187.3 ± 13.1
+0.0149
+15.41 ± 0.08
+white
+205.63
+209.62
+3.94
+146.1 ± 10.3
+0.0145
+15.68 ± 0.08
+white
+209.64
+213.62
+3.93
+158.1 ± 11.1
+0.0148
+15.59 ± 0.08
+white
+213.63
+217.62
+3.94
+147.3 ± 10.4
+0.0149
+15.67 ± 0.08
+white
+217.63
+221.63
+3.94
+134.1 ± 9.6
+0.0149
+15.77 ± 0.08
+white
+221.64
+225.62
+3.93
+135.8 ± 9.7
+0.0147
+15.76 ± 0.08
+white
+225.63
+229.63
+3.94
+117.3 ± 8.5
+0.0145
+15.92 ± 0.08
+white
+229.64
+233.62
+3.93
+118.3 ± 8.6
+0.0146
+15.91 ± 0.08
+white
+233.63
+237.63
+3.94
+114.9 ± 8.4
+0.0143
+15.94 ± 0.08
+white
+237.64
+239.56
+1.90
+102.2 ± 10.9
+0.0145
+16.07 ± 0.12
+H/Ld
+u
+3627.5
+3827.3
+196.6
+0.0194 ± 0.1266
+0.0384
+> 20.74
+b
+3832.6
+4032.4
+196.6
+0.1181 ± 0.1624
+0.0635
+> 19.95
+white
+4037.3
+4237.1
+196.6
+0.4286 ± 0.1266
+0.1187
+21.90 ± 0.32
+w2
+4242.9
+4442.6
+196.6
+−0.3402 ± 0.0613
+0.0012
+> 21.54
+v
+4447.6
+4647.4
+196.6
+0.4469 ± 0.0617
+0.0151
+18.66 ± 0.15
+m2
+4652.4
+4852.1
+196.6
+0.0015 ± 0.0168
+0.0006
+> 21.25
+w1
+4857.5
+5057.3
+196.6
+0.0099 ± 0.0308
+0.0019
+> 21.33
+u
+5062.2
+5201.4
+137.0
+0.0659 ± 0.0730
+0.0080
+> 20.90
+white
+10266
+11051
+765
+0.3408 ± 0.0502
+0.0602
+22.15 ± 0.16
+v
+21543
+22361
+798
+0.1786 ± 0.0280
+0.0141
+19.66 ± 0.17
+white
+27727
+28549
+802
+0.1635 ± 0.0396
+0.0594
+22.95 ± 0.26
+v
+39724
+40112
+378
+0.1419 ± 0.0340
+0.0138
+19.91 ± 0.26
+white
+44845
+46039
+1137
+0.1662 ± 0.0492
+0.0601
+22.93 ± 0.36
+white
+50868
+61521
+1528
+0.1369 ± 0.0530
+0.1073
+23.13 ± 0.42
+white
+66833
+85013
+5485
+0.0202 ± 0.0378
+0.1000
+> 23.33
+a. Signal photon count rates have been corrected for coincidence losses (30,51) and long-term sensitivity correction.
+b. Magnitudes are based on Swift/UVOT zeropoints (30), errors are adjusted by a binomial distribution (52), limiting magnitudes are 3σ upper
+limits. These values have not been corrected for the Galactic extinctions of E(B − V ) = 0.0483 (41).
+c. These data have been analyzed in ring apertures as introduced in supplementary materials.
+d. Images taken before are in high resolution, our photometry is in 5′′ aperture, after are in low resolution, our photometry is in 3′′ aperture.
+Table S5: Photometry for Swift UVOT observations of GRB 220101A.
+
+GRB
+z
+Band
+Mpeak
+AV,MW
+β
+Aλa
+DMc
+Mabs
+Ref.
+AB
+fν ∝ ν−β
+AB
+990123
+1.600
+White to V
+MV = 8.86 ± 0.02
+0.04
+0.67
+0.04
+45.42
+-36.60
+(3)
+050904
+6.295
+White to 9500 ˚A
+M= 12.13 ± 0.24b
+0.16
+1.0
+1.25
+48.97
+-38.09
+(23)
+080319B
+0.937
+White to V
+MV = 5.34 ± 0.04
+0.03
+0.5
+0.03
+43.99
+-38.68
+(9)
+220101A
+4.618
+White to V
+Mwh = 13.56 ± 0.19
+0.15
+0.7
+4.78
+48.19
+-39.41
+This work
+a. Aλ= Aλ,MW+Aλ,host+Aλ,IGM is derived from photometric SED fit.
+b. Converted from fλ=9500 ˚A = 17 ± 4 × 10−15 erg cm−2 s−1 ˚A−1.
+c. Absolute Magnitude at the peak Mpeak,abs = Mpeak − DM − Aλ.
+Table S6: Properties of extremely bright GRB flares at the peak.
+
+Supplementary Figures
+(a) GRB 130427A WH
+(b) GRB 220101A WH
+deep exposure
+(c) GRB 220101A WH
+deep exposure mask map
+(d) GRB 220101A WH
+T-T0=103.63s, M~14.7
+(e) GRB 220101A WH
+T-T0=115.63s, M~13.56
+(f) GRB 220101A WH
+T-T0=167.63s, M~15.06
+Figure S1: Swift/UVOT white (WH) band images demonstrating the halo ring photometry
+method. Panel (a) is the White band image of GRB 130427A, where the solid circle represents
+the standard aperture of UVOT with a radius of 5 arcsec. The dotted square region strongly
+suffered from coincidence loss with a typical side length of ∼ 20 arcsec. Dashed annulus with
+an inner radius of 15 arcsec and an out radius of 25 arcsec is the halo ring region defined in
+this work, for which the ˙Nring is derived. Panel (b) shows the deep exposure of GRB 220101A
+field in White band, which reveals 2 faint sources in the halo ring region, hence we masked the
+annulus region from 95◦ to 150◦, as shown in panel (c). In addition, images of panel (b) and (c)
+have a pixel scale of 1.004 arcsec/pixel instead of 0.502 arcsec/pixel for other 4 images. Panel
+(d), (e) and (f) show some images around the peak time of GRB 220101A. We measured count
+rates in unmasked annulus region and corrected it to the whole annulus region.
+
+V-0.8
+M2+0.8
+B-0.4
+W2+1.2
+U
+B-0.4 Maselli
+W1+0.4
+U Maselli
+3×102
+5×102
+1×103
+2×103
+3×103
+10
+11
+12
+13
+14
+15
+16
+t-t0[s]
+AB Magnitude
+Epoch 1 -0.75
+Epoch 5 +0.25
+Epoch 2 -0.5
+Epoch 6 +0.5
+Epoch 3 -0.25
+Epoch 7 +0.75
+Epoch 4
+Epoch 8 +1.
+3×1014
+5×1014
+1×1015
+2×1015
+3×1015
+10
+11
+12
+13
+14
+15
+16
+Frequency[Hz]
+AB Magnitude
+a
+b
+Figure S2: The UVOT lightcurves (left) as well as the SEDs (right) of GRB 130427A. In
+the left panel, the vertical grey regions mark the observation periods of the White filter. Note
+that the second U-band data is saturated, which was however a detection point in Maselli et
+al. (31) if only event data in the last 6s was measured, hence the filled and the empty green
+triangles coincide. The shaded colorful regions across photometry points are our interpolation
+results of light curve. The right panel presents the optical to ultraviolet SEDs at the White
+band observation times constructed with the extrapolated UVOT narrow band data. A single
+power-law spectrum well reproduces the data, as anticipated in the fireball external forward
+shock afterglow model, with which a reliable evaluation of the White band emission is yielded,
+as reported in the last column of Table S2.
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Nring[count/s]
+0
+500
+1000
+1500
+2000
+2500
+3000
+Naper[count/s]
+a
+Fit Func: y=k*x
+k=22.22±0.84
+2/d.o.f=0.90
+GRB130427A extracted
+GRB220101A
+stars in GRB 220101A field
+GRB130427A Maselli
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Nring[count/s]
+0
+500
+1000
+1500
+2000
+2500
+3000
+Naper[count/s]
+b
+Fit Func: y=k*x
+k=20.62±0.43
+2/d.o.f=0.37
+GRB080319B Wozniak
+GRB080319B Page
+GRB080319B
+Figure S3: Photon count rates in 5′′ aperture ˙Naper (directly measured or inferred from
+the intrinsic value ˙Nint) and 15 − 25′′ ring ˙Nring (coincidence loss corrected) for some
+Swift/UVOT white and V band measurements. The left panel is for the White band. The dark
+green upward triangles represent the two unsaturated measurements of GRB 220101A in the
+tail phase of the flash. The filled squares are for three bright stars in the filed of GRB 220101A.
+The light green downward empty triangle represents inferred ˙Naper with the photometry result
+of GRB 130427A derived with readout streak method (31). As for orange points, the vertical
+coordinate represents the White-band emission of GRB 130427A inferred from measurements
+in other UVOT bands (see Fig. S2), while the horizontal coordinate is the ˙Nring (see Fig. S1 for
+definition). Black squares are 3 unsaturated stars in GRB 220101A field. The right panel is for
+the V band. Empty dark green triangles are unsaturated measurements of GRB 080319B with
+HEASoft and empty light green squares are photometry results of GRB 080319B derived with
+the readout streak method (20). As for orange points, the vertical coordinate represents the pho-
+tometry result of GRB 080319B observed with RAPTOR-T (32) when the UVOT observations
+were ongoing, while the horizontal axis represents ˙Nring in the corresponding UVOT V-band
+image. The linear fit is just for filled points in both panels, and the correlation coefficients of
+filled points are 0.990 and 0.998 for the left and right panel, respectively. Black dashed lines
+represent the saturation count rate (coincidence loss corrected, ∼ 372 count s−1) of UVOT.
+
+10-1
+100
+101
+102
+103
+102
+103
+104
+105
+UVOT count rate (count/s)
+Time since trigger (s)
+a
+white
+v
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+ 70
+ 80
+ 90
+ 2000
+ 3000
+ 4000
+ 5000
+ 6000
+ 7000
+ 8000
+Effective area (cm2)
+Wavelength (Å)
+b
+Lyα
+Lyman limit
+V
+B
+U
+W1
+M2
+W2
+White
+Figure S4: The similarity of Swift/UVOT White and V band observations of GRB 220101A.
+The left panel shows that the photon count rates in White band are almost the same as that in
+V band. This is because the photons with wavelengths below the Lyman limit (in the observer
+frame, it is 5124 ˚A; see the right panel) are almost fully absorbed, and the photons near the
+Lyman α may also suffer from strong absorption (see Fig. S5 for this effect). Therefore the
+collected photons are mainly within the V band.
+
+ 17
+ 18
+ 19
+ 20
+ 21
+ 22
+ 23
+ 24
+ 25
+1014
+1015
+1016
+1017
+1018
+10−6
+10−5
+10−4
+Magnitude (AB)
+Flux density (Jy)
+Frequency (Hz)
+GRB 000131 −4 Mag
+GRB 100219A −3 Mag
+GRB 220101A
+ 
+Figure S5: Optical to X-ray SED of GRB 220101A. Swift XRT, UVOT and g, r, i, z observa-
+tions of Liverpool telescope in the time interval of t ∼ 0.62−0.68 day after the burst (38). Such
+a set of ground-based telescope observation data are chosen because they are almost simulta-
+neous with one UVOT White exposure. Neither the X-ray nor the optical emission displays a
+flare. Therefore, we construct the optical SED with the data collected at t ∼ 0.625 day. We find
+that the absorption correction is AWh = 4.78 mag for intrinsic optical to X-ray spectrum with
+index βoX = 0.65, it is well consistent with X-ray spectrum βX = 0.63 ± 0.09. The central fre-
+quency of the White band observation has been taken as the same as that of the V band because
+of the serious absorption of the bluer photons, as demonstrated in Fig. S4. The optical SEDs
+of other two GRBs 000131 (39) and 100219A (40) at similar redshifts (z = 4.500 and 4.667,
+respectively) are also shown for comparison.
+
+ 10
+ 15
+ 20
+ 25
+ 30
+102
+103
+104
+105
+106
+10−9
+10−8
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Magnitude (AB)
+Flux density (Jy)
+Time since trigger (s)
+z−6mag
+i−4mag
+r−2mag
+Swift white
+Swift v
+g+2mag
+BAT 10 keV
+XRT 10 keV
+Figure S6: Fit to the multi-band afterglow lightcurves of GRB 220101A. The Swift XRT
+and UVOT data are analyzed in this work, and the other optical data are adopted from Liv-
+erpool telescope (38, 49). The total extinction corrections, including Galactic extinction and
+interstellar-medium extinction are AWh = 4.78, Av = 1.88, Ag = 3.51, Ar = 1.46, Ai = 0.24
+and Az = 0.10, respectively. The dashed and dash-dotted lines represents forward and reverse
+shock emission arising from the weak/slow and main/fast outflow collision. Solid and dotted
+lines are the regular external forward and reverse shock emission of the outflow. In our calcula-
+tion, the main/fast outflow was launched 92 seconds after the BAT trigger. Note that the X-ray
+emission at t ≤ 170 sec was attributed to the low energy part of the prompt emission and has
+not been addressed in our modeling.
+
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 3000
+ 4000
+ 5000
+ 6000
+ 7000
+ 8000
+ 9000
+ 10000  11000
+Response (%)
+Wavelength (Å)
+Lyα
+Lyβ
+Lyγ
+Lyman limit
+B
+B
+R
+R
+Figure S7: The response of the SVOM/VT and the Lyman absorption of the high redshift
+(∼ 6) event. The optical/ultraviolet flash will surfer from strong absorption by intergalactic
+medium. Following Moller & Jakobsen (50), we find that AB ∼ 5 mag (the received photons
+are mainly caused by red leak of blue filter) and AR ∼ 1 mag for a source at z = 6, based on
+the responses of SVOM/VT blue and red channels (i.e., B and R). If the initial flash is as bright
+as that detected in GRB 080319B and GRB 220101A, the absorbed one would still be caught
+by SVOM/VT with a dynamic range of 9 − 18 mag for the shortest exposure of 1s. Therefore
+SVOM/VT is a suitable equipment to detected extremely bright optical flares of GRBs at z ∼ 6.
+

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+Selection of Centrality Measures Using Self-Consistency
+and Bridge Axioms
+Pavel Chebotarev∗
+Moscow Institute of Physics and Technology
+9 Inststitutskii per., Dolgoprudny, Moscow Region, 141700 Russia
+January 3, 2023
+Abstract
+We consider several families of network centrality measures induced by graph ker-
+nels. The Self-consistency and Bridge axioms that appeared earlier in the literature
+turn out to be closely related to two of these families. We obtain a necessary and suffi-
+cient condition of Self-consistency, a sufficient condition of the Bridge axiom, indicate
+specific measures that satisfy these axioms and show that under some additional con-
+ditions they are incompatible. It is also shown that PageRank centrality violates most
+conditions under consideration, and has a property that, according to some authors,
+is hardly imaginable for a centrality measure. Adopting such conditions as the Self-
+consistency or Bridge axioms allows one to dramatically reduce the length of a survey
+for selecting the most appropriate centrality measures in the culling method proposed
+in [1].
+Keywords: network | centrality measure | axiomatic approach | self-consistency | bridge
+axiom | PageRank
+1
+Introduction
+The number of network centrality measures studied in the literature exceeds 400 [2] and many
+new measures appear every year. This diversity needs to be structured. The main means
+of structuring it is to establish a correspondence between the measures and their properties,
+some of which can be considered as normative conditions or axioms. The purpose of this
+paper is to advance this work by studying two natural axiomatic conditions, namely, the Self-
+consistency and Bridge axioms, which are closely related to special classes of kernel-based
+centrality measures. We establish a sufficient condition of the Bridge axiom, a necessary
+and sufficient condition of Self-consistency, and indicate centralities, some of which are well
+known and others are new, that satisfy these axioms.
+Very often, centrality is identified with structural importance [3–7]. However, there are
+concepts of importance that are not reducible to centrality. Say, in a chain of moving people
+modeled by a path graph, the most important actors may be the leader and the trailer, i.e.,
+∗[email protected]
+1
+arXiv:2301.00084v1  [physics.soc-ph]  31 Dec 2022
+
+the least central end elements of the chain. Moreover, the central elements of such a chain
+may not be of particular importance. Thus, the importance of nodes in networks is not
+necessarily manifested through centrality.
+Anyway, each point centrality measures some structural capital of the nodes. It turns out
+that the types of capital accounted for by the centralities that satisfy the Bridge axiom on the
+one hand and by centralities satisfying the conjunction of Self-consistency and Monotonicity
+on the other hand are different, and therefore these conditions are incompatible, provided
+that Equivalence is assumed.
+Similarly, the Bridge axiom is incompatible with Transit
+monotonicity.
+PageRank is a centrality measure that attracts a lot of attention.
+In this paper, we
+show that it does not satisfy the most of the conditions under consideration and give an
+explanation of this phenomenon.
+The paper is organized as follows. After introducing the basic notation in Section 2,
+in Section 3 we consider several families of centralities associated with graph kernels. In
+Section 4, the Bridge and Self-consistency axioms are introduced.
+Section 5 presents a
+sufficient condition of the Bridge axiom as well as a number of measures that satisfy it.
+In Section 6, we prove a necessary and sufficient condition of Self-consistency and present
+centralities that satisfy it. In Section 7, simple general properties of centrality measures are
+discussed. Axioms of Monotonicity and Transit monotonicity are considered in Section 8 and
+we prove that the addition of these axioms is sufficient to ensure the properties of Section 7
+and to form conditions incompatible with the Bridge axiom. In the concluding Section 9,
+we propose some interpretations of the results obtained.
+2
+Notation
+Let G = (V, E) be an undirected graph with node set V = V (G) and edge set E = E(G).
+The order of G is |V | = n. Graph nodes will be denoted by letters u, v, w, ui, vi, etc., numbers
+0, 1, 2, . . . , or names: Medici, Pazzi, etc. We consider graphs with n > 1, without loops and
+multiple edges. Since some centrality measures under study are applicable only to connected
+graphs, we confine ourselves to them.
+Nodes u and v of G are neighbors iff {u, v} ∈ E(G). Let Nu denote the set of neighbors
+of node u.
+The adjacency matrix of G is denoted by A = A(G) = (auv)n×n: auv = 1 when u and v
+are neighbors and auv = 0, otherwise. Let ρ(A) be the spectral radius of A.
+The degree du of a node u is the number of neighbors of u: du = |Nu|. The vector of node
+degrees is d = (d1, . . . , dn)T = A1, where 1 = (1, . . . , 1)T. A leaf is a node that has exactly
+one neighbor. Nodes u and v are equivalent in G if there exists an automorphism of G that
+takes u to v; in this case we write u ∼ v.
+The Laplacian matrix of G is
+L = diag(A1) − A,
+where diag(x) is the diagonal matrix with vector x on the diagonal.
+2
+
+The union of graphs G = G1 ∪ G2 (not necessarily disjoint) is defined by: V (G) =
+V (G1) ∪ V (G2) and E(G) = E(G1) ∪ E(G2).
+Given a graph G, a centrality measure (or centrality; sometimes, point centrality) f
+attaches a real number f(v) to each node v ∈ V (G). Thus, f depends on G, however, for
+simplicity we do not reflect this dependence in the notation. In most cases G is fixed, and
+when it is not, we explicitly specify the graph to which centrality applies. Formally, for a
+fixed graph G, a centrality on G is a function f : V (G) → R+ ∪ {0}. It associates a non-
+negative real number f(v) with every node v ∈ V (G) based only on the graph structure [4].
+Various conceptions of centrality are quite diverse.
+In this regard, there is no generally
+accepted definition of centrality that would semantically distinguish it from other types of
+point structural measures. On some attempts to make such a distinction, see Section 7.
+When a centrality measure f(·) on G is fixed, we will write u ≻ v, u ⪰ v, and u ∼= v as
+short versions of f(u) > f(v), f(u) ≥ f(v), and f(u) = f(v), respectively. Moreover, if, for
+instance, V = {1, . . . , 7}, then ({1, 6}, {2, 3, 4}, 5, 7) is an example of centrality ranking of
+nodes 1 to 7 in which f(1) = f(6) > f(2) = f(3) = f(4) > f(5) > f(7).
+3
+Centrality measures induced by graph kernels
+In this section, we consider several families of centrality measures.
+Let d(u, v) be the shortest path distance [8] between nodes u and v in a graph, i.e., the
+length of a shortest path between u and v. Two popular1 distance based centrality measures
+are the [Shortest path] Closeness [10,11]
+f(u) =
+� �
+v∈V
+d(u, v)
+�−1
+(1)
+and [Shortest path] Eccentricity [10,12]
+f(u) = (max
+v∈V d(u, v))−1.
+(2)
+General classes of Closeness and Eccentricity centralities are defined by (1) and (2) with
+d(u, v) being arbitrary distances for graph nodes. In the literature, several classes of such
+distances and, more generally, dissimilarity measures have been proposed (see, e.g., [13,14]).
+Substituting them in (1) and (2) provides centralities whose properties may vary. Most of
+the alternative distances and dissimilarity measures are defined via graph kernels. Let us
+consider several of them.
+1. The parametric Katz [15] kernels (also referred to as Walk [16] or Neumann diffusion
+[17] kernels) are defined as
+P Walk(t) =
+∞
+�
+k=0
+(tA)k = (I − tA)−1
+(3)
+1For example, in the recent study [9], the authors come to the conclusion that in the infection source
+identification problem “a combination of eccentricity and closeness... generally performs better than several
+state-of-the-art source identification techniques, with higher accuracy and lower average hop error.”
+3
+
+with 0 < t < (ρ(A))−1.
+2. The Communicability kernels [18,19] are
+P Comm(t) =
+∞
+�
+k=0
+(tA)k
+k!
+= exp(tA),
+t > 0.
+Two other classes of kernels are defined similarly via the Laplacian matrix L = L(G).
+3. The Forest kernels or regularized Laplacian kernels [20,21] are
+P For(t) = (I + tL)−1, where t > 0.
+(4)
+4. The Heat kernels are the Laplacian exponential diffusion kernels [22]
+P Heat(t) =
+∞
+�
+k=0
+(−tL)k
+k!
+= exp(−tL),
+t > 0.
+By Schoenberg’s theorem [23,24], if matrix P = (puv) is a kernel (i.e., is positive semidef-
+inite), then it produces a Euclidean distance d(u, v) by means of the transformation
+d(u, v) =
+� 1
+2(puu + pvv − puv − pvu)
+� 1
+2,
+u, v ∈ V,
+(5)
+where factor 1
+2 determines the scale.
+Thus, all Walk, Communicability, Forest, and Heat kernels with appropriate parameters t
+provide distances that can be substituted in (1) and (2) to obtain Closeness and Eccentricity
+centralities. We will denote them by Closeness(Kernel) and Eccentricity(Kernel) with the
+corresponding kernels substituted.
+Furthermore, if Pn×n = (puv) determines a proximity measure (which means that for any
+x, y, z ∈ V, pxy + pxz − pyz ≤ pxx, and the inequality is strict whenever z = y and y ̸= x),
+then [25] transformation
+d(u, v) = 1
+2(puu + pvv − puv − pvu),
+u, v ∈ V
+(6)
+provides a distance function that satisfies the axioms of a metric. The Forest kernel with
+any t > 0 produces a proximity measure, while kernels in the remaining three families do
+so when t is sufficiently small [14]. The centralities obtained from a Proximity measure by
+transformation (6) and substitution of the resulting distance into (1) and (2) will be denoted
+by Closeness∗(Proximity) and Eccentricity∗(Proximity), respectively.
+Moreover, if P represents a strictly positive transitional measure on G (i.e., pxy pyz ≤
+pxz pyy for all nodes x, y, and z, with pxy pyz = pxz pyy whenever every path in G from x to z
+visits y), then transformation
+ˆpuv = ln puv,
+u, v ∈ V
+produces [13,26] a proximity measure. In this case, (6) applied to ˆP = (ˆpuv) reduces to
+d(u, v) = 1
+2(ln puu + ln pvv − ln puv − ln pvu)
+(7)
+4
+
+and generates [13] a cutpoint additive distance d(u, v), viz., such a distance that d(u, v) +
+d(v, w) = d(u, w) whenever v is a cutpoint between u and w in G (or, equivalently, whenever
+all paths connecting u and w visit v). The centralities obtained from anyTransitional Measure
+by transformation (7) and substitution of the resulting distance into (1) and (2) will be
+denoted by Closeness∗(logTransitionalMeasure) and Eccentricity∗(logTransitionalMeasure),
+respectively.
+Since the Walk and Forest kernels determine [26] strictly positive transitional measures,
+transformation (7) applied to them generates cutpoint additive distances. Substituting them
+into (1) and (2) produces Closeness∗(logForest), Closeness∗(logWalk) and the corresponding
+Eccentricity∗(·) centrality measures.
+Thus, based on the above results, we define Closeness and Eccentricity centrality mea-
+sures obtained by substituting the:
+• Forest kernel;
+• Heat kernel;
+• logarithmic Forest kernel;
+• logarithmic Walk kernel;
+• logarithmic Heat kernel, and
+• logarithmic Communicability kernel
+transformed by (5) or (6) into (1) and (2).
+These centralities were used in the survey
+proposed in [1] with parameter t = 1 for the Forest, Heat, and Communicability kernels and
+t = (ρ(A) + 1)−1 for the Walk kernel.
+While the above measures are promising kernel-based centralities, they do not exhaust all
+kernels and transformations [14,17] that can be used to obtain such measures. To mention
+some alternative constructions, note that every distance on graph nodes can be integrated
+in the p-Means framework [27] or in the framework developed in [28].
+The Closeness(Forest) centrality was examined in [29] with the conclusion that “forest
+distance centrality has a better discriminating power than alternate metrics such as be-
+tweenness, harmonic centrality, eigenvector centrality, and PageRank.” Along with this, the
+authors note that the order of node importance induced by forest distances on some simple
+graphs is consistent with their intuition.
+In addition to the above approaches, kernels and similarity/proximity measures can be
+used to obtain centralities directly, without transformations into distances.
+An example
+of such measures is the Estrada subgraph centrality [18].
+This index of a graph node u
+is equal to the diagonal entry pComm
+uu
+of the Communicability kernel, so we denote it by
+Communicability(Kii). Similarly, Walk(Kii) is the measure f(u) = pWalk
+uu
+, u ∈ V determined
+by the diagonal entries of the Walk kernel.
+One more type of centrality measures is constructed by summing the non-diagonal entries
+of the rows of a kernel matrix. We consider the measures of this kind Communicability(Kij)
+and Walk(Kij) defined by f(u) = �
+v̸=u pComm
+uv
+and f(u) = �
+v̸=u pWalk
+uv
+, u ∈ V, respectively.
+Finally, Total communicability [30] is obtained by summing all row entries of the Commu-
+nicability kernel: f(u) = �
+v∈V pComm
+uv
+; it can be described [31] in terms of “potential gain,”
+as well as the corresponding Walk measure.
+5
+
+The existence of hundreds of types and subtypes of centralities compounded by the
+existence of infinite families of them highlights the need for powerful tools for comparing
+centrality measures and choosing the most appropriate ones. The axiomatic approach is
+indispensable in this regard.
+4
+Axioms of Bridge and Self-consistency
+The axioms considered in this section determine the relation between the centrality values of
+two nodes in a graph of a special structure. As mentioned above, the measures under study
+assign centrality to nodes based solely on the graph structure. The Equivalence axiom is a
+partial embodiment of this idea (cf. [32, axiom A3]).
+Axiom E (Equivalence).
+If u, v ∈ V (G) and u ∼ v, then f(u) = f(v).
+All measures under consideration satisfy axiom E; it will be assumed by default.
+Among the most appealing axioms characterizing various classes of “reasonable” centrality
+measures are those of an ordinal nature. Such axioms allow one to compare the centrality of
+some nodes, but they do not determine specific computational algorithms. In other words,
+they are not fingerprints of particular centrality measures.
+Positive responsiveness is a type of axiom, which is of primary importance in many
+axiomatic constructions. The template of these axioms is as follows: “an increase in input
+(making a node more central from some point of view) leads to an increase in output (i.e.,
+raises its centrality).” Now we present two axioms of this kind. In the next two sections, we
+will find centrality measures that satisfy them.
+Recall that a bridge in a graph is an edge whose deletion increases the number of graph’s
+connected components. The following axiom [33] relates the centrality of the endpoints of
+any bridge.
+Axiom B (Bridge).
+If edge {u, v} is a bridge in G, i.e., the removal of {u, v} from E(G)
+separates G into two connected components (with node sets Vu ∋ u and Vv ∋ v), then
+|Vu| < |Vv| ⇔ f(u) < f(v).
+A strengthening of this axiom is the Ratio property [34], which holds when under the
+same premise, f(w) > 0 for all w ∈ V and f(u)/f(v) = |Vu|/|Vv|.
+The idea of the second axiom is quite different. We assume that the vector of centrality
+values of the neighbors of any node u carries a lot of information about the centrality of
+u itself (cf. Consistency in [35]). A more specific form of this idea is that “the higher the
+centrality values of a node’s neighbors, the higher the centrality of the node itself.”
+This is in line with the justification given by Bonacich and Lloyd [36] to the Eigenvector
+centrality, a measure satisfying (Section 6) the axiom we are going to introduce: “The eigen-
+vector is an appropriate measure when one believes that actors’ status is determined by those
+with whom they are in contact. This conception of importance or centrality makes sense in
+a variety of circumstances. Social status rubs off on one’s associates. Receiving information
+6
+
+from knowledgeable sources adds more to one’s own knowledge. However, eigenvectors can
+give weird and misleading results when misapplied.”
+The final step in refining this concept leads to the axiom of Self-consistency. In the case
+of directed graphs that express paired comparisons, it appeared in [37–39]; for undirected
+graphs, in [40, 41] under the name of Structural consistency. It strengthens Preservation
+of neighborhood-inclusion [42], whose directed version goes back to Preservation of cover
+relation [43].
+Axiom S (Self-consistency).
+If for u, v ∈ V, there is a bijection between Nu to Nv such
+that every element of Nu is, according to f(·), no more central than the corresponding element
+of Nv, then f(u) ≤ f(v). If “no more” is actually “less” at least once, then f(u) < f(v).
+Both the Bridge and Self-consistency axioms belong to the class of positive responsive-
+ness axioms, however, the positivity requirement in the premise of Self-consistency is not
+objective: it reduces to positivity in terms of f(·). This implies that when f(·) satisfies ax-
+iom S and the values of ¯f(·) are ordered oppositely to those of f(·), then ¯f(·) also satisfies S.
+Consequently, the sole axiom S allows in some cases to conclude that f(u) = f(v), but never
+that f(u) > f(v). In particular, if f(u) = f(v) for all u, v ∈ V, then f(·) satisfies S for any
+graph. Therefore, Self-consistency is usually combined with other axioms indicating how
+centrality is related to the graph structure itself rather than to the neighbors’ centrality.
+In the following two sections, we present several results on the centrality measures that
+satisfy the Bridge or Self-consistency axioms.
+5
+Centrality measures satisfying the Bridge axiom
+In the statements of this section, the notion of a cutpoint additive distance and the Close-
+ness∗(logForest) and Closeness∗(logWalk) measures are those introduced in Section 3.
+The Connectivity centrality [34] of vertex u is equal to the number of permutations
+π = (π1, . . . , π|V |) of V (G) such that π1 = u and for every j ∈ {2, . . . , |V | − 1}, the induced
+subgraph of G with node set {π1, . . . , πj} is connected.
+Lemma 1. Any Closeness centrality of the form (1) such that the corresponding distance
+d(·, ·) is cutpoint additive satisfies axiom B.
+Proof.
+For any connected G, consider a Closeness centrality f(u) =
+��
+v∈V d(u, v)
+�−1,
+where d(·, ·) is a cutpoint additive distance.
+Let {u, v} be a bridge in G. Since v is a
+cutpoint between u and any node w ∈ Vv∖{v}, it holds that
+(f(u))−1
+=
+�
+w∈V (G)
+d(u, w) =
+�
+w∈Vu
+d(u, w) +
+�
+w∈Vv
+d(u, w)
+=
+�
+w∈Vu
+d(u, w) + |Vv| d(u, v) +
+�
+w∈Vv
+d(v, w).
+7
+
+Figure 1: A tree on which Betweenness violates axiom B.
+Similarly, (f(v))−1 = �
+w∈Vv d(v, w) + |Vu| d(v, u) + �
+w∈Vu d(u, w). Hence
+(f(u))−1 − (f(v))−1 = (|Vv| − |Vu|) d(u, v),
+consequently, f(u) < f(v) ⇔ (f(v))−1 < (f(u))−1 ⇔ |Vu| < |Vv|. Therefore, f(·) satisfies the
+Bridge axiom.
+Proposition 1. The Shortest path Closeness, Connectivity, Closeness∗(logWalk), and
+Closeness∗(logForest) centralities satisfy the Bridge axiom.
+Proof. The fulfilment of the Bridge axiom for the Shortest path Closeness is due to Skibski
+and Sosnowska [33]. Alternatively, it follows from Lemma 1.
+The Bridge axiom holds for Connectivity since this centrality measure satisfies the
+stronger Ratio property [34].
+The Walk (3) and Forest (4) kernels represent [26] strictly positive transitional measures
+on any connected graph. Therefore, definition (7) transforms [13] them into cutpoint additive
+distances d(u, v). By Lemma 1 this implies that the Closeness centralities corresponding
+to these distances, namely, the Closeness∗(logWalk) and Closeness∗(logForest) centralities,
+satisfy the Bridge axiom.
+Similarly, other strictly positive transitional measures [26] and cutpoint additive distances
+also produce centralities that satisfy the Bridge axiom.
+Remark 1. It is worth noting that the Betweenness centrality [44] satisfies the Bridge axiom
+for many graphs, however, generally this is not the case. The simplest graph on which Be-
+tweenness violates this axiom is shown in Fig. 1. Here, axiom B requires that the centralities
+of nodes 0 and 5 are equal since |V0| = |V5|. However, the Betweenness centrality of node 0
+is higher than that of node 5, as 0 lies on the shortest path from 1 to 2.
+6
+Centrality measures satisfying Self-consistency
+To formulate a necessary and sufficient condition of Self-consistency, we introduce two defi-
+nitions.
+8
+
+Definition 1. A function ϕ : Mk → R, where Mk = {M : 0 < |M| < k}, M being a
+multiset2 of real numbers, will be called a scoring function if ϕ(M) is strictly increasing in
+any element of M, while the remaining elements, including those equal to the varying one,
+are fixed.
+Definition 2. A centrality vector x = (x1, . . . , xn)T assigned to a connected graph G with
+V (G) = {1, . . . , n} (xu = f(u), u ∈ V (G), where f is the corresponding centrality measure)
+has a monotonic neighborhood representation if there exists a scoring function ϕ such that
+x satisfies the system of equations
+xu = ϕ({xw : w ∈ Nu}),
+u = 1, . . . , n.
+(8)
+In Definition 2, {xw : w ∈ Nu} is the multiset of the components of x that correspond to
+the neighbors of node u in G. If a centrality vector has a monotonic neighborhood represen-
+tation, then finding this vector reduces to solving the system (8).
+Lemma 2. A centrality measure on G satisfies Self-consistency if and only if the centrality
+vector this measure attaches to G has a monotonic neighborhood representation.
+Proof.
+Suppose that the centrality vector x = (x1, . . . , xn)T associated with G has a
+monotonic neighborhood representation (8). Let the premise of Self-consistency be true for
+nodes u and v. Consider the equations (8) corresponding to u and v:
+xu
+=
+ϕ({xw : w ∈ Nu}),
+(9)
+xv
+=
+ϕ({xw : w ∈ Nv}).
+(10)
+Since there is a bijection that maps each element of Nu to an element of Nv with a greater
+or equal centrality, step by step replacing in (9) the xw value of each element of Nu by the
+x component of the corresponding element of Nv and using the definition of monotonic
+neighborhood representation, we get a growth or preservation of the value of ϕ(·) at each
+step, yielding the value xv in the last step. This implies that xu ≤ xv, or, stronger, xu < xv
+whenever xw has been strictly increased at least once. Therefore, Self-consistency is satisfied.
+Conversely, suppose that a centrality measure on G is Self-consistent. Let us construct a
+scoring function ϕ(·) that provides a monotonic neighborhood representation of the centrality
+vector x = (x1, . . . , xn)T associated with G. First, we set ϕ({xw : w ∈ Nu})
+def
+= xu for all
+u ∈ {1, . . . , n}. Whenever {xw : w ∈ Nu} = {xw : w ∈ Nv} for some u, v ∈ V, Self-consistency
+implies xu = xv, i.e., the above definition of ϕ(·) on the set of multisets P = {{xw : w ∈ Nu},
+1 ≤ u ≤ n} ⊂ Mk is not contradictory. Thus, we defined the function ϕP(·) on P. Now,
+to obtain a monotonic neighborhood representation of x, it suffices to extend ϕP(·) to the
+entire set Mk (k = max{|Nu|, 1 ≤ u ≤ n}) of multisets of real numbers in such a way that
+the resulting ϕ(·) is strictly increasing on Mk.
+2A finite multiset is an equivalence class of vectors such that two vectors z and z′ are equivalent whenever
+z′ can be obtained from z by permuting its components. As distinct from a set, a multiset may contain
+several copies of the same element, as the components of a vector may be equal.
+9
+
+By the definition of a scoring function, the strict increase of ϕ(·) is required with respect
+to the following preorder ≽ on Mk: for X, Y ∈ Mk, X ≽ Y ⇔ [there is a bijection between
+X to Y such that every element of Y does not exceed the corresponding element of X]. The
+condition of strict increase reduces to the implication [X ≽ Y and Y ̸≽ X] ⇒ ϕ(X) > ϕ(Y ),
+since the second necessary implication [X ≽ Y and Y ≽ X] ⇒ ϕ(X) = ϕ(Y ) is trivial as its
+premise implies X = Y.
+Observe that the preorder ≽ has a numerical [utility] representation. This means that
+there exists a function u: Mk → R such that for all X, Y ∈ Mk, X ≻ Y ⇒ u(X) > u(Y ),
+where, by definition, X ≻ Y ⇔ [X ≽ Y and Y ̸≽ X]. Indeed, u(X) can be defined, say,
+as the sum of the elements of multiset X. Then X ≻ Y ⇒ u(X) > u(Y ) and so u(·) is a
+numerical representation of ≽.
+By Self-consistency, ϕP(·) strictly increases on P, i.e., ϕP(·) is a numerical representation
+of ≽P, the restriction of ≽ to P. Since ≽ has a numerical representation, it follows from [45,
+Theorem 1] that ϕP(·) has a strictly increasing extension to Mk if and only if ϕP(·) is gap-
+safe increasing, i.e., is weakly increasing and for any X, Y ∈ Mk ∪ {−∞, +∞}, Y ≻ X
+implies
+inf{ϕP(Z) : Z ≽ Y, Z ∈ P} > sup{ϕP(Z) : X ≽ Z, Z ∈ P},
+(11)
+where, by convention, sup ∅ = −∞ and inf ∅ = +∞.
+To prove that ϕP(·) is gap-safe increasing, first observe that since P is finite, sup and inf
+in (11) can be replaced by max and min, respectively, under the convention that max ∅ = −∞
+and min ∅ = +∞. Then, if the [multi]sets on the left-hand and right-hand sides of (11) are
+both nonempty, then for any Z′′ and Z′ minimizing ϕP(Z) on the left and maximizing ϕP(Z)
+on the right, respectively, Z′′ ≽ Y ≻ X ≽ Z′ holds, and by the “mixed” strict transitivity3 of
+≽, Z′′ ≻ Z′. By Self-consistency this implies ϕP(Z′′) > ϕP(Z′) and (11) is valid. Otherwise,
+if some multiset in (11) is empty, then we have +∞ on the left or/and −∞ on the right, in a
+possible combination with a finite number on one of the sides. In all these cases, (11) is valid,
+hence ϕP(·) is gap-safe increasing. Therefore, by [45, Theorem 1], ϕP(·) can be extended
+to Mk so that its extension ϕ(·) is a strictly increasing function and therefore, provides
+a monotonic neighborhood representation of the centrality vector x = (x1, . . . , xn)T. This
+completes the proof. The extension of ϕP(·) to Mk can be made, in particular, using the
+approach proposed in [45].
+The following propositions involve five centrality measures; we now recall their definitions
+using the notation introduced in Section 2.
+For a connected graph G of order n, vector x = (x1, . . . , xn)T presents:
+• the Walk centrality [15] if
+x =
+∞
+�
+k=1
+(tA)k1 = ((I − tA)−1 − I)1,
+(12)
+where t ∈ R is a parameter such that 0 < t < (ρ(A))−1;
+3This means that for any X, Y, Z ∈ Mk, Z ≽ Y ≻ X ⇒ Z ≻ X and Y ≻ X ≽ Z ⇒ Y ≻ Z.
+10
+
+• the Bonacich centrality [46] with real parameters α and β > 0 if x satisfies the system
+of equations
+xu =
+�
+w∈Nu
+(α + βxw),
+u = 1, . . . , n;
+(13)
+• the Generalized Degree centrality [47] if x satisfies the system of equations
+(I + εL)x = d,
+(14)
+where ε > 0 is a real parameter;
+• the Eigenvector centrality [48,49] if x is positive and satisfies the equation
+Ax = ρ(A)x;
+(15)
+• the PageRank centrality [50] if x is positive and satisfies the equation4
+x =
+�
+αAT(diag(A1))−1 + (1 − α)J
+�
+x,
+(16)
+where J = 1
+n11T, while α ∈ R is the “teleportation” parameter such that 0 < α < 1.
+Proposition 2. The Generalized Degree, Walk, Eigenvector, and Bonacich centralities sat-
+isfy Self-consistency.
+Proof. 1.
+Since for any u, du = |Nu|, Eq. (14) can be written in component form as
+xu(1 + ε|Nu|) − ε
+�
+w∈Nu
+xw = |Nu|,
+u = 1, . . . , n,
+which is equivalent to
+xu = (1 + ε|Nu|)−1 �
+w∈Nu
+(1 + εxw),
+u = 1, . . . , n.
+(17)
+Eq. (17) is a monotonic neighborhood representation of vector x, therefore, by Lemma 2,
+the Generalized Degree centrality satisfies Self-consistency.
+2. It follows from (12) that
+(I − tA)x = td,
+from which
+xu = t
+�
+w∈Nu
+(1 + xw),
+u = 1, . . . , n.
+(18)
+Since for any t > 0, (18) is a monotonic neighborhood representation of x, Lemma 2
+implies that the Walk centrality satisfies Self-consistency.
+3.
+A component form of (15) is
+xu = (ρ(A))−1 �
+w∈Nu
+xw,
+u = 1, . . . , n,
+(19)
+4In the case of simple graphs considered in this paper, AT = A.
+11
+
+which is a monotonic neighborhood representation of x. Hence, by Lemma 2, the Eigenvector
+centrality satisfies Self-consistency.
+4.
+The equations (13) of the Bonacich centrality provide a monotonic neighborhood
+representation of x. By Lemma 2, these centralities satisfy Self-consistency. It follows from
+the comparison of (18) and (13) that the Walk centralities are the Bonacich centralities with
+α = β = t.
+To prove that a centrality measure satisfies Self-consistency, it suffices to find its mono-
+tonic neighborhood representation, as we did, e.g., for the Walk centrality. Disproving the
+hypothesis of the Self-consistency of some measure reduces to giving a refuting example, i.e.,
+an appropriate pair of nodes in some network. Here, among others, the famous network of
+Florentine ruling families (Fig. 2) can be of help, as we show in Lemma 3 and Proposition 3.
+Figure 2: Marriage network of the Florentine ruling families of the 15th century (without
+the isolated Pucci family).
+Let f(·) be a centrality measure on a graph G. We say that two arrays (u1, . . . , uk)
+and (v1, . . . , vk) of the nodes of G are f(·) order equivalent iff for any i, j ∈ {1, . . . , k},
+sign(f(ui) − f(uj)) = sign(f(vi) − f(vj)).
+Lemma 3. If a centrality measure f(·) satisfies axiom S, then for the Florentine families
+graph of Fig. 2, the following arrays of nodes are f(·) order equivalent:
+(a) (Tornabuoni, Albizzi) and (Ridolfi, Ginori);
+(b) (Bischeri, Peruzzi) and (Guadagni, Castellani);
+(c) (Bischeri, Castellani) and (Guadagni, Barbadori);
+12
+
+Lamberteschi
+Ginori
+Guadagni
+Albizzi
+Bischeri
+Tornabuoni
+Acciaiuoli
+Medici
+Ridolfi
+Strozzi
+Peruzzi
+Salviati
+Barbadori
+Castellahi
+Pazzi(d) (Peruzzi, Castellani) and (Bischeri, Barbadori);
+(e) (Tornabuoni, Ridolfi) and (Guadagni, Strozzi);
+(f) (Barbadori, Salviati) and (Castellani, Pazzi);
+(g) (Ginori, Aciaiuoli, Pazzi, Lamberteschi) and (Albizzi, Medici, Salviati, Guadagni).
+Proof. (a) Observe that Tornabuoni and Albizzi have three neighbors each, and they share
+two neighbors. Therefore, by S, the relation between them is the same as the relation between
+the remaining neighbors, Ridolfi and Ginori. (b) Bischeri and Peruzzi are adjacent and have
+a common neighbor Strozzi; in addition, Bischeri has a neighbor Guadagni, while Peruzzi has
+a neighbor Castellani. Due to S, the relation between Bischeri and Peruzzi coincides with
+that between Guadagni and Castellani. Indeed, it is easy to see that the edge {Bischeri,
+Peruzzi} cannot correct the violation of Self-consistency that may occur in the absence of
+this edge. This completes the proof of (b). The remaining parts are proved similarly.
+The following proposition demonstrates that Lemma 3 can be quite useful in proving
+that certain measures violate Self-consistency.
+Proposition 3. Walk(Kii), Communicability(Kii), Closeness(Forest), Closeness(Heat),
+Closeness∗(logWalk), Closeness∗(logCommunicability), Closeness∗(logForest), and Close-
+ness∗(logHeat) centralities violate axiom S.
+Proof.
+For the graph in Fig. 2, Walk(Kii) and Communicability(Kii) provide a central-
+ity ranking in which Peruzzi ≻ Bischeri, but Guadagni ≻ Castellani. Thus, by part (b)
+of Lemma 3, these measures violate Self-consistency. Measures Closeness(Forest), Close-
+ness∗(logWalk), Closeness∗(logCommunicability), and Closeness∗(logHeat) provide rankings
+in which Ridolfi ≻ Tornabuoni, but Guadagni ≻ Strozzi. Thus, by part (e) of Lemma 3, these
+measures violate Self-consistency. Measures Closeness(Heat), and Closeness∗(logForest) pro-
+vide rankings in which Castellani ≻ Peruzzi, but Bischeri ≻ Barbadori. Thus, by part (d)
+of Lemma 3, these measures violate Self-consistency.
+7
+On core intuition behind centrality
+The best example of a “central” node is the center of a star of order more than 2.
+A star of order n is a graph with one node (the center) having degree n − 1 and n − 1
+nodes of degree 1. The edges of a star are sometimes called rays.
+As Freeman [51] noted, “one general intuitive theme seems to have run through all the
+earlier thinking about point centrality in social networks: the point at the center of a star
+[...] is the most central possible position.”
+Definition 3. We say that a centrality measure on a star G with n ≥ 3 nodes satisfies the
+star condition if it assigns maximum centrality to the center of this star.
+For an example of a centrality measure that violates the star condition, see [1, Section 1].
+13
+
+Self-consistency is a strong axiom, however, as was noted, it is not comprehensive. One
+of its features is that it only applies to nodes of the same degree. Therefore, it does not
+imply the star condition. As distinct from it, the Bridge axiom implies this condition.
+Proposition 4. On a star with two or more rays, any centrality measure that satisfies
+axiom B also satisfies the star condition.
+Proof. This is true as each ray of a star is a bridge, and among the components formed after
+its removal, the component containing a leaf is smaller than that containing the center.
+However, axiom B does not imply that the centrality of all leaves of a star is the same,
+which is immediate from Self-consistency (or from axiom E, as the leaves are equivalent).
+Roy and Tredan [6], trying to capture the intuition underlying the concept of centrality
+claim that for a path graph with nodes 1, . . . , n, where each node u such that 1 < u < n is
+linked to u − 1 and u + 1, it is (converting to our notation) “hard to imagine a centrality f
+such that, given a node u (u ̸= n+1
+2 ), we have f(u) ̸∈ [f(u − 1), f(u + 1)].”
+Definition 4. Let G be a path graph where each node u such that 1 < u < n is linked to
+u − 1 and u + 1. A centrality measure f on G is said to satisfy the
+• Roy-Tredan (RT) condition if for any node u, u ̸= n+1
+2
+⇒ f(u) ∈ [f(u − 1), f(u + 1)];
+• path centripetal condition if the centrality of a node strictly increases with increasing
+shortest path distance from the nearest leaf.
+Obviously, the path centripetal condition is generally stronger than the RT condition.
+Proposition 5 states that the path centripetal condition is fulfilled for all centralities that
+satisfy axioms B and E.
+Proposition 5. For a path graph, any centrality measure that satisfies axioms B and E also
+satisfies the path centripetal condition.
+Proof. Let f(·) satisfy axioms B and E. Consider the path graph 1—2—· · · —n, where “—”
+denotes an edge. Let 1 ≤ u = v − 1 < n. Suppose that v ≤ n+1
+2 . Then {u, v} ∈ E is a
+bridge and by axiom B, f(u) < f(v), since |Vu| < |Vv|. Hence for such u and v, the centrality
+strictly increases with increasing distance from the nearest leaf 1. The case of u ≥ n+1
+2
+is
+considered similarly. Finally, if u − 1 = n − v, i.e., u and v have the same distance from the
+nearest leaf, then u ∼ v and by axiom E, f(u) = f(v).
+It is all the more remarkable that PageRank, one of the most popular5 centrality measures,
+according to Roy and Tredan, is “hard to imagine” as it violates the RT condition.
+Proposition 6. For the path graph 1—2—3—4—5, the PageRank centrality f PR
+α (·) with any
+parameter α violates the RT condition. Namely, f PR
+α (2) > f PR
+α (1) and f PR
+α (2) > f PR
+α (3).
+5According to [52], “PageRank centrality is probably the most well-known and frequently used measure.”
+14
+
+Proof.
+For the path graph 1—2—3—4—5, let us search the solution of (16) in the form
+x = (x1, x2, x3, x2, x1)T, where x1 = 1. Then the first two equations of (16) have the form
+α
+2 x2 + 1 − α
+5
+(2 + 2x2 + x3)
+=
+1;
+α + α
+2 x3 + 1 − α
+5
+(2 + 2x2 + x3)
+=
+x2.
+The solution of this system is:
+x2
+=
+2(−α2 + 2α + 4)−1(3α + 2);
+x3
+=
+2(−α2 + 2α + 4)−1(α2 + 2α + 2).
+Since both differences x2 − x1 = α(α + 4)(−α2 + 2α + 4)−1 and x2 − x3 = 2α(1 − α)(−α2 +
+2α + 4)−1 are strictly positive for all α ∈ (0, 1), PageRank centralities with all appropriate
+parameter values violate the RT condition. Namely, x2 > x1 and x2 > x3.
+Node 3 of the 1—2—3—4—5 path can be considered as its center. It follows from the
+proof of Proposition 6 that PageRank never assigns maximum centrality to this center. It
+can be shown that PageRank centrality also violates the RT condition on paths with n > 5.
+It is worth noting that if the user considers the Self-consistency or Bridge axiom as an
+indispensable property of a centrality measure, then this leads to a dramatic reduction of
+the set of candidate measures (see [1], where the corresponding reduced surveys for choosing
+the most appropriate centrality measure are shown in Figures 7 and 8).
+8
+Combinations with monotonicity axioms
+In this section, we focus on edge-monotonicity conditions, which, as well as the Self-
+consistency and Bridge axioms, belongs to the class of positive responsiveness axioms. It is
+proved that together with axiom E (and once S) they imply the star and path centripetal
+conditions and contradict axiom B, while PageRank violates axioms B, S, and Transit mono-
+tonicity.
+The edge-monotonicity axioms of this section involve two graphs: an original graph G0
+and a graph G obtained from G0 by adding an extra edge (extra edges). These axioms
+restrict a universal centrality measure fG(·) operating on any connected graph G. The word
+“universal” in the formulations of Propositions 7 to 10 is implied, not explicit.
+Axiom M (Monotonicity). Suppose that u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, and G =
+G0 ∪ G′ ̸= G0, where V (G′) = {v, w}, E(G′) = {{v, w}}, and w ̸= u. Then fG(u) < fG(v).
+According to Monotonicity, if u is no more central than v and a new edge not adjacent
+to u is attached to v, then v becomes or remains more central than u.
+Similar axioms called Adding rank monotonicity and Strict rank-monotonicity have been
+proposed in [47] and [53] (for directed graphs), respectively. Item 1.2 of Dynamic monotonic-
+ity in [54] is the corresponding condition for directed graphs representing paired comparisons.
+Monotonicity together with axiom E imply the star condition.
+15
+
+Proposition 7. For a star with two or more rays, any centrality measure that satisfies
+axioms E and M also satisfies the star condition and assigns the same centrality to all leaves.
+Proof.
+By E, the centrality of the two nodes of a 1-ray star is the same. By M, adding one
+more node adjacent to the “center” of the 1-ray star makes the centrality of the center greater
+than the same centrality of the leaves, and attaching additional leaves preserves this.
+Transit monotonicity is a natural strengthening of M.
+Axiom T (Transit monotonicity).
+If u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, G =
+G0 ∪ G′ ̸= G0, and any path in G from a node of G′ to u contains v, then fG(u) < fG(v).
+According to Transit monotonicity, if u is no more central than v and v is a cutpoint
+between the new edges and u, then v becomes or remains more central than u.
+Together with E, Transit monotonicity implies the path centripetal condition.
+Proposition 8. For a path graph, any centrality measure that satisfies axioms E and T also
+satisfies the path centripetal condition.
+Proof.
+By E, the conclusion holds for the 1—2 path graph on two nodes. Assume that it
+holds for the path graph 1—· · · —2k. Then for all i ∈ {1, . . . , k}, f(i) ≤ f(i + 1). Attaching
+a new node 2k +1 and the edge {2k, 2k +1} provides the path graph 1—· · · —(2k +1). Since
+any path in the new graph from 2k + 1 to i contains i + 1, axiom T implies f(i) < f(i + 1).
+Therefore, the centrality of the nodes i ∈ {1, . . . , k + 1} of the new graph strictly increases
+with the increase of the shortest path distance from the nearest leaf. This is also the case
+for the remaining nodes i ∈ {k + 2, . . . , 2k + 1} by axiom E, since for them i ∼ (2k + 2 − i).
+Thus, the conclusion of Proposition 8 is true for the 1—· · · —(2k + 1) graph. Adding node
+2k + 2 and edge {2k + 1, 2k + 2} to it, we similarly derive that this conclusion also holds for
+the resulting 1—· · · —(2k + 2) graph. This completes the proof by induction.
+As a corollary of Proposition 8 we obtain that PageRank centrality violates axiom T.
+Moreover, it does not satisfy axioms B and S.
+Proposition 9. The PageRank centrality with any parameter α violates axioms T, B, and S.
+Proof. PageRank centrality violates axiom T since otherwise, by Proposition 8, PageRank,
+obeying axiom E, satisfies the path centripetal condition and therefore the RT condition,
+which is not true by Proposition 6.
+By Proposition 5, axioms B and E imply the path centripetal condition. Thus, PageRank
+centrality similarly violates axiom B.
+In the path graph 1—2—3—4—5, node 2 has neighbors 1 and 3, 3 has neighbors 2 and 4;
+by Proposition 6, for any α ∈ (0, 1), f PR
+α (2) > f PR
+α (1) and f PR
+α (4) = f PR
+α (2) > f PR
+α (3), i.e.,
+the neighbors of 3 have higher centrality values than the corresponding neighbors of 2. In
+this case, axiom S requires f PR
+α (3) > f PR
+α (2), which is not the case. Therefore, axiom S is
+violated.
+On some other peculiarities of the PageRank centrality, see [1, Section 1].
+We conclude by showing that under Equivalence, the conjunction of M and S is incom-
+patible with axiom B, and so is T.
+16
+
+Proposition 10. If a centrality measure satisfies axioms E, M, and S or axioms E and T,
+then it violates axiom B.
+Proof.
+Let a universal centrality measure satisfy axioms E and B. For the graph G in
+Fig. 3a, fG(4) = fG(3) by B. For the graph G0 in Fig. 3b, fG0(4) = fG0(3) by E. Observe
+that G = G0 ∪ G′, where V (G′) = {0, 1} and E(G′) = {{0, 1}}.
+(a)
+(b)
+Figure 3: (a) A graph G on which axiom B is incompatible with E&M&S as well as with
+E&T; (b) G0, a subgraph of G used in the proof of Proposition 10.
+Assume that this universal centrality measure satisfies axioms M and S. By E and M,
+fG(0) = fG(1) > fG(2) = fG(5). Therefore, by S, fG(4) > fG(3), a contradiction.
+Now assume that, instead of M&S, this centrality measure satisfies axiom T. Since all
+paths in G from 0 or 1 to 3 contain 4, by T, fG(4) > fG(3) holds, a contradiction.
+Axioms B and T are incompatible under Equivalence for the following reason. Suppose
+that {u, v} is a bridge in G and |Vu| = |Vv|. Then B implies f(u) = f(v). However, if
+the restriction of E(G) to Vu is sparse, while its restriction to Vv is dense, then T requires
+f(u) < f(v). The logic of axiom S is similar to that of T in terms of transferring influences,
+however, S is not “grounded” as it does not require any direct effect of density on centrality.
+In the conjunction M&S, axiom M provides this “grounding.”
+9
+Discussion
+Each point centrality measures some structural capital of the nodes.
+According to the
+Bridge axiom, one end-node of a bridge is more central than the other if and only if the
+removal of the bridge leaves the first one in a greater (in terms of the number of nodes)
+component. In this sense, the corresponding capital is node-based: it does not depend on
+the density of the components. Self-consistency states that a node’s capital increases with
+the capital of its neighbors. By the Monotonicity axiom, edges incident to a node contribute
+to its capital, i.e., the corresponding capital is locally edge-based. The conjunction of the
+Self-consistency and Monotonicity makes this impact of edges global.
+As a result, this
+conjunction turns out to be incompatible (under Equivalence) with the node-based Bridge
+axiom (Proposition 10). Similarly, by the same proposition, the Bridge axiom is incompatible
+with the Transit monotonicity axiom, which postulates the edge nature of the capital globally.
+17
+
+5
+4
+3
+2
+05
+4
+3
+2
+0
+GoAn additional subject of this paper is the properties of the PageRank centrality measure
+related to the main topic. It turns out that this measure violates most of the conditions
+we consider and even has a property that, according to some authors, “is hard to imag-
+ine” for a measure of centrality. The reason for this is the stochastic normalization used
+in PageRank. In the path graph 1—2—3—4—5 used in Proposition 6, nodes 2 and 4 have
+maximum PageRank centrality as they are linked to the leaves: these links receive a maxi-
+mum weight of 1, since normalization does not change them. This maximum weight can be
+interpreted as the specific importance of these links for the leaves, and not for the nodes 2
+and 4, which profit from this weight. It is this counterintuitive normalization that violates
+the RT condition.
+The axioms of Self-consistency and Bridge are quite strong, so the adoption of either
+of them dramatically reduces the set of centrality measures under consideration. This fact
+is used in [1], where the “culling” method for determining the most appropriate centrality
+measure is proposed.
+This method consists in compiling and completing a survey that
+allows the user to find a measure that matches their underlying concept of centrality. In the
+framework of this method, adopting a certain axiom results in compiling a shorter survey
+on the set of measures that satisfy this axiom. In [1], the surveys reduced to the measures
+obeying the Self-consistency or Bridge axioms are shown in Figures 7 and 8, respectively.
+Acknowledgement
+The author thanks Anna Khmelnitskaya and Dmitry Gubanov for helpful discussions.
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+21
+

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@@ -0,0 +1,871 @@
+Encrypted Data-driven Predictive Cloud Control with Disturbance
+Observer
+Qiwen Li, Runze Gao and Yuanqing Xia∗
+Abstract— In data-driven predictive cloud control tasks, the
+privacy of data stored and used in cloud services could be leaked
+to malicious attackers or curious eavesdroppers. Homomorphic
+encryption technique could be used to protect data privacy
+while allowing computation. However, extra errors are intro-
+duced by the homomorphic encryption extension to ensure the
+privacy-preserving properties, and the real number truncation
+also brings uncertainty. Also, process and measure noise existed
+in system input and output may bring disturbance. In this work,
+a data-driven predictive cloud controller is developed based
+on homomorphic encryption to protect the cloud data privacy.
+Besides, a disturbance observer is introduced to estimate and
+compensate the encrypted control signal sequence computed in
+the cloud. The privacy of data is guaranteed by encryption and
+experiment results show the effect of our cloud-edge cooperative
+design.
+Index Terms— Cloud Control Systems, Data-Driven Predic-
+tive Control, Disturbance Observer, Homomorphic Encryption.
+I. INTRODUCTION
+Cloud computing provides enormous computing and stor-
+age resources for the implementation of control applications,
+which brings the concept of cloud control systems (CCSs)
+[1]–[3]. In CCSs, control algorithms are outsourced and
+executed on cloud platforms to offer control services for
+local plants. With the development of CCSs, there is an
+emerging requirement of cloud control for complex systems.
+However, the complexity and scale of control systems bring
+new difficulty in designing model-based cloud control laws,
+since system models are difficult to obtain. As a kind of
+model-free control approach, data-driven predictive control
+(DPC) [4] directly computes control sequences based on the
+input-output data of the system, which avoids the process
+of system modeling. Therefore, the combination of CCSs
+and DPC, i.e., data-driven predictive cloud control (DPCC)
+[5]–[7], takes advantage of data storage and computation in
+the cloud, as well as the model-free manner in control of
+complex systems, becoming a potential candidate in CCSs.
+However, in DPCC scenarios, the input-output data and
+control law of systems are stored and computed in the
+cloud with no data privacy protection, leading to the risk
+of privacy leakage. To be specific, an eavesdropper could
+get access to the private system data through communication
+channel, cloud storage and memory. The eavesdropper could
+consequently infer the state and model of the system for
+malicious purposes, such as advanced persistent threat (APT)
+Q. Li, R. Gao and Y. Xia are with School of Automation, Beijing
+Institute of Technology, Beijing 100081, P. R. China. (Corresponding
+author: Yuanqing Xia). E-mail address: [email protected] (Q. Li),
+runze [email protected] (R. Gao), xia [email protected] (Y. Xia).
+design and system state tracking. Thus, the privacy issues in
+DPCC should be seriously considered.
+As a solution, we use homomorphic encryption (HE)
+approaches to protect data privacy while computing the
+DPCC control law, since HE schemes allow computations
+on encrypted data. Specifically, we use CKKS scheme [8],
+which is a RLWE-based HE protocol that ensures the privacy
+of the scheme through introducing errors to satisfy the
+hardness of the RLWE problem. In CKKS scheme, complex-
+number vectors are mapped to integer-coefficient polyno-
+mials through interpolation, amplification and truncation.
+Consequently, the addition and multiplication of ciphertext
+in polynomials are homomorphically equivalent to element-
+wise addition and multiplication of plaintext in vectors. In
+this work, we design a privacy-preserving DPCC controller
+based on CKKS scheme to compute control sequences while
+keeping system information invisible to potential attackers.
+When performing the privacy-preserving DPCC tasks
+described above, we should consider the effects on the
+control quality induced by system noise and uncertainty.
+Firstly, errors are introduced to the privacy-preserving DPCC
+procedure through HE scheme. To be specific, errors are
+introduced to public keys in CKKS scheme to protect the
+semantic security properties. Moreover, the amplification and
+truncation procedure bring noises into ciphertexts. Besides,
+measurement noise, process noise and system uncertainty are
+ubiquitous in control systems, which consequently influence
+the control effect of data-driven approaches.
+Hence, disturbance observer (DOB) [5], [9], [10] is used
+to guarantee the control accuracy under the uncertainty,
+including system noise and errors induced by HE scheme.
+The function of DOB is to estimate the effects performed
+on a system based on an auxiliary system. If estimated, the
+system uncertainty could be properly compensated with a
+suitable magnitude.
+Motivated by the above reasons, the main contributions of
+the privacy-preserving DPCC based on HE scheme are listed
+as follows:
+• We design a private DPCC protocol based on CKKS
+scheme, which preserves the privacy of sensitive system
+input-output data.
+• We apply the DOB technique to estimate and com-
+pensate for the uncertainty induced by the HE scheme
+and system noise under the privacy-preserving DPCC
+scenario.
+• A numerical example shows the effectiveness of
+privacy-preserving DPCC with DOB, compared to un-
+encrypted non-DOB and encrypted non-DOB condi-
+arXiv:2301.00322v1  [eess.SY]  1 Jan 2023
+
+tions.
+The remainder of this work is shown as follows. DPCC
+approaches and their privacy issues are briefly surveyed in
+Section II, based on which we develop a privacy-preserving
+data-driven control protocol in Section III. In Section IV,
+a disturbance observer is proposed to compensate for the
+error induced by encryption and data noise. In Section V
+a numerical example of our proposed method is shown to
+demonstrate its effectiveness. Section VI concludes the paper.
+II. RELATED WORKS
+Showing potential in model-free control scenarios, DPC
+approaches compute the control input directly from the input-
+output data of the system, and have been widely used in
+extended situations. [11] propose a model-free approach for
+linear parameter-varying systems. A data-driven error model
+is learned with precollected data in [12] to achieve accurate
+position tracking with a robot arm.
+DPC approaches may require extensive data to estimate
+system models or generate control inputs, in which cases the
+computation time of system input may become the bottleneck
+of implementation. Thus cloud computing and distributed
+computing are gathering more and more attention in DPC
+tasks for the possibility of computation acceleration by prop-
+erly utilizing elastic resources in the cloud. [6], [7] develop a
+cloud-edge-endpoint DPC prototype, showing the feasibility
+of cloud-based control systems. To optimize the effort of
+subspace identification task, which is the basis of data-driven
+control, [13] decomposes the identification algorithm to inter-
+connected containerized tasks through parallel computing.
+A further implementation of cloud-edge cooperative DPCC
+[5] uses workflow-based parallel cloud control and edge
+compensation.
+The privacy of data and models could be leaked through
+outsourced tasks, since the communication channel and
+execution environment could be eavesdropped by untrusted
+third-parties. Therefore, encrypted control approaches have
+been widely studied since it could simultaneously allow
+the computation of control signals and preservation of data
+privacy. Encrypted linear feedback controllers are realized
+in [14]. Moreover, the encrypted realization of more ef-
+ficient and complex control schemes are proposed to fit
+integrated cloud scenarios. In [15], a privacy-preserving sub-
+space identification approach based on partially HE scheme
+is proposed. Alexandru et al. [16] offer offline and online
+encrypted cloud control designs, both based on HE, to
+protect the input-output data of DPC based on a single cloud
+server. Subsequently a privacy-preserving distributed alter-
+nating direction method of multipliers approach is designed
+to perform the system estimation process in ciphertexts [17].
+III. PRELIMINARIES
+In this section, we sketch the preliminaries of DPC and
+RLWE-based HE.
+A. Implementation of data-driven predictive control
+We consider a state-space expression of discrete linear
+time-invariant (LTI) system:
+x(k + 1) =Ax(k) + Bu(k) + ϵp,
+y(k) =Cx(k) + ϵs,
+(1)
+where x(k) ∈ Rn, u(k) ∈ Rm, y(k) ∈ Rp are the state,
+input and output vector of the system, ϵp, ϵs are process
+noise and measure noise of suitable shapes, respectively. In
+the following statements, vectors are all viewed as column
+vectors, except for additional specifications.
+In DPC, we cannot access the specific parameter A, B
+and C mentioned in (1). Therefore, data-driven approaches
+are used to infer the system information and perform control
+task. Specifically, we have the input-output data series of the
+system through time:
+{u(n), y(n), n = 1, 2, ..., T}.
+At every time step k, we use some slices of the input-
+output data series as prior information of the system for
+identification, which are denoted as:
+uf(k) =
+�
+����
+u(k)
+u(k + 1)
+...
+u(k + N − 1)
+�
+���� , yf(k) =
+�
+����
+y(k)
+y(k + 1)
+...
+y(k + N − 1)
+�
+���� ,
+up(k) =
+�
+����
+u(k − N)
+u(k − N + 1)
+...
+u(k − 1)
+�
+���� , yp(k) =
+�
+����
+y(k − N)
+y(k − N + 1)
+...
+y(k − 1)
+�
+���� ,
+(2)
+and
+vp(k) =
+� yp(k)
+up(k)
+�
+,
+(3)
+where the subscript ”p” and ”f” indicate ”past” and ”future”,
+respectively.
+Based on the slices shown above, we can fit the implicit
+system expression with linear regression:
+yf(k) = Lvvp(k) + Luuf(k) + e(k),
+(4)
+where Lv and Lu are coefficient matrices to be fit with
+appropriate shapes that contain system information, e(k) is
+a noise vector.
+Aiming at sufficiently utilizing prior information, we con-
+catenate the slices of data into the form of Hankel matrix:
+Uf(k) = [uf(N) uf(N + 1) · · · uf(N + j − 1)],
+(5)
+Yf(k) = [yf(N) yf(N + 1) · · · yf(N + j − 1)],
+(6)
+Vp(k) = [vp(N) vp(N + 1) · · · vp(N + j − 1)].
+(7)
+Thus the linear regression problem (4) can be viewed as:
+Yf(k) = LvVp(k) + LuUf(k) + E(k).
+(8)
+
+After solving this linear regression problem, i.e. Lv, Lu
+being obtained, we consider an optimal control problem with
+the loss function
+J = (rf(k) − yf(k))⊤Q(rf(k) − yf(k)) + uf(k)⊤Ruf(k),
+(9)
+where Q and R are positive-definite matrices of appropriate
+shapes, rf is the reference signal. Problem (9) could be
+solved by taking derivative of J with respect to uf after
+substituting (4) to (9):
+uf(k) = (R + L⊤
+u QLu)−1L⊤
+u Q(rf − Lvvp(k)),
+(10)
+where uf(k) is a sequence of predicted control signals.
+B. Lattice-based HE
+HE schemes enable addition and/or multiplication on en-
+crypted data, which is ensured by a homomorphism between
+ciphertext space and plaintext space [18]. HE schemes can
+be divided into three categories [16]: partially HE, somewhat
+HE and fully HE. Partially HE schemes only support addition
+or multiplication. Levelled or somewhat HE schemes extend
+the functionality of partially HE and enable both addition and
+multiplication, with limited times of computation. Fully HE
+schemes allow infinite times of addition and multiplication,
+thus support evaluating arbitrary computable functions. Some
+levelled HE schemes could be converted to fully HE schemes
+with the use of a refresh algorithm called bootstrapping [19].
+In this work, we use CKKS scheme [8], [19], a typical
+public key encryption scheme which is levelled homomor-
+phic on complex vectors. CKKS scheme supports addition,
+finite times element-wise multiplication on real vectors, to
+protect the privacy of data-driven control. Besides, CKKS
+scheme utilizes key-switching technique to support advanced
+operation like element-wise vector rotation and relineariza-
+tion after multiplication. Also, CKKS scheme supports ci-
+phertext rescaling to control the noise expansion caused by
+specific operations.
+A brief description of CKKS scheme is shown in Fig.
+1. Denote N be power of 2 and QL be a big modulus
+that equals to the product of a series of positive integers
+{q0, q1, ..., qL}. In CKKS scheme, a complex vector m with
+at most N/2 elements is interpolated into a polynomial.
+Then the embedded polynomial is multiplied by a large
+scaling factor ∆ and truncated to get plaintext p, which is a
+polynomial in ZQL [X] /(XN + 1), for further encryption.
+Vector
+Plaintext Polynomial
+Ciphertext
+Vector
+Plaintext Polynomial
+Ciphertext
+Interpolation
+Evaluation
+Encryption
+Decryption
+Addition
+Multiplication
+Rotation
+……
+�/�
+��
+�
+�
+�
+��
+�
+�
+�
+�
+��
+�
+�
+��
+�
+�/�
+Fig. 1.
+A brief description of CKKS scheme.
+The plaintext p will be encrypted into the form of cipher-
+text c = (c0, c1) such that c0 + c1s = p + e (mod Ql),
+where s is the secret key and e is the error. Here, ciphertext
+c ∈ Z2
+Ql [X] /(XN + 1) is denoted to be at level l with
+Ql = �l
+i=0 qi for l = 1, ..., L + 1. The plaintext p could be
+encrypted both by the secret key s and the public key but
+could be only decrypted with the secret key. The security
+properties of CKKS scheme are ensured by the hardness of
+the RLWE problem [18]. Specifically, all the public keys
+are in the form of RLWE example (−as + e, a), where
+random polynomial a and error e safely seal the secret
+key s according to the hardness of the RLWE problem.
+Besides, extra public keys in CKKS scheme are available to
+perform advanced operations like relinearization and rotation
+to support the design of elaborated computations.
+The noise bound in ciphertexts explodes when performing
+multiple homomorphic multiplications since the noise is
+exponentially amplified by the extra scaling factor ∆. As
+shown in Fig. 2, the multiplication result c at level l could
+be rescaled by dividing ql, and the level is consequently
+reduced to l − 1. Therefore, the noise bound explosion
+could be reduced to linear expansion, which allows more
+multiplications to be performed.
+�
+�
+���
+�
+Multiplication 
+& 
+Relinearization
+Rescalation
+Fig. 2.
+Illustrated procedure of the scale limitation in CKKS scheme.
+IV. PRIVACY-PRESERVING DPCC DESIGN WITH
+DOB
+In DPCC scenarios, we assume that the public cloud envi-
+ronment and potential eavesdroppers are honest but curious,
+which means that they will perform the specified compu-
+tation or communication correctly, but they want to access
+the system information to infer the current state and system
+dynamics. Therefore, the untrusted part placed in the cloud
+should be encrypted. In this process, the encryption module
+may introduce new uncertainty. Based on this consideration,
+the DOB-based privacy-preserving DPCC solution requires
+the cooperation of three general components: public cloud,
+trustable edge and plant, respectively. The system design
+is shown in Fig. 3. In the public cloud, an encrypted con-
+troller is deployed, maintaining some encrypted matrices to
+compute encrypted control input sequences. On the trustable
+edge platform, the HE module is equipped to encrypt and
+decrypt data, along with a DOB to perform control signal
+compensation. The plant feeds the modified control input
+to the system and returns the current output to the edge
+side. The encrypted data in the cloud controller could be
+periodically updated to fit the current system dynamics.
+
+Edge
+Plant
+Public Cloud
+Encrypted Data
+Predicted control 
+inputs (encrypted)
+Historical information  
+(encrypted)
+System Dynamics
+HE Module
+DOB-based Compensator
+Encryption
+Decryption
+Compensated inputs
+Trustable
+Untrustable
+Fig. 3.
+Design of privacy-preserving DPCC.
+A. Privacy-preserving DPC
+The privacy of the system behavior, including input-output
+data, should be protected. Similar to [16], an offline privacy-
+preserving DPC solution is introduced based on CKKS
+homomorphic encryption scheme.
+We could observe that the computation of (10) is realized
+by specified matrix-vector multiplications. In practice, denote
+matrix Mr := (R + L⊤
+u QLu)−1L⊤
+u Q and Mv := (R +
+L⊤
+u QLu)−1L⊤
+u QLv, which are 2 terms in (10). Since we
+could compute Lv and Lu in advance, Mr and Mv could
+be consequently computed offline on a trustable platform,
+which could be encrypted and uploaded to the cloud, then
+updated periodically.
+Then, the cloud receives the ciphertexts of Mr and Mv,
+and the control input could be consequently computed:
+uf = Mrrf − Mvvp,
+(11)
+where vp is the same as in (3) and timestamp t is omitted
+for convenience. For the efficiency of computation, matrices
+Mr and Mv would be reused for a given interval and then
+updated, which is a trade-off in the computation overhead.
+Consequently, the computing procedure could be reduced
+to a matrix-vector multiplication in ciphertext space. Here,
+a diagonal computation method is utilized to perform the
+computation [19]. To implement the encrypted matrix-vector
+computation Mx, the matrix M ∈ RK×L and vector x ∈ RL
+should firstly be rewritten in an encryption-friendly way,
+which are illustrated in upper part of Fig. 4(a). The modified
+matrix Mmod of matrix M and repeated vector xdup =
+�
+x⊤ x⊤ ... x⊤�⊤ of x are provided, which are encrypted
+and sent to the cloud computing component.
+Denote the encrypted columns of matrix Mmod ∈ RK×L
+as M (i)
+mod, and we need to homomorphically compute matrix-
+vector multiplication y = Mx in the form of ciphertexts. The
+matrix-vector multiplication in ciphertext is shown as below:
+y =
+L−1
+�
+i=0
+M (i)
+mod ∗ rot(xdup, i),
+(12)
+where the function rot(xdup, i) is the rotation operation
+supported by the CKKS scheme, meaning that rotating vector
+xdup i steps to the left. The computation procedures are
+illustrated in Fig. 4(b).
+Based on above description, the whole encrypted matrix-
+vector computation procedure is described in Algorithm 1.
+Algorithm 1 Encryption-friendly matrix-vector multiplica-
+tion.
+Input: Matrix M ∈ Rm×n, vector x ∈ Rn.
+Output: Encrypted result of Mx.
+1: Initialization: build a full zero matrix Mmod with the
+same shape as M.
+2: for i := 0 to n − 1 do
+3:
+for j := 0 to m − 1 do
+4:
+Mmod[j][i] = M[j][(i + j) mod n].
+5:
+end for
+6: end for
+7: xdup := Encryption of
+�
+x⊤ x⊤ ... x⊤�⊤.
+8: M (0)
+mod, ... M (n−1)
+mod
+:= Encryption of Mmod’s columns
+9: Compute matrix-vector multiplication through (12).
+B. DOB and DOB-based cooperative control design
+As analyzed in III, CKKS scheme introduces error to pro-
+tect its security, meanwhile the amplification and truncation
+procedures bring error to the system. Besides, the process
+and measurement noise may also impact the control effect.
+For reducing the uncertainty and disturbance existed in HE
+scheme and system dynamics, we adopt the solution in [5],
+which uses a cloud-edge cooperative control design with a
+data-driven DOB to estimate the uncertainty and disturbance
+brought by the cloud. The estimation result obtained by
+data-driven DOB could be added to the control input for
+compensation with a proper gain.
+Assume that only the first term in the decrypted uf is fed
+to the system, which is denoted as uc, as the cloud control
+signal. We take the nominal input-output relationship into
+consideration without noise and disturbance:
+ˆy(k + 1) =
+N
+�
+i=1
+ˆgiy(k + i − N)
++
+N
+�
+i=1
+ˆhiu(k + i − N) + ˆb(k)uc(k + 1),
+(13)
+where ˆgi and ˆhis form the first block row of ˆ
+Lv and ˆ
+Lu,
+i.e. the disturbed term of Lv and Lu, respectively. (13) is
+actually the first p rows of the HE implementation of (4).
+If uncertainty and disturbance are considered, the real
+system dynamics should be:
+y(k + 1) =
+N
+�
+i=1
+ˆgiy(k + i − N)
++
+N
+�
+i=1
+ˆhiu(k + i − N) + ˆbuc(k) + ˆb(k)d(k),
+(14)
+where d(k) = ∆u(k) is the input disturbance.
+Then, a DOB is introduced with the form
+ˆd(k) = P(k) + Ky(k),
+(15)
+where the disturbance d(k) is estimated by ˆd(k), K is the
+observer amplification matrix to be designed, and P(k) is an
+
+Repeat & Concatenate
+Duplicate
+Reform
+xdup
+Mmod
+M0,0
+M0,1
+M1,0
+M1,1
+M2,0
+M2,1
+M3,0
+M3,1
+M0,2
+M1,2
+M2,2
+M3,2
+M0,0
+M0,1
+M1,0
+M1,1
+M2,0
+M2,1
+M3,0
+M3,1
+M0,2
+M1,2
+M2,2
+M3,2
+M0,0
+M0,1
+M1,0
+M1,1
+M2,0
+M2,1
+M3,0
+M3,1
+M0,2
+M1,2
+M2,2
+M3,2
+M0,0
+M0,1
+M1,1
+M1,2
+M2,2
+M2,0
+M3,0
+M3,1
+M0,2
+M1,0
+M2,1
+M3,2
+x0
+x1
+x2
+x0
+x1
+x2
+x0
+x1
+x2
+x0
+x1
+x2
+(a) Reformation of matrix and vector.
+Mul
+Mul
+Mul
+Sum
+Rotate(1)
+Rotate(1)
+M0,0
+M0,1
+M1,1
+M1,2
+M2,2
+M2,0
+M3,0
+M3,1
+M0,2
+M1,0
+M2,1
+M3,2
+*
+*
+*
+*
+*
+*
+x2
+x0
+x1
+x2
+*
+*
+x0
+x1
+x2
+x0
+x1
+x2
+x1
+x2
+x0
+x1
+x2
+*
+(Mx)0,1,2,3
+*
+(b) Matrix-vector multiplication procedure.
+Fig. 4.
+Encryption-friendly matrix-vector multiplication: an illustrative example.
+auxiliary vector which is updated as below:
+P(k + 1) = −K(
+N
+�
+i=1
+ˆgi(k)y(k + i − N)
++
+N
+�
+i=1
+ˆhi(k)u(k + i − N)
++ˆbuc(k) + ˆb ˆd(k)).
+(16)
+From (16), one can obtain
+ˆd(k + 1) = Kˆb(d(k) − ˆd(k)).
+(17)
+Now, define the estimation error as ∆d(k) = d(k) − ˆd(k)
+and we have the residue system:
+∆d(k + 1) = −Kˆb∆d(k) + d(k + 1).
+(18)
+In this system, the edge-compensated input ue is added to
+the cloud control signal uc, i.e. u = uc+ue, to get the DPCC
+cloud-edge co-design. Since the uncertainty caused by HE is
+viewed as a part of input disturbance, ue is designed to be
+ue(k) = − ˆd(k),
+(19)
+and
+ˆd(k) = K
+�
+y(k) −
+N
+�
+i=1
+ˆgi(k − 1)y(k − N + i − 1)
+−
+N−1
+�
+i=0
+ˆhi(k − 1)u(k − N + i − 1)
+− ˆb(k − 1)uc(k − 1)
+�
+(20)
+when k = N + 1, N + 2, ....
+When k = 1, 2, ..., N, the DOB-based edge compensator
+do not have enough data in the DPC stage, and ue could be
+set to 0 in this time interval, i.e. u = uc.
+V. NUMERICAL EXAMPLES
+We consider a typical 2-order discrete LTI system control
+problem with parameters
+A =
+�2
+−1
+1
+0
+�
+,
+(21)
+B =
+�1
+0
+�
+,
+(22)
+and
+C =
+�0.00014
+0.00014�
+.
+(23)
+The control input u is clipped between -0.15 and 0.15,
+and the measure output y is clipped between 0 and 0.4.
+The system parameters are: N = 20, j = 1000, K = 62,
+λ = 0.009. The system state is initialized at [0 0]⊤ and the
+whole control procedure is divided into 2 stages, i.e. data
+precollection stage and data-driven control stage. In the data
+precollection stage, the system is controlled through a PID
+controller with Kp = Kd = 9 and Ki = 3. The control
+reference is yr = 0.2 in the first 2N + j = 1040 steps.
+In the data-driven control stage, Lw and Lu are computed
+and updated periodically every 50 iterations based on newly
+collected data. In this stage, the control reference is set to
+0.1.
+The whole experiment is realized in a standard Hyper
+Elastic Cloud Server (HECS) in Huawei Cloud with 2GB
+RAM and 1 CPU. We implement the private-preserving
+part of the whole algorithm using the RLWE-based HE
+library Microsoft SEAL [20]. The security parameter λ is
+chosen to be 128-bit, meaning an encryption scheme could
+be infiltrated with a probability of 2−128. The ring dimension
+is chosen to be 4096, which controls the packing capability
+of vectors and multiplication depth. The truncation error,
+which is related to the scaling factor and modulus bits,
+influences the effect of control. The scaling factor determines
+the multiplication level, which is bounded by the 128-bit
+security requirement. The multiplication depth is chosen to
+be 2, since in this experiment only one multiplication depth is
+performed in each step. The scaling factor of CKKS scheme
+is chosen to be 222 and 225, based on which the influence of
+floating point number truncation is researched. The process
+noise and measurement noise are set to be Gaussian with the
+variance of 0.0027.
+The experiment is performed to show the control effect of
+the privacy-preserving DPCC with a DOB-based compen-
+sator in three circumstances for comparison, i.e. data-driven
+control in plaintext, data-driven control in ciphertext with
+and without DOB-based compensator.
+The experimental results are illustrated in Fig. 5(a) and
+Fig. 5(b). As shown in these figures, the DOB-based com-
+pensator effectively removes the error induced by system
+uncertainty, encryption error and external noise. Specifically,
+in Fig 5(a), the scaling factor is set to be 222, i.e. about
+
+0
+250
+500
+750
+1000 1250 1500 1750 2000 2250 2500
+Time Step
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Output
+Unencrypted without DOB
+Encrypted without DOB
+Encrypted with DOB
+Switching line
+DPC Reference
+(a) Control results with 22-bit scaling factor.
+0
+250
+500
+750
+1000 1250 1500 1750 2000 2250 2500
+Time Step
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Output
+Unencrypted without DOB
+Encrypted without DOB
+Encrypted with DOB
+Switching line
+DPC Reference
+(b) Control results with 25-bit scaling factor.
+Fig. 5.
+Simulation results of the privacy-preserving DPCC.
+4 million, which truncates too much information from the
+plaintext such that compromises the system performance.
+The system is out of control without compensation. In
+contrast, DOB-based compensator successfully compensates
+the uncertainty and disturbance, which improves the control
+quality. In Fig 5(b), the scaling factor is 8 times bigger
+than 222, reducing the truncation error by 8 times, which
+leads to a similar performance compared to the unencrypted
+and uncompensated benchmark. In this case, the uncertainty
+mainly appears in encryption and noise, which could be well
+estimated and compensated.
+VI. CONCLUSION
+In this work, we design a privacy-preserving DPCC so-
+lution. Based on HE, we implement a privacy-preserving
+cloud controller to ensure the data privacy using the CKKS
+scheme. Also, the uncertainty and disturbance in HE-based
+control systems are considered, a DOB-based compensator
+is designed on a trustable edge to estimate and compensate
+the uncertainty and disturbance. A numerical example shows
+the effect of our proposed privacy-preserving DPCC design.
+In the future, the computation efficiency problem of privacy-
+preserving cloud control solutions would be studied.
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+tification method based on containerised cloud workflow processing
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+Mar 2022. Microsoft Research, Redmond, WA.
+

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+arXiv:2301.02430v1  [math.GM]  6 Jan 2023
+Some Solitons on Homogeneous Almost
+α-Cosymplectic 3-Manifolds and Harmonic
+Manifolds
+Naeem Ahmad Pundeer, Paritosh Ghosh, Hemangi Madhusudan
+Shah and Arindam Bhattacharyya
+Abstract. In this paper, we investigate the nature of Einstein solitons,
+whether it is steady, shrinking or expanding on almost α-cosymplectic
+3-manifolds. We also prove that a simply connected homogeneous al-
+most α-cosymplectic 3-manifold, admitting a contact Einstein soliton,
+is an unimodular semidirect product Lie group. Finally, we show that a
+harmonic manifold admits a Ricci soliton if and only if it is flat.
+Mathematics Subject Classification (2010). 53B40, 58B20, 53C25,
+53D15.
+Keywords. Almost α-cosymplectic manifold, Harmonic manifold, Ricci
+soliton, Einstein soliton.
+1. Introduction
+The study of solitons, in particular Ricci solitons, on Riemannian man-
+ifolds play a vital role in understanding the geometry of underlying mani-
+fold. It is very interesting to study Ricci and Einstein solitons on almost α-
+cosymplectic 3-manifolds. Recently, Jin and Ximin [9] showed that a simply
+connected homogeneous almost α-cosymplectic 3-manifold, admitting con-
+tact Ricci solitons, is cosymplectic; and the manifold under consideration is
+an unimodular semidirect product Lie group R2⋊AR, where A =
+�
+0
+b
+−b
+0
+�
+,
+equipped with a flat left invariant cosymplectic structure.
+Motivated by this result we show in this paper that, if a simply con-
+nected homogeneous almost α-cosymplectic 3-manifold, with some additional
+hypothesis, admits a contact Einstein soliton, then the manifold is an uni-
+modular semidirect product Lie group G of type G0bb = R2 ⋊A R, where
+
+2
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+A =
+�
+0
+b
+−b
+0
+�
+̸= 0. And also G is the Lie group ˜E2 equipped with its
+flat left invariant cosymplectic structure (see Corrollary 3.5). In order to
+prove this result, we first obtain a characterization of almost α-cosymplectic
+3-manifold admitting contact Einstein solitons, which is the main theorem
+(Theorem 3.4) of Section 3. To establish this aforementioned theorem we
+derive an identity (Lemma 3.3) involving scalar curvature, Lie derivative of
+the metric and Ricci operator on a Riemannian manifold admitting Einstein
+soliton. We also give some conditions on α for contact Einstein solitons to
+be steady, shrinking or expanding on almost α-cosymplectic 3-manifolds (see
+Theorem 3.1).
+Another interesting topic in the differential geometry is the geometry
+of harmonic manifolds. In 1965, Tashiro [19] showed that if a complete Rie-
+mannian manifold admits a Gaussian, then it is either flat or a complete
+warped product manifold or a sphere. In this paper, we show that a har-
+monic manifold admits a Gaussian if and only if it is flat; thus confirming
+Tashiro’s result for harmonic manifolds. Moreover, we prove that flat har-
+monic manifold admits Ricci solitons of steady, expanding or shrinking type.
+We also determine the corresponding potential function. In fact, Busemann
+function on Rn turns to be the potential function in case of steady solitons
+(see Theorem 4.1 of Section 4).
+The paper is divided into four sections. Section 2 is devoted to the
+preliminaries about Ricci soliton, Einstein soliton, almost α-cosymplectic 3-
+manifolds and harmonic manifolds. In Section 3, we prove our main results
+on almost α-cosymplectic 3-manifold admitting contact Einstein solitons, as
+stated above. In the last section, we prove the main flatness result about
+harmonic manifolds admitting Ricci solitons.
+2. Preliminaries
+In this section, we discuss some notions required to prove the results of this
+paper.
+2.1. Ricci solitons
+Ricci solitons are the self similar solutions of the Ricci flow. The concept of
+Ricci flow was first introduced by Hamilton [7] in (1982), motivated by the
+work of Eells and Sampson [6] on harmonic map and the flow was given by
+the equation
+∂g
+∂t = −2S,
+where S is the Ricci tensor.
+Ricci solitons are the generalizations of the Einstein metrics and are the
+solutions of the equation
+Ric(g) + 1
+2LXg = λg,
+(1)
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+3
+where Ric(X, Y ) = S(X, Y ) is the Ricci curvature tensor, LX is the Lie
+derivative along the direction of the vector field X and λ is a real constant.
+The soliton is said to be shrinking if λ > 0, steady if λ = 0 and expanding if
+λ < 0.
+Tashiro [15] proved very important result for complete Einstein manifolds
+admitting Ricci solitons.
+Theorem 2.1. [15] Let (M, g) be a complete Riemannian n-manifold admit-
+ting a nontrivial function f such that Hess f = λg, then (M, g) is isometric
+to a complete warped product metric and must have one of the three forms:
+1. M = R × N, g = dr2 + ρ2(r)gN,
+2. M = Rn, g = dr2 + ρ2(r)ds2
+n−1, r ≥ 0,
+3. M = Sn, g = dr2 + ρ2(r)ds2
+n−1, r ∈ [a, b].
+2.2. Einstein solitons
+The Einstein solitons are the generalization of the Ricci solitons, was first
+introduced by Catino and Mazzieri [3] in (2016). They are the solutions of
+the equation
+LV g + 2S = (2λ + r)g,
+(2)
+where, Ricci tensor S(X, Y ) = g(X, QY ), Q being the Ricci operator, r is
+the scalar curvature, λ ∈ R is a constant and V is known as potential vector
+field.
+Einstein solitons are the self-similar solutions of the Einstein flow,
+∂
+∂tg + 2S = rg.
+It is said to be steady if λ = 0, shrinking if λ > 0 and expanding if λ < 0.
+2.3. Almost contact metric manifolds
+In order to define contact metric manifolds, we need the concept of Reeb
+vector field.
+Reeb vector field [2]: A global vector field ξ on a contact manifold M 2n+1,
+equipped with a global 1-form η, is called Reeb vector field or characteristic
+vector field, if any vector field X satisfies η(ξ) = 1 and dη(X, ξ) = 0.
+Almost contact manifold [2]: Let M be a Riemannian manifold of di-
+mension (2n + 1), n ≥ 1. M 2n+1 is said to have an almost contact structure
+(ϕ, ξ, η), if there exists a (1, 1)-tensor ϕ, a global vector field ξ and a 1-form
+η such that
+ϕ2X = −X + η(X)ξ, η(ξ) = 1,
+(3)
+for any vector field X on M, where ξ is the Reeb vector field. The manifold
+M equipped with the structure (ϕ, ξ, η) is called an almost contact manifold.
+Almost contact metric manifold [2]: A Riemannian metric g is said to
+be compatible with an almost contact structure (ϕ, ξ, η), if
+g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ),
+(4)
+
+4
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+holds for any X, Y ∈ χ(M) and (M, ϕ, ξ, η, g) is called an almost contact
+metric manifold.
+Normal almost contact metric manifold [2]: An almost contact met-
+ric manifold is said to be normal, if for any X, Y ∈ χ(M) the tensor field
+N = [ϕ, ϕ]+ 2dη ⊗ ξ vanishes everywhere on the manifold, where [ϕ, ϕ] is the
+Nijenhuis tensor of ϕ.
+Homogeneous almost contact metric manifold [9]: An almost contact
+metric manifold (M, ϕ, ξ, η, g) is said to be homogeneous, if there exists a con-
+nected Lie group G of isometries acting transitively on M leaving η invariant.
+2.4. Cosymplectic manifolds
+A (2n + 1)-dimensional manifold is said to be a cosymplectic manifold [10],
+if it admits a closed, 1-form η and 2-form Φ such that η ∧ Φn is a volume
+element, where Φ(X, Y ) = g(ϕX, Y ) is a 2-form on M 2n+1.
+Almost cosymplectic manifold [10]: If η and Φ are not closed but η ∧ Φn
+is a volume form, then the manifold is called almost cosymplectic manifold.
+α-cosymplectic manifold [12]: An almost cosymplectic manifold is said
+to be α-cosymplectic if dη = 0 and dΦ = 2αη ∧ Φ for some constant α.
+Almost α-cosymplectic manifold [10]: An almost α-cosymplectic manifold
+is defined as an almost contact metric manifold with dη = 0 and dΦ = 2αη∧Φ,
+for any constant α. In particular, the almost α-cosymplectic manifold is
+• almost α-Kenmotsu if α ̸= 0,
+• almost cosymplectic if α = 0,
+• almost Kenmotsu if α = 1.
+Harmonic vector field [14]: A characteristic vector field ξ on an almost
+α-cosymplectic manifold is harmonic if and only if ξ is an eigenvector field
+of the Ricci operator Q.
+2.5. Almost α-cosymplectic 3-manifold
+In this article, we will mainly focus on 3-dimensional almost α-cosymplectic
+manifold. In what follows, we will be using the following results.
+Theorem 2.2. [12] An almost α-cosymplectic 3-manifold is α-cosymplectic
+if and only if Lξh = 0, where h = 1
+2Lξϕ.
+Any almost α-cosymplectic 3-manifold satisfies important relationships be-
+tween Φ, ξ and h.
+Lemma 2.3. [12] Let M 2n+1 be an almost α-cosymplectic 3-manifold, then
+we have,
+∇ξϕ = 0, ∇ξ = 0, hϕ + ϕh = 0, hξ = 0,
+(5)
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+5
+with
+∇Xξ = −αϕ2X − ϕhX.
+(6)
+We would require some identities on the ϕ-bases [2] and the following table
+of the Levi-Civita connection.
+Proposition 2.4. [12] On almost α-cosymplectic 3-manifold, there exists
+ϕ-bases satisfying
+he = σe, hϕe = −σϕe, hξ = 0,
+with σ a local smooth eigen-function of h.
+Theorem 2.5. [12] The Levi-Civita connection on almost α-cosymplectic
+3-manifold are given by,
+
+
+
+
+
+∇ee = −aϕe − αξ, ∇ϕee = −bϕe + σξ, ∇ξe = µϕe,
+∇eϕe = ae + σξ, ∇ϕeϕe = be − αξ, ∇ξϕe = −µe,
+∇eξ = αe − σϕe, ∇ϕeξ = −σe + αϕe, ∇ξξ = 0,
+(7)
+where a = g(∇eϕe, e), b = −g(∇ϕee, ϕe) and µ = g(∇ξe, ϕe) are smooth
+functions.
+The Ricci operator on almost α-cosymplectic 3-manifold is known explicitly
+[12].
+Proposition 2.6. [12] The Ricci operator Q on almost α-cosymplectic 3-
+manifold is given by,
+
+
+
+
+
+Qξ = −(2α2 + tr h2)ξ + (2bσ − e(σ))ϕe − (2aσ + (ϕe)(σ))e,
+Qϕe = (2bσ − e(σ))ξ + (α2 + r
+2 + tr h2
+2
++ 2σµ)ϕe + (ξ(σ) + 2ασ)e,
+Qe = −(2aσ + (ϕe)(σ))ξ + (ξ(σ) + 2ασ)ϕe + (α2 + r
+2 + tr h2
+2
+− 2σµ)e.
+(8)
+Furthermore, the scalar curvature r = tr Q is given by
+r = −6α2 − tr h2 − 2(a2 + b2) − 2(ϕe)(a) + 2e(b).
+(9)
+The structure of simply-connected, homogeneous almost α-cosymplectic 3-
+manifold, admitting a contact Ricci soliton, is very well known.
+Theorem 2.7. [9] Let M be a simply-connected, homogeneous almost α-
+cosymplectic 3-manifold admitting a contact Ricci soliton. Then M is an
+unimodular semidirect product Lie group G of type G0bb = R2 ⋊A R, where
+A =
+�
+0
+b
+−b
+0
+�
+, equipped with a flat left invariant cosymplectic structure.
+Moreover, we have the following:
+1. If A = 0, i.e., b = 0, G is the abelian Lie group R3 equipped with its flat
+left invariant cosymplectic structure.
+2. If A ̸= 0, i.e., b ̸= 0, G is the Lie group ˜E2 equipped with its flat left
+invariant cosymplectic structure.
+
+6
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+2.6. Harmonic manifolds
+A complete Riemannian manifold (M n, g) is said to be harmonic, if for any
+p ∈ M, the volume density ωp(q) =
+�
+det(gij(q)) in normal coordinates,
+centered at any p ∈ M is a radial function [1]. Thus,
+Θ(r) = rn−1�
+det(gij(q))
+is density of geodesic sphere, is a radial function. It is known that harmonic
+manifolds are Einstein [1]. They are naturally classified as per the sign of the
+Ricci constant. Let r be the constant scalar curvature of M.
+• If r = 0, then M is flat, that is (M, g) = (Rn, Can) (Lemma 4.4).
+• If r > 0, then by Bonnet-Myer’s theorem M is compact with finite
+fundamental group. They are compact rank one symmetric spaces by a
+well known result of Szabo (cf. [18]).
+• If r < 0, then M is non-compact harmonic manifold. They are rank one
+symmetric spaces of non-compact type, if dimension of M is atmost 5.
+The main result in the theory of harmonic spaces is the Lichnerowicz
+Conjecture: Any simply connected, complete harmonic manifold is either flat
+or a rank one symmetric space. By the above classification, we see that the
+conjecture is resolved for compact harmonic manifolds and is open for non-
+compact harmonic manifolds of dimension 6. There are counter examples to
+the conjecture when dimension is atleast 7, known as the Damek-Ricci spaces
+or NA spaces. See for more details references in [18].
+In the category of non-compact harmonic manifolds, we will be con-
+sidering simply connected, complete, non-compact harmonic manifolds. It
+follows that, these spaces don’t have conjugate points (cf. [18]). Hence, by
+the Cartan-Hadamard theorem,
+expp : TpM → M
+is a diffeomorphism and every geodesic of M is a line. That is, if γv : R → M
+is a geodesic of M with v ∈ SpM, γ′
+v(0) = v, then d(γv(t), γv(s)) = |t − s|.
+Busemann function: Let γv be a geodesic line, then the two Busemann
+functions associated to γv are defined as [15]:
+b+
+v (x) = lim
+t→∞ d(x, γv(t)) − t,
+b−
+v (x) =
+lim
+t→−∞ d(x, γv(t)) − t.
+3. Einstein Solitons on Almost α-Cosymplectic
+3-Manifolds
+In this section, we examine the nature of a contact Einstein soliton on al-
+most α-cosymplectic manifold. We also show that, the characteristic vector
+field ξ is harmonic on almost α-cosymplectic 3-manifold admitting a contact
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+7
+Einstein soliton. Finally, we generalize Theorem 2.7 using these results.
+Contact Einstein soliton: Let (M 2n+1, g) be a Riemannian manifold of
+dimension 2n + 1 (n ≥ 1). Consider the Einstein soliton (2), with potential
+vector field V , on an almost contact metric manifold (M, ϕ, ξ, η, g). Then the
+soliton is called contact Einstein soliton, if V = ξ that is, the potential vector
+field is the characteristic vector field.
+The potential vector field V is called transversal, if it is orthogonal to the
+characteristic vector field, that is V ⊥ ξ.
+Theorem 3.1. Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold,
+admitting a contact Einstein soliton. Then the soliton is:
+1. steady, if α2 = σ2 − (a2 + b2) − (ϕe)(a) + e(b),
+2. shrinking, if α2 > σ2 − (a2 + b2) − (ϕe)(a) + e(b),
+3. expanding, if α2 < σ2 − (a2 + b2) − (ϕe)(a) + e(b).
+Proof. If the soliton is contact Einstein soliton, using V = ξ in (2), we have
+g(∇Xξ, Y ) + g(X, ∇Y ξ) + 2g(X, QY ) = (2λ + r)g(X, Y ),
+(10)
+for any vector fields X, Y on M.
+Substituting X = Y = ξ in the above equation and using (8), we obtain
+λ = −2α2 − 2σ2 − r
+2.
+(11)
+From the expression of r (9), we get
+λ = α2 − σ2 + (a2 + b2) + (ϕe)(a) − e(b),
+(12)
+from which we can conclude the proof.
+□
+Theorem 3.2. Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold,
+admitting a contact Einstein soliton. Then the characteristic vector field ξ is
+harmonic.
+Proof. From (10), we get for X = ξ and Y = e,
+(ϕe)(σ) = −2aσ.
+(13)
+And for X = ξ and Y = ϕe, from (10) we have
+e(σ) = 2bσ.
+(14)
+Now, using (13) and (14) in the expression of Qξ in (8), we obtain
+Qξ = −(2α2 + 2σ2)ξ,
+which shows that ξ is an eigenvector field of the Ricci operator Q concluding
+the fact that ξ is harmonic.
+□
+We derive the identity involving the Lie derivative of the metric, Ricci oper-
+ator, the potential vector field V .
+
+8
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+Lemma 3.3. Let (M, g) be a Riemannian manifold of scalar curvature r,
+admitting an Einstein soliton (2). Then
+∥LV g∥2 = 2V (r) + 4 div
+��
+λ + r
+2
+�
+V − QV
+�
+,
+(15)
+where Q is the Ricci operator.
+Proof. In local coordinate system, (2) leads to
+LV gij + Sij = (2λ + r)gij.
+Therefore,
+∥LV g∥2 = − SijLV gij + (2λ + r)gijLV gij.
+= − LV r + gijLV Sij − (2λ + r)gijLV gij.
+(16)
+Now,
+gijLV Sij =gij∇V Sij − gij∇αViSαj − gij∇αVjSiα
+=2V (r) − 2 div QV.
+(17)
+Observing that gijLV gij = −2 div V and using (16) and (17), we get the
+required result.
+□
+Now we derive the main result of this section.
+Theorem 3.4. Consider M to be an almost α-cosymplectic 3-manifold, ad-
+mitting a contact Einstein soliton. Then the following hold.
+1. If σ ̸= 0, then α = a2 + b2 − 2λ2 + (ϕe)(a) − e(b).
+2. If σ = 0, then M is cosymplectic.
+Proof. Replacing X by e and Y by ϕe, from (10) we get
+g(∇eξ, ϕe) + g(e, ∇ϕeξ) + 2g(e, Qϕe) = (2λ + r)g(e, ϕe).
+Using (7) and (8), after simplification we acquire,
+ξ(σ) = σ − 2ασ.
+(18)
+Now putting X = e = Y in (10) and using (7), (8), (9) and (12), we get
+6α2 + 6σ2 − 4σµ + 2α + r = 0.
+(19)
+Similarly, putting X = ϕe = Y in (10) and using (7), (8), (9) and (12), we
+also obtain
+6α2 + 6σ2 + 4σµ + 2α + r = 0.
+(20)
+So comparing (19) and (20), we have σµ = 0. If σ ̸= 0, then from (20), we
+obtain the required result using (9).
+Now suppose σ = 0, then M is α-cosymplectic. From [12], recall that an
+almost α-cosymplectic manifold M is α-cosymplectic if and only if for any
+X ∈ χ(M),
+QX =
+�
+α2 + r
+2
+�
+X −
+�
+3α2 + r
+2
+�
+η(X)ξ.
+(21)
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+9
+Since ∇ξ is symmetric, (10) becomes
+g(∇Xξ, Y ) + g(X, QY ) =
+�
+λ + r
+2
+�
+g(X, Y ).
+(22)
+Using (6) and (21), we have from (22), for any X, Y ∈ χ(M),
+(α2 + α − λ)g(X, Y ) −
+�
+3α2 + α + r
+2
+�
+η(X)η(Y ) = 0,
+which implies α2 + α − λ = 0 and 3α2 + α + r
+2 = 0.
+That is λ = α2 + α and r = −6α2 − 2α = constant, so that, λ + r
+2 = −2α2.
+Also, from (21), we have Qξ = −2α2ξ which implies (λ + r
+2)ξ − Qξ = 0.
+Therefore, using Lemma 3.3 (15), we can say that ξ is a Killing vector field,
+that is, ∇ξ is skew-symmetric. But in our case ∇ξ is symmetric, which implies
+∇ξ = 0, that is, α = 0, proving the fact that M is cosymplectic.
+□
+Corollary 3.5. Consider M to be a simply-connected, homogeneous, almost
+α-cosymplectic 3-manifold, admitting a contact Einstein soliton with σ = 0.
+Then M is an unimodular semidirect product Lie group G of type G0µµ =
+R2 ⋊A R, where A =
+�
+0
+µ
+−µ
+0
+�
+̸= 0, is a real matrix. Moreover, G is the
+Lie group ˜E2 equipped with its flat left invariant cosymplectic structure.
+Proof. The proof follows from Theorem 2.7 and Theorem 3.4.
+□
+4. Ricci Solitons on Harmonic Manifolds
+In this section, we study Ricci solitons on complete, simply connected, har-
+monic manifolds. We prove a Lichnerowicz type result that, a harmonic man-
+ifold admits a Ricci soliton if and only if M is flat. More precisely, we show
+that compact harmonic manifolds and non-flat harmonic manifolds do not
+admit Ricci solitons. But flat harmonic manifold do admit steady, shrinking,
+expanding Ricci solitons.
+In the sequel, harmonic manifold means complete, simply connected harmonic
+manifold. The main theorem of this section is:
+Theorem 4.1. Let (M, g) be a harmonic manifold. Then M admits Ricci
+soliton if and only if M is flat. In this case, the steady Ricci soliton is Killing
+given by X = ∇bv
+−; where b−
+v (x) = −⟨x, v⟩, the Busemann function, is a
+potential function on M. In case, the Ricci soliton is shrinking or expanding,
+the potential function is given by f(x) = λd(x, p)2 +f(p), for constant λ ̸= 0;
+and point p is minimum or maximum of f and X = ∇f is the corresponding
+Ricci soliton.
+We begin with the following important proposition.
+
+10
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+Proposition 4.2. Every Ricci soliton is a gradient soliton on complete man-
+ifold. Hence, in particular on any harmonic manifold Ricci soliton is a gra-
+dient soliton. Consequently, any harmonic manifold admits a Gaussian.
+Proof. Perelman showed that, Ricci soliton on any complete manifold is al-
+ways a gradient soliton [11]. Hence, in this case X = ∇f, for some smooth
+function f : M → R. As L∇fg = ∇2f, (1) reduces to
+Ric + 1
+2∇2f = λg.
+(23)
+As (M, g) is harmonic and hence Einstein, then it follows that
+∇2f = 2(λ − r)g,
+(24)
+where r is a constant scalar curvature of M. Thus f is a Gaussian, that is it
+satisifes (24).
+□
+Remark 4.3. Note that because any harmonic manifold is Einstein, trivial
+solitons X = 0 and X a Killing vector field are solutions of (1) with λ = r.
+Lemma 4.4. Ricci flat harmonic manifold is flat.
+Proof. It can be shown that any harmonic manifold (M, g) is asymptotically
+harmonic [18]. That is there exists a constant h ≥ 0 such that
+∆bv
+± = h.
+Let L = ∇2bv
++ denote the second fundamental form of horospheres, b−1
+v (t).
+Then L satisfies the Riccati equation, that is for x ∈ v⊥,
+L′(x) + L2(x) + R(x, v)v = 0.
+Tracing the above equation, we obtain that tr L2 = 0, as Ricci(v, v) = 0. But
+L is a symmetric operator on v⊥. This implies that L = 0 for any v ∈ SM.
+Consequently, R(x, v)v = 0 for any x ∈ v⊥. Thus (M, g) is flat.
+□
+Lemma 4.5. Let X = ∇f be a Killing vector field on compact harmonic
+manifold, then X is trivial. Solitons of Killing type do not exist on non-
+compact, non-flat harmonic manifold. On flat harmonic manifold, Killing
+vector field is X = ∇bv
+−, where b−
+v (x) = −⟨x, v⟩ is a Busemann function on
+Rn.
+Proof. Because X = ∇f is a non-trivial Killing vector field, we have
+∇2f = 0.
+Therefore, ∥∇f∥ = constant ̸= 0, consequently, f has no critical points.
+Any Killing vector field of constant norm satisfies (p. 164-167, [15]):
+∥∇2f∥
+2 = Ric(∇f, ∇f).
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+11
+Therefore,
+0 =∥∇2f∥ = r∥∇f∥2
+This implies that for f non-constant, r = 0 and therefore Ric ≡ 0 and hence
+harmonic manifold must be flat (Lemma 4.4).
+We have ∥∇f∥ = constant. We may assume that ∥∇f∥ = 1, therefore f is
+distance function which is harmonic function on (Rn, Can). By Proposition
+5.1 of [18], it follows that
+f(x) = b−
+v (x) = −⟨x, v⟩,
+is a Busemann function on Rn [15].
+If M is compact, ∇2f = 0 implies that f is a harmonic function. Hence, f
+must be a constant function.
+□
+Proposition 4.6. Let (M, g) be a compact harmonic manifold, then a Ricci
+soliton on M is trivial.
+Proof. We have,
+∇2f = 2(λ − r)g.
+Therefore, ∆f = 2(λ − r)n implies by the Bochner’s formula that,
+1
+2∆(∥∇f∥2) = 4(λ − r)2n2 + r(∥∇f∥2).
+(25)
+Therefore,
+4(λ − r)2n2 Vol(M) = −r
+�
+M
+∥∇f∥2 < 0.
+This implies that ∥∇f∥ = 0, therefore f is constant.
+□
+Lemma 4.7. Let (M, g) be a non-compact, non-flat harmonic manifold.
+Then Ricci solitons on M don’t exist. In case, (λ − r) ̸= 0, implies that
+M is flat and r = 0. In this case the potential function is given by f(x) =
+λd(p, x)2 + f(p), for some p ∈ M.
+Proof. We have,
+∇2f = 2(λ − r)g.
+Therefore, f is either convex or concave function. Consequently, the only
+possible critical point of f is either maximum or minimum of f. Suppose
+that p is a critical point of f. Note that along any unit speed geodesic of M
+starting from p,
+f ′′(t) = 2(λ − r).
+(26)
+Therefore, f ′(t) = 2(λ − r)t + c. Hence, there is exactly one critical point,
+and hence c = 0. Thus, f(t) = (λ − r)t2 + f(p), consequently f is a radial
+function. This implies that,
+∆f = f ′′ + Θ′
+Θ f ′ = 2(λ − r)n.
+
+12
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+Therefore,
+f ′′ + Θ′
+Θ 2(λ − r)t = 2(λ − r)n.
+Consequently by (26),
+Θ′(t)
+Θ(t) = n − 1
+t
+.
+Comparing with the series expansion (see (4.4) of [18]),
+Θ′(t)
+Θ(t) = n − 1
+r
+− r
+3 + · · · ,
+we obtain r = 0, hence M is flat. Finally, f(x) = λd(p, x)2 + f(p) follows
+from section 1 of [4].
+□
+Finally we come to the proof of Theorem 4.1.
+Proof of Theorem 4.1: If M is compact, then the Ricci soliton on M is
+trivial (Proposition 4.6). If (λ−r) = 0, then M is flat and X = ∇bv
+− (Lemma
+4.5). If (λ−r) ̸= 0, then M is flat, and X = ∇f, where f(x) = λd(p, x)2+f(p),
+for some p ∈ M (Lemma 4.7).
+□
+Remark: We have shown that Theorem 4.1 confirms Theorem 2.1 in case of
+harmonic manifolds. Also Theorem 4.1 implies that there are no non-trivial
+deformation of non-flat harmonic manifolds. This indicates a result support-
+ing the conjecture that, there are no non-trivial deformations of harmonic
+manifolds; and hence there should be only finitely many classes of harmonic
+manifolds.
+5. Acknowledgements
+Dr. Naeem Ahmad Pundeer would like to thank to U.G.C. for its Dr. D.S.
+Kothari Postdoctoral Fellowship. The corresponding author, Mr. Paritosh
+Ghosh, thanks UGC Junior Research Fellowship of India. The authors also
+would like to thank Mr. Dipen Ganguly for his wishful help in this research.
+References
+[1] Besse, A.L. Manifolds all of whose geodesics are closed, Berlin Heidel-
+berg, Springer-Verlag, (1978).
+[2] Blair, D.E. Riemannian geometry of contact and symplectic manifolds,
+Progress in Mathematics, Birkh¨auser, New York, (2010).
+[3] Catino, G. and Mazzieri, L. Gradient Einstein solitons, Nonlinear
+Anal., 132, 66–94, (2016).
+[4] Cheeger, J. and Colding, T. Lower bounds on Ricci curvature and
+the almost rigidity of warped products, Ann. Math., 144(1), 189-237,
+(1996).
+
+On harmonic and homogeneous almost α-cosymplectic 3-manifolds
+13
+[5] Cunha, A.W. and Griffin, E. On non-compact gradient solitons,
+arXiv:2207.05822, (2022).
+[6] Eells, J. and Sampson, J.H. Harmonic Mappings of Riemannian Man-
+ifolds, Amer. J. Math., 86, 109-160, (1964).
+[7] Hamilton, R.S. Three manifolds with positive Ricci curvature, J. Diff.
+Geom., 17, 255-306, (1982).
+[8] Hu, Q., Xu, G. and Yu, C. The rigidity and stability of gradient esti-
+mates, J. Geom. Anal., 32, 1-13, (2022).
+[9] Li, J. and Liu, X. Ricci solitons on homogeneous almost α-cosymplectic
+three-Manifolds, Mediterr. J. Math., 19, 1-12, (2022).
+[10] Libermann, P. Sur les automorphismes infinit´esimaux des structures
+symplectiques et des structures de contact, Colloque G´eom. Diff. Glob-
+ale, 37–59, (1959).
+[11] Perelman, G. The entropy formula for the Ricci flow and its geometric
+applications, arXiv:math 211159, (2002).
+[12] Perrone, D. Classification of homogeneous almost α-coK¨ahler three-
+manifolds, Diff. Geom. Appl., 59, 66–90, (2018).
+[13] Perrone, D. Classification of homogeneous almost cosymplectic three
+manifolds, Diff. Geom. Appl., 30, 49–58, (2012).
+[14] Perrone, D. Left-invariant almost α-co K¨ahler structures on 3D semidi-
+rect product Lie groups, Int. J. Geom. Meth. Mod. Phys., 16, 1-18,
+(2018).
+[15] Petersen, P. Riemannian geometry, New York, Springer-Verlag, (2006).
+[16] Petersen, P. and Wylie, W. Rigidity of gradient Ricci solitons, Pac. J.
+Math., 241, 329-345, (2009).
+[17] Ranjan, A. and Shah, H. Harmonic manifolds with minimal horo-
+spheres, J. Geom. Anal., 12, 683-694, (2002).
+[18] Ranjan, A. and Shah, H. Busemann functions in a harmonic manifold,
+Geom. Dedicata, 101, 167-183, (2003).
+[19] Tashiro, Y. Complete Riemannian manifolds and some vector fields,
+Trans. Amer. Math. Soc., 117, 251-275, (1965).
+Naeem Ahmad Pundeer
+Department of Mathematics
+Jadavpur University
+Kolkata-700032, India.
+e-mail: [email protected]
+Paritosh Ghosh
+Department of Mathematics
+Jadavpur University
+Kolkata-700032, India.
+e-mail: [email protected]
+
+14
+N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya
+Hemangi Madhusudan Shah
+Harish-Chandra Research Institute
+A CI of Homi Bhabha National Institute
+Chhatnag Road, Jhunsi, Prayagraj-211019, India.
+e-mail: [email protected]
+Arindam Bhattacharyya
+Department of Mathematics
+Jadavpur University
+Kolkata-700032, India
+e-mail: [email protected]
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='02430v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='GM]  6 Jan 2023  Some Solitons on Homogeneous Almost  α-Cosymplectic 3-Manifolds and Harmonic  Manifolds  Naeem Ahmad Pundeer, Paritosh Ghosh, Hemangi Madhusudan  Shah and Arindam Bhattacharyya  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In this paper, we investigate the nature of Einstein solitons,  whether it is steady, shrinking or expanding on almost α-cosymplectic  3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We also prove that a simply connected homogeneous al-  most α-cosymplectic 3-manifold, admitting a contact Einstein soliton,  is an unimodular semidirect product Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Finally, we show that a  harmonic manifold admits a Ricci soliton if and only if it is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' 53B40, 58B20, 53C25,  53D15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Almost α-cosymplectic manifold, Harmonic manifold, Ricci  soliton, Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Introduction  The study of solitons, in particular Ricci solitons, on Riemannian man-  ifolds play a vital role in understanding the geometry of underlying mani-  fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' It is very interesting to study Ricci and Einstein solitons on almost α-  cosymplectic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Recently, Jin and Ximin [9] showed that a simply  connected homogeneous almost α-cosymplectic 3-manifold, admitting con-  tact Ricci solitons, is cosymplectic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' and the manifold under consideration is  an unimodular semidirect product Lie group R2⋊AR, where A =  �  0  b  −b  0  �  ,  equipped with a flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Motivated by this result we show in this paper that, if a simply con-  nected homogeneous almost α-cosymplectic 3-manifold, with some additional  hypothesis, admits a contact Einstein soliton, then the manifold is an uni-  modular semidirect product Lie group G of type G0bb = R2 ⋊A R, where  2  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  A =  �  0  b  −b  0  �  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' And also G is the Lie group ˜E2 equipped with its  flat left invariant cosymplectic structure (see Corrollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In order to  prove this result, we first obtain a characterization of almost α-cosymplectic  3-manifold admitting contact Einstein solitons, which is the main theorem  (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4) of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' To establish this aforementioned theorem we  derive an identity (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3) involving scalar curvature, Lie derivative of  the metric and Ricci operator on a Riemannian manifold admitting Einstein  soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We also give some conditions on α for contact Einstein solitons to  be steady, shrinking or expanding on almost α-cosymplectic 3-manifolds (see  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Another interesting topic in the differential geometry is the geometry  of harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In 1965, Tashiro [19] showed that if a complete Rie-  mannian manifold admits a Gaussian, then it is either flat or a complete  warped product manifold or a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In this paper, we show that a har-  monic manifold admits a Gaussian if and only if it is flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' thus confirming  Tashiro’s result for harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Moreover, we prove that flat har-  monic manifold admits Ricci solitons of steady, expanding or shrinking type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  We also determine the corresponding potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In fact, Busemann  function on Rn turns to be the potential function in case of steady solitons  (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1 of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  The paper is divided into four sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Section 2 is devoted to the  preliminaries about Ricci soliton, Einstein soliton, almost α-cosymplectic 3-  manifolds and harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In Section 3, we prove our main results  on almost α-cosymplectic 3-manifold admitting contact Einstein solitons, as  stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In the last section, we prove the main flatness result about  harmonic manifolds admitting Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Preliminaries  In this section, we discuss some notions required to prove the results of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ricci solitons  Ricci solitons are the self similar solutions of the Ricci flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The concept of  Ricci flow was first introduced by Hamilton [7] in (1982), motivated by the  work of Eells and Sampson [6] on harmonic map and the flow was given by  the equation  ∂g  ∂t = −2S,  where S is the Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Ricci solitons are the generalizations of the Einstein metrics and are the  solutions of the equation  Ric(g) + 1  2LXg = λg,  (1)  On harmonic and homogeneous almost α-cosymplectic 3-manifolds  3  where Ric(X, Y ) = S(X, Y ) is the Ricci curvature tensor, LX is the Lie  derivative along the direction of the vector field X and λ is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  The soliton is said to be shrinking if λ > 0, steady if λ = 0 and expanding if  λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Tashiro [15] proved very important result for complete Einstein manifolds  admitting Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [15] Let (M, g) be a complete Riemannian n-manifold admit-  ting a nontrivial function f such that Hess f = λg, then (M, g) is isometric  to a complete warped product metric and must have one of the three forms:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M = R × N, g = dr2 + ρ2(r)gN,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M = Rn, g = dr2 + ρ2(r)ds2  n−1, r ≥ 0,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M = Sn, g = dr2 + ρ2(r)ds2  n−1, r ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Einstein solitons  The Einstein solitons are the generalization of the Ricci solitons, was first  introduced by Catino and Mazzieri [3] in (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' They are the solutions of  the equation  LV g + 2S = (2λ + r)g,  (2)  where, Ricci tensor S(X, Y ) = g(X, QY ), Q being the Ricci operator, r is  the scalar curvature, λ ∈ R is a constant and V is known as potential vector  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Einstein solitons are the self-similar solutions of the Einstein flow,  ∂  ∂tg + 2S = rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  It is said to be steady if λ = 0, shrinking if λ > 0 and expanding if λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Almost contact metric manifolds  In order to define contact metric manifolds, we need the concept of Reeb  vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Reeb vector field [2]: A global vector field ξ on a contact manifold M 2n+1,  equipped with a global 1-form η, is called Reeb vector field or characteristic  vector field, if any vector field X satisfies η(ξ) = 1 and dη(X, ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Almost contact manifold [2]: Let M be a Riemannian manifold of di-  mension (2n + 1), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M 2n+1 is said to have an almost contact structure  (ϕ, ξ, η), if there exists a (1, 1)-tensor ϕ, a global vector field ξ and a 1-form  η such that  ϕ2X = −X + η(X)ξ, η(ξ) = 1,  (3)  for any vector field X on M, where ξ is the Reeb vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The manifold  M equipped with the structure (ϕ, ξ, η) is called an almost contact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Almost contact metric manifold [2]: A Riemannian metric g is said to  be compatible with an almost contact structure (ϕ, ξ, η), if  g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ),  (4)  4  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  holds for any X, Y ∈ χ(M) and (M, ϕ, ξ, η, g) is called an almost contact  metric manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Normal almost contact metric manifold [2]: An almost contact met-  ric manifold is said to be normal, if for any X, Y ∈ χ(M) the tensor field  N = [ϕ, ϕ]+ 2dη ⊗ ξ vanishes everywhere on the manifold, where [ϕ, ϕ] is the  Nijenhuis tensor of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Homogeneous almost contact metric manifold [9]: An almost contact  metric manifold (M, ϕ, ξ, η, g) is said to be homogeneous, if there exists a con-  nected Lie group G of isometries acting transitively on M leaving η invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Cosymplectic manifolds  A (2n + 1)-dimensional manifold is said to be a cosymplectic manifold [10],  if it admits a closed, 1-form η and 2-form Φ such that η ∧ Φn is a volume  element, where Φ(X, Y ) = g(ϕX, Y ) is a 2-form on M 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Almost cosymplectic manifold [10]: If η and Φ are not closed but η ∧ Φn  is a volume form, then the manifold is called almost cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  α-cosymplectic manifold [12]: An almost cosymplectic manifold is said  to be α-cosymplectic if dη = 0 and dΦ = 2αη ∧ Φ for some constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Almost α-cosymplectic manifold [10]: An almost α-cosymplectic manifold  is defined as an almost contact metric manifold with dη = 0 and dΦ = 2αη∧Φ,  for any constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In particular, the almost α-cosymplectic manifold is  almost α-Kenmotsu if α ̸= 0,  almost cosymplectic if α = 0,  almost Kenmotsu if α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Harmonic vector field [14]: A characteristic vector field ξ on an almost  α-cosymplectic manifold is harmonic if and only if ξ is an eigenvector field  of the Ricci operator Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Almost α-cosymplectic 3-manifold  In this article, we will mainly focus on 3-dimensional almost α-cosymplectic  manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In what follows, we will be using the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [12] An almost α-cosymplectic 3-manifold is α-cosymplectic  if and only if Lξh = 0, where h = 1  2Lξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Any almost α-cosymplectic 3-manifold satisfies important relationships be-  tween Φ, ξ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [12] Let M 2n+1 be an almost α-cosymplectic 3-manifold, then  we have,  ∇ξϕ = 0, ∇ξ = 0, hϕ + ϕh = 0, hξ = 0,  (5)  On harmonic and homogeneous almost α-cosymplectic 3-manifolds  5  with  ∇Xξ = −αϕ2X − ϕhX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (6)  We would require some identities on the ϕ-bases [2] and the following table  of the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [12] On almost α-cosymplectic 3-manifold, there exists  ϕ-bases satisfying  he = σe, hϕe = −σϕe, hξ = 0,  with σ a local smooth eigen-function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [12] The Levi-Civita connection on almost α-cosymplectic  3-manifold are given by,  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ∇ee = −aϕe − αξ, ∇ϕee = −bϕe + σξ, ∇ξe = µϕe,  ∇eϕe = ae + σξ, ∇ϕeϕe = be − αξ, ∇ξϕe = −µe,  ∇eξ = αe − σϕe, ∇ϕeξ = −σe + αϕe, ∇ξξ = 0,  (7)  where a = g(∇eϕe, e), b = −g(∇ϕee, ϕe) and µ = g(∇ξe, ϕe) are smooth  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  The Ricci operator on almost α-cosymplectic 3-manifold is known explicitly  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [12] The Ricci operator Q on almost α-cosymplectic 3-  manifold is given by,  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  Qξ = −(2α2 + tr h2)ξ + (2bσ − e(σ))ϕe − (2aσ + (ϕe)(σ))e,  Qϕe = (2bσ − e(σ))ξ + (α2 + r  2 + tr h2  2  + 2σµ)ϕe + (ξ(σ) + 2ασ)e,  Qe = −(2aσ + (ϕe)(σ))ξ + (ξ(σ) + 2ασ)ϕe + (α2 + r  2 + tr h2  2  − 2σµ)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (8)  Furthermore, the scalar curvature r = tr Q is given by  r = −6α2 − tr h2 − 2(a2 + b2) − 2(ϕe)(a) + 2e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (9)  The structure of simply-connected, homogeneous almost α-cosymplectic 3-  manifold, admitting a contact Ricci soliton, is very well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [9] Let M be a simply-connected, homogeneous almost α-  cosymplectic 3-manifold admitting a contact Ricci soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then M is an  unimodular semidirect product Lie group G of type G0bb = R2 ⋊A R, where  A =  �  0  b  −b  0  �  , equipped with a flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Moreover, we have the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If A = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=', b = 0, G is the abelian Lie group R3 equipped with its flat  left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If A ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=', b ̸= 0, G is the Lie group ˜E2 equipped with its flat left  invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  6  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Harmonic manifolds  A complete Riemannian manifold (M n, g) is said to be harmonic, if for any  p ∈ M, the volume density ωp(q) =  �  det(gij(q)) in normal coordinates,  centered at any p ∈ M is a radial function [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Thus,  Θ(r) = rn−1�  det(gij(q))  is density of geodesic sphere, is a radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' It is known that harmonic  manifolds are Einstein [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' They are naturally classified as per the sign of the  Ricci constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let r be the constant scalar curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  If r = 0, then M is flat, that is (M, g) = (Rn, Can) (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  If r > 0, then by Bonnet-Myer’s theorem M is compact with finite  fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' They are compact rank one symmetric spaces by a  well known result of Szabo (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  If r < 0, then M is non-compact harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' They are rank one  symmetric spaces of non-compact type, if dimension of M is atmost 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  The main result in the theory of harmonic spaces is the Lichnerowicz  Conjecture: Any simply connected, complete harmonic manifold is either flat  or a rank one symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' By the above classification, we see that the  conjecture is resolved for compact harmonic manifolds and is open for non-  compact harmonic manifolds of dimension 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' There are counter examples to  the conjecture when dimension is atleast 7, known as the Damek-Ricci spaces  or NA spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' See for more details references in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  In the category of non-compact harmonic manifolds, we will be con-  sidering simply connected, complete, non-compact harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' It  follows that, these spaces don’t have conjugate points (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Hence, by  the Cartan-Hadamard theorem,  expp : TpM → M  is a diffeomorphism and every geodesic of M is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' That is, if γv : R → M  is a geodesic of M with v ∈ SpM, γ′  v(0) = v, then d(γv(t), γv(s)) = |t − s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Busemann function: Let γv be a geodesic line, then the two Busemann  functions associated to γv are defined as [15]:  b+  v (x) = lim  t→∞ d(x, γv(t)) − t,  b−  v (x) =  lim  t→−∞ d(x, γv(t)) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Einstein Solitons on Almost α-Cosymplectic  3-Manifolds  In this section, we examine the nature of a contact Einstein soliton on al-  most α-cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We also show that, the characteristic vector  field ξ is harmonic on almost α-cosymplectic 3-manifold admitting a contact  On harmonic and homogeneous almost α-cosymplectic 3-manifolds  7  Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Finally, we generalize Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='7 using these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Contact Einstein soliton: Let (M 2n+1, g) be a Riemannian manifold of  dimension 2n + 1 (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Consider the Einstein soliton (2), with potential  vector field V , on an almost contact metric manifold (M, ϕ, ξ, η, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then the  soliton is called contact Einstein soliton, if V = ξ that is, the potential vector  field is the characteristic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  The potential vector field V is called transversal, if it is orthogonal to the  characteristic vector field, that is V ⊥ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold,  admitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then the soliton is:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' steady, if α2 = σ2 − (a2 + b2) − (ϕe)(a) + e(b),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' shrinking, if α2 > σ2 − (a2 + b2) − (ϕe)(a) + e(b),  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' expanding, if α2 < σ2 − (a2 + b2) − (ϕe)(a) + e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If the soliton is contact Einstein soliton, using V = ξ in (2), we have  g(∇Xξ, Y ) + g(X, ∇Y ξ) + 2g(X, QY ) = (2λ + r)g(X, Y ),  (10)  for any vector fields X, Y on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Substituting X = Y = ξ in the above equation and using (8), we obtain  λ = −2α2 − 2σ2 − r  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (11)  From the expression of r (9), we get  λ = α2 − σ2 + (a2 + b2) + (ϕe)(a) − e(b),  (12)  from which we can conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold,  admitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then the characteristic vector field ξ is  harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' From (10), we get for X = ξ and Y = e,  (ϕe)(σ) = −2aσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (13)  And for X = ξ and Y = ϕe, from (10) we have  e(σ) = 2bσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (14)  Now, using (13) and (14) in the expression of Qξ in (8), we obtain  Qξ = −(2α2 + 2σ2)ξ,  which shows that ξ is an eigenvector field of the Ricci operator Q concluding  the fact that ξ is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  We derive the identity involving the Lie derivative of the metric, Ricci oper-  ator, the potential vector field V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  8  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, g) be a Riemannian manifold of scalar curvature r,  admitting an Einstein soliton (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then  ∥LV g∥2 = 2V (r) + 4 div  ��  λ + r  2  �  V − QV  �  ,  (15)  where Q is the Ricci operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In local coordinate system, (2) leads to  LV gij + Sij = (2λ + r)gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Therefore,  ∥LV g∥2 = − SijLV gij + (2λ + r)gijLV gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  = − LV r + gijLV Sij − (2λ + r)gijLV gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (16)  Now,  gijLV Sij =gij∇V Sij − gij∇αViSαj − gij∇αVjSiα  =2V (r) − 2 div QV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (17)  Observing that gijLV gij = −2 div V and using (16) and (17), we get the  required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Now we derive the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Consider M to be an almost α-cosymplectic 3-manifold, ad-  mitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If σ ̸= 0, then α = a2 + b2 − 2λ2 + (ϕe)(a) − e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If σ = 0, then M is cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Replacing X by e and Y by ϕe, from (10) we get  g(∇eξ, ϕe) + g(e, ∇ϕeξ) + 2g(e, Qϕe) = (2λ + r)g(e, ϕe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Using (7) and (8), after simplification we acquire,  ξ(σ) = σ − 2ασ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (18)  Now putting X = e = Y in (10) and using (7), (8), (9) and (12), we get  6α2 + 6σ2 − 4σµ + 2α + r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (19)  Similarly, putting X = ϕe = Y in (10) and using (7), (8), (9) and (12), we  also obtain  6α2 + 6σ2 + 4σµ + 2α + r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (20)  So comparing (19) and (20), we have σµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If σ ̸= 0, then from (20), we  obtain the required result using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Now suppose σ = 0, then M is α-cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' From [12], recall that an  almost α-cosymplectic manifold M is α-cosymplectic if and only if for any  X ∈ χ(M),  QX =  �  α2 + r  2  �  X −  �  3α2 + r  2  �  η(X)ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (21)  On harmonic and homogeneous almost α-cosymplectic 3-manifolds  9  Since ∇ξ is symmetric, (10) becomes  g(∇Xξ, Y ) + g(X, QY ) =  �  λ + r  2  �  g(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (22)  Using (6) and (21), we have from (22), for any X, Y ∈ χ(M),  (α2 + α − λ)g(X, Y ) −  �  3α2 + α + r  2  �  η(X)η(Y ) = 0,  which implies α2 + α − λ = 0 and 3α2 + α + r  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  That is λ = α2 + α and r = −6α2 − 2α = constant, so that, λ + r  2 = −2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Also, from (21), we have Qξ = −2α2ξ which implies (λ + r  2)ξ − Qξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Therefore, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3 (15), we can say that ξ is a Killing vector field,  that is, ∇ξ is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' But in our case ∇ξ is symmetric, which implies  ∇ξ = 0, that is, α = 0, proving the fact that M is cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Consider M to be a simply-connected, homogeneous, almost  α-cosymplectic 3-manifold, admitting a contact Einstein soliton with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Then M is an unimodular semidirect product Lie group G of type G0µµ =  R2 ⋊A R, where A =  �  0  µ  −µ  0  �  ̸= 0, is a real matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Moreover, G is the  Lie group ˜E2 equipped with its flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The proof follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='7 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ricci Solitons on Harmonic Manifolds  In this section, we study Ricci solitons on complete, simply connected, har-  monic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We prove a Lichnerowicz type result that, a harmonic man-  ifold admits a Ricci soliton if and only if M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' More precisely, we show  that compact harmonic manifolds and non-flat harmonic manifolds do not  admit Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' But flat harmonic manifold do admit steady, shrinking,  expanding Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  In the sequel, harmonic manifold means complete, simply connected harmonic  manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The main theorem of this section is:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, g) be a harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Then M admits Ricci  soliton if and only if M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In this case, the steady Ricci soliton is Killing  given by X = ∇bv  −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' where b−  v (x) = −⟨x, v⟩, the Busemann function, is a  potential function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In case, the Ricci soliton is shrinking or expanding,  the potential function is given by f(x) = λd(x, p)2 +f(p), for constant λ ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  and point p is minimum or maximum of f and X = ∇f is the corresponding  Ricci soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  We begin with the following important proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  10  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Every Ricci soliton is a gradient soliton on complete man-  ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Hence, in particular on any harmonic manifold Ricci soliton is a gra-  dient soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Consequently, any harmonic manifold admits a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Perelman showed that, Ricci soliton on any complete manifold is al-  ways a gradient soliton [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Hence, in this case X = ∇f, for some smooth  function f : M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' As L∇fg = ∇2f, (1) reduces to  Ric + 1  2∇2f = λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (23)  As (M, g) is harmonic and hence Einstein, then it follows that  ∇2f = 2(λ − r)g,  (24)  where r is a constant scalar curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Thus f is a Gaussian, that is it  satisifes (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Note that because any harmonic manifold is Einstein, trivial  solitons X = 0 and X a Killing vector field are solutions of (1) with λ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ricci flat harmonic manifold is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' It can be shown that any harmonic manifold (M, g) is asymptotically  harmonic [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' That is there exists a constant h ≥ 0 such that  ∆bv  ± = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Let L = ∇2bv  + denote the second fundamental form of horospheres, b−1  v (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Then L satisfies the Riccati equation, that is for x ∈ v⊥,  L′(x) + L2(x) + R(x, v)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Tracing the above equation, we obtain that tr L2 = 0, as Ricci(v, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' But  L is a symmetric operator on v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' This implies that L = 0 for any v ∈ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Consequently, R(x, v)v = 0 for any x ∈ v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Thus (M, g) is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let X = ∇f be a Killing vector field on compact harmonic  manifold, then X is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Solitons of Killing type do not exist on non-  compact, non-flat harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' On flat harmonic manifold, Killing  vector field is X = ∇bv  −, where b−  v (x) = −⟨x, v⟩ is a Busemann function on  Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Because X = ∇f is a non-trivial Killing vector field, we have  ∇2f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Therefore, ∥∇f∥ = constant ̸= 0, consequently, f has no critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Any Killing vector field of constant norm satisfies (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' 164-167, [15]):  ∥∇2f∥  2 = Ric(∇f, ∇f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  On harmonic and homogeneous almost α-cosymplectic 3-manifolds  11  Therefore,  0 =∥∇2f∥ = r∥∇f∥2  This implies that for f non-constant, r = 0 and therefore Ric ≡ 0 and hence  harmonic manifold must be flat (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  We have ∥∇f∥ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We may assume that ∥∇f∥ = 1, therefore f is  distance function which is harmonic function on (Rn, Can).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' By Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1 of [18], it follows that  f(x) = b−  v (x) = −⟨x, v⟩,  is a Busemann function on Rn [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  If M is compact, ∇2f = 0 implies that f is a harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Hence, f  must be a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, g) be a compact harmonic manifold, then a Ricci  soliton on M is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We have,  ∇2f = 2(λ − r)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Therefore, ∆f = 2(λ − r)n implies by the Bochner’s formula that,  1  2∆(∥∇f∥2) = 4(λ − r)2n2 + r(∥∇f∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (25)  Therefore,  4(λ − r)2n2 Vol(M) = −r  �  M  ∥∇f∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  This implies that ∥∇f∥ = 0, therefore f is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Let (M, g) be a non-compact, non-flat harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Then Ricci solitons on M don’t exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In case, (λ − r) ̸= 0, implies that  M is flat and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' In this case the potential function is given by f(x) =  λd(p, x)2 + f(p), for some p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' We have,  ∇2f = 2(λ − r)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Therefore, f is either convex or concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Consequently, the only  possible critical point of f is either maximum or minimum of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Suppose  that p is a critical point of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Note that along any unit speed geodesic of M  starting from p,  f ′′(t) = 2(λ − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  (26)  Therefore, f ′(t) = 2(λ − r)t + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Hence, there is exactly one critical point,  and hence c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Thus, f(t) = (λ − r)t2 + f(p), consequently f is a radial  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' This implies that,  ∆f = f ′′ + Θ′  Θ f ′ = 2(λ − r)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  12  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  Therefore,  f ′′ + Θ′  Θ 2(λ − r)t = 2(λ − r)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Consequently by (26),  Θ′(t)  Θ(t) = n − 1  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Comparing with the series expansion (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='4) of [18]),  Θ′(t)  Θ(t) = n − 1  r  − r  3 + · · · ,  we obtain r = 0, hence M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Finally, f(x) = λd(p, x)2 + f(p) follows  from section 1 of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Finally we come to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1: If M is compact, then the Ricci soliton on M is  trivial (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If (λ−r) = 0, then M is flat and X = ∇bv  − (Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' If (λ−r) ̸= 0, then M is flat, and X = ∇f, where f(x) = λd(p, x)2+f(p),  for some p ∈ M (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  □  Remark: We have shown that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1 confirms Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1 in case of  harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Also Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='1 implies that there are no non-trivial  deformation of non-flat harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' This indicates a result support-  ing the conjecture that, there are no non-trivial deformations of harmonic  manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' and hence there should be only finitely many classes of harmonic  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Acknowledgements  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Naeem Ahmad Pundeer would like to thank to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' for its Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Kothari Postdoctoral Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The corresponding author, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Paritosh  Ghosh, thanks UGC Junior Research Fellowship of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The authors also  would like to thank Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Dipen Ganguly for his wishful help in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  References  [1] Besse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content=' Harmonic Mappings of Riemannian Man-  ifolds, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content=' Three manifolds with positive Ricci curvature, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=', 17, 255-306, (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  [8] Hu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content=' and Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' The rigidity and stability of gradient esti-  mates, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content=' Complete Riemannian manifolds and some vector fields,  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
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+page_content='  Naeem Ahmad Pundeer  Department of Mathematics  Jadavpur University  Kolkata-700032, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  e-mail: pundir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='naeem@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='com  Paritosh Ghosh  Department of Mathematics  Jadavpur University  Kolkata-700032, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  e-mail: paritoshghosh112@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='com  14  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content=' Bhattacharyya  Hemangi Madhusudan Shah  Harish-Chandra Research Institute  A CI of Homi Bhabha National Institute  Chhatnag Road, Jhunsi, Prayagraj-211019, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='  e-mail: hemangimshah@hri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='in  Arindam Bhattacharyya  Department of Mathematics  Jadavpur University  Kolkata-700032, India  e-mail: bhattachar1968@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}
+page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'}

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@@ -0,0 +1,1010 @@
+Logically at Factify 2023: A Multi-Modal Fact
+Checking System Based on Evidence Retrieval
+techniques and Transformer Encoder Architecture
+Pim Jordi Verschuuren, Jie Gao, Adelize van Eeden, Stylianos Oikonomou and
+Anil Bandhakavi
+Brookfoot Mills, Brookfoot Industrial Estate, Brighouse, HD6 2RW, United Kingdom
+Abstract
+In this paper, we present the Logically submissions to De-Factify 2 challenge (DE-FACTIFY 2023) on
+the task 1 of Multi-Modal Fact Checking. We describes our submissions to this challenge including
+explored evidence retrieval and selection techniques, pre-trained cross-modal and unimodal models, and
+a cross-modal veracity model based on the well established Transformer Encoder (TE) architecture which
+is heavily relies on the concept of self-attention. Exploratory analysis is also conducted on this Factify
+2 data set that uncovers the salient multi-modal patterns and hypothesis motivating the architecture
+proposed in this work. A series of preliminary experiments were done to investigate and benchmarking
+different pre-trained embedding models, evidence retrieval settings and thresholds. The final system, a
+standard two-stage evidence based veracity detection system, yields weighted avg. 0.79 on both val set
+and final blind test set on the task 1, which achieves 3rd place with a small margin to the top performing
+system on the leaderboard among 9 participants.
+Keywords
+fact verification, multimodal representation learning, multimodal entailment, text entailment, Multi-head
+Attention
+1. Introduction
+Misinformation and fake news can spread rapidly and cause harm at various levels. One way to
+protect ourselves from these negative impacts is through fact-checking and debunking false
+information with evidence-based reporting. However, this process can be resource-intensive and
+time-consuming. To address this issue, researchers have developed automated fact-checking
+systems using deep learning techniques, which can handle tasks such as claim detection,
+claim matching, evidence retrieval, and veracity prediction using natural language processing
+techniques on textual content. While there has been progress in this area, there is still a need
+for multimodal approaches that can handle both text and image inputs. To address this gap,
+this paper presents a multimodal veracity prediction system for automated fact-checking and is
+De-Factify: Workshop on Multimodal Fact-Checking and Hate Speech Detection, co-located with AAAI 2023. 2023
+Washington DC, USA
+� [email protected] (P. J. Verschuuren); [email protected] (J. Gao); [email protected] (A. v. Eeden);
+[email protected] (S. Oikonomou); [email protected] (A. Bandhakavi)
+� https://www.logically.ai/team/leadership/anil-bandhakavi (A. Bandhakavi)
+� 0000-0002-3610-8748 (J. Gao)
+© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
+CEUR
+Workshop
+Proceedings
+http://ceur-ws.org
+ISSN 1613-0073
+CEUR Workshop Proceedings (CEUR-WS.org)
+arXiv:2301.03127v1  [cs.CL]  9 Jan 2023
+
+developed as part of the Factify 2 competition organized by De-Factify@AAAI 2023.
+The remainder of the paper is structured as follows: Section 2 presents a brief overview of
+related work and section 3 describes our general framework and model architecture. Section
+4 discusses the dataset supplied by the Factify 2 competition followed by an overview of
+our experiments in section 5. Section 6 and 7 present the final results and our conclusions,
+respectively.
+2. Related Work
+As an essential part of automated fact verification, effective techniques for modeling claim-
+evidence for veracity prediction have been a hot topic and key research questions in existing
+fact-checking methods. Most of the recent work focuses on using textual evidence in veracity
+prediction and there are mainly two lines of work. One direction [1, 2, 3] is to use a single
+document (such as is provided in the Factify task dataset) with long text evidence and through
+leveraging models constructed for long sequences. Examples such as BigBird [4], Longformer[5]
+and recent advancements in the ConvNets architecture witnessed in the Long Range Arena
+leaderboard (e.g., Mega [6], S5[7]) are seen to obtain top results in a wide range of tasks and
+leader boards. The benefits of exploiting long-sequence model at document level is a) the
+simplicity of the overall architecture; b) allows to accommodate for more context of the whole
+article into modeling and natural language inference. An optimal setup of the maximum length
+for both claim (or query) and document sequence, and the document level veracity labels is
+commonly required [8, 1, 3]. The advantage of incorporating lots of context into inference is also
+seen in modeling question answering (QA) tasks [4, 5], for which the document-level veracity
+labels are relatively "cheap" to obtain. The downside of using a simple long-text model technique
+at document-level is the lack of interpretability (w.r.t. evidence selection), it is computational
+expensive, the limitation in dealing with the complexity of certain (multi-hop) claims [9], and
+lack of diversity and scalability when dealing with a large amount of diverse documents in a
+real-world application. These constraints were more apparent in open domain fact checking
+task that make use of web data extracted with commercial search engines as building blocks
+in fact-checking system in order to incorporate more diverse sources. It is worth to note that
+long-sequence model can be adapted for the purpose of evidence selection e.g., through framing
+the task as a token-level prediction task. For instance, as one of the top systems in SciFact
+leaderboard 1, LongChecker [10] used LongFormer [5] for scientific claim verification with
+paragraph-level evidence selection. In their method, every sentences is inserted with a [CLS]
+token with global attention, which allows the model to predict on this sentence-level token as
+evidence. Most of these works focus on a limited context such as a few Wikipedia documents, a
+single article and abstracts or text snippets from research literature or a small synthetic corpus.
+Another line of work widely adopted and one of the key tasks in FEVER [11, 12] is to involve
+evidence retrieval and selection. The framework exploits larger document context to extract
+evidentiary (or rationales) passages as first step and veracity prediction is then modeled to
+condition on the claim and the selected rationales. The evidentiary passages can be either at
+sentence-level or paragraph-level and report the findings to the claim which can be used to
+1https://leaderboard.allenai.org/scifact/submissions/public
+
+justify each veracity label. Despite the revolutionary breakthroughs with Large-Scale Language
+Models (LSLMs), such as GPT-3[13] and ChatGPT2, and their impressive generative capabilities,
+these large models are still lacking key zero-shot or few-shot learning capabilities needed for
+fact checking tasks. This is mainly due to their incorrectly retrieved, incomplete or outdated
+knowledge stored in their weights which makes these techniques susceptible to hallucinations
+[14, 15], which is conflicting with fact checking tasks that require factuality as an essential
+element in modeling. Moreover, an efficient approach to keep LSLMs up-to-date and grounded
+to ever-growing factual and new information is imperative but still unresolved to date. Recent
+work [15, 16] shows that lightweight methods with fine-tuned and smaller models outperform
+these big models in a range of knowledge-intensive NLP tasks including Natural Language
+Inference (NLI), Recognizing textual entailment (RTE), Reading Comprehension (RC), QA, etc.
+Sentence-BERT (SBERT) [17] is one of the most popular techniques based on the BERT language
+model [18] used for evidence selection [19, 20] which can be framed as a sentence-pair regression
+task. SBERT models are used to encode contextualized representations for each of the evidence
+passages which are then ranked according to their semantic similarity with the contextualized
+representation of the corresponding claim. In the final step, top 𝑘 evidentiary passages are
+selected for veracity prediction. The challenge of this multi-staged verification framework is
+1) the rationales extracted out-of-context may lack information required to make a prediction
+(e.g., acronyms, unresolved coreferences); 2) the evidence extraction (through passages ranking)
+requires high quality training data that is costly to obtain with domain experts from both closed
+and open domain tasks [21]. Various efforts to address the constraints have been undertaken
+to explore 1) paragraph level train data from scientific literature with paper title as claim and
+abstract as evidence as high-precision heuristics (e.g., SciFact [1]); 2) QA dataset with question
+and answer considered as claim and evidence respectively [22]; 3) NLI dataset with hypothesis
+as the claim and premise as evidence [23]. We follow a second line of work for which the
+evidence retrieval component is implemented in our system following current SoTA methods.
+Automated multi- or cross-modal fact checking is an under developed field compared to
+text-based techniques. Recent developments have shown that cross-modal pre-trained models
+(e.g.,VideoBERT [24], VisualBERT [25], Uniter [26], CLIP [27]) has achieved significant results
+in downstream cross-modal tasks [28, 29, 30] with great transferability for zero-shot or few-shot
+scenarios. Our work is inspired by [31], which one of the initial explorations in multimodal
+fact-checking task. In their proposed method, Contrastive Language–Image Pre-training (CLIP)
+model [27]) is adopted as encoder to learn joint language-image embedding between each
+image and the input claim text. Top-5 candidate image evidences are taken as input along
+with multi-modal claim for multimodal claim verification model with a simple cross-attention
+network. It is worth noting that CLIP model allows to model image-text contextual alignment at
+coarse-grained contextual (global) level but ignores the compositional matching of disentangled
+concepts (i.e., finer-grained cross-modal alignment at region-word level)[30, 32, 32].
+2https://openai.com/blog/chatgpt/
+
+3. Methodology
+3.1. Problem statement
+We frame the Factify 2 problem as a multimodal entailment task as in the previous submission
+[3], which considers a multimodal claim 𝑐 = 𝑐𝑡𝑒𝑥𝑡 + 𝑐𝑖𝑚𝑎𝑔𝑒 as hypothesis and a multimodal
+document 𝑑 = 𝑑𝑡𝑒𝑥𝑡 + 𝑑𝑖𝑚𝑎𝑔𝑒 as premise. The goal is to learn a function 𝑓(𝑐, 𝑑) that infers one
+of the five entailment categories including "Support_Multimodal", "Support_Text", "Refutes",
+"Insufficient_Multimodal" and "Insufficient_Text". Additional details on the task can be found in
+[33].
+3.2. General Architecture
+Our system architecture follows a standard two-stage claim verification approach as established
+through various shared tasks in recent years, typically FEVER[34], FEVER 2.0 [35], FEVEROUS
+[36] and SCIVER [37]. First, a textual evidence retrieval component identifies from a given
+document the evidence passages most relevant to the corresponding claim text. Then, a trans-
+former based cross-modal model is trained on all the input across multi-modalities including
+selected evidence passages text, claim text, claim image, document image, claim OCR text
+and document OCR text to predict five multimodal entailment categories with respect to the
+multimodal claim. A pre-trained cross-modal model (i.e. CLIP) and a pre-trained text embedding
+model are both employed in the embedding layer in order to learn a cross-modal matching
+model using both unified-multimodal and unimodal representations. Overall, the implemented
+architecture adopts listwise concatenation strategy [38] which is one of common strategies in
+most recent sequence-to-sequence SoTA veracity prediction models.
+Figure 1: Logically General System Architecture
+
+Cross-modal veracity prediction model
+Text Encoder
+ Cross-modal 
+(CLIP)
+Claim text embed
+Transformer Encoder
+ Self-Attention
+Evidence Retrieval
+max
+Multihead
+embedding layer
+Masked
+Semantic Search
+Image Encoder
+(cosine similarity)
+00
+(CLIP)
+Sofmax
+4
+Re-rank & Concatenate
+Text passages dense
+Top K evidence
+W
+representations
+candidates
+Claim text dense
+Text Embedding Layer
+representations
+Transformer E ncoder
+im+Doc text embedding
+Self-Attention
+M utihead
+xe W
+Wr2V
+Masked
+ Pooling
+SBERT
+(MPNet-QA dense retriever)
+Doc
+Claim text
+Claim Image
+Doc Image
+Claim ORC text
+Doc ORC text3.3. Evidence Retrieval
+In evidence retrieval, ‘multi-qa-mpnet-base-dot-v1‘ 3 is employed to compute embeddings for
+both claim text and document text at passage level. In terms of passage granularity, both
+paragraph-level retrieval and sentence-level retrieval are experimented (see Section 5). This
+is a SBERT model based on the MPNet architecture [39] and is trained on a Question-Answer
+(QA) dataset with 215M QA pairs from diverse sources. The model was tuned for a semantic
+search using a dot-product score function in order to find relevant passages corresponding to a
+given query. The model encodes text into a 768-d vector and supports 512 maximum number of
+tokens.
+Regarding the similarity computation and semantic search, we use a simple dot product with
+the normalised SBERT embeddings (as proxy to cosine similarity) which enables a quick and
+efficient passage ranking and scalability of up to about 1 Million entries.
+Top 𝐾 passages obtained from the semantic search are then re-ranked based on their relevancy
+to the claim text and concatenated into a longer text snippet before being fed into the cross-modal
+veracity prediction model.
+3.4. Embedding Layer
+Our embedding layer consists of a cross-modal encoder and a unimodal text encoder. We
+hypothesize that modeling solely on text-to-text interaction (i.e., text premise and hypothesis)
+can supplement the modeling solely on cross-modal premise and hypothesis interaction and
+vice versa. This architecture facilitates the measuring of multi-modal semantic relatedness in
+this multi-modal fact checking task by mapping more textual alignment signals into subse-
+quent semantic space. This considers that text specific models can capture more accurate and
+semantically meaningful word-level or sentence level alignment.
+The cross-modal encoder is implemented with a pre-trained CLIP model that aims to map
+visual and text embeddings into a common space. The ViT-B/32 variant (ViT-Base with patch
+size 32) is chosen in this work because of its smaller amount of parameters, less FLOPS and
+greater inference speed. ViT-B/32 consists of a text encoder and an image encoder which
+are used to encode text inputs (including claim text, evidentiary passage and two images
+OCR text) and image inputs (including claim image and document image) respectively before
+concatenating into a 6 × 512 matrix as a single input to subsequent transformer encoder.
+The CLIP architecture allows for a maximum input text length of 77 tokens. The pre-trained
+Word2vec model ("Word2vec Google News 300") [40] is adopted as a unimodal text encoder. It
+encodes the concatenated text sequence of claim and document evidentiary passage text, and
+obtains a 300-D feature vector for each token. Zero-padding is applied to match the longest
+sentence in the training set. Both the pre-trained CLIP and Word2Vec embedding model were
+not fine-tuned.
+3The model is available in on the Hugging Face hub and accessible via https://huggingface.co/sentence-
+transformers/multi-qa-mpnet-base-dot-v1
+
+3.5. Cross-modal veracity prediction
+The second component of veracity prediction is based on the well established Transformer
+Encoder (TE) architecture, which heavily relies on the concept of self-attention [41] to effectively
+model higher-order interactions and context in an input. Recent research has shown that multi-
+head self-attention mechanisms and transformer architectures are computationally efficient and
+accurate in this regard. The self-attention mechanisms of the TE encoder allows for simple but
+powerful reasoning that can identify hidden relationships between vector entities, regardless of
+whether they are visual or textual in nature. Therefore, our cross-modal veracity prediction
+model is implemented based on self-attention mechanisms to learn the joint distribution of
+text representations of claim-document text pair and cross-modal feature representations of all
+modalities contained in claim and document .
+Specifically, the claim and document embeddings of joint input by CLIP and text input by
+text embedding layer are passed through two separate transformer encoder [41] consisting
+of 𝑁 identical sequential blocks of a multi-head attention (MHA) and a fully connected feed-
+forward network (FFN). Within each transformer encoder, multiple blocks allows for a deeper
+understanding of the inputs. For each block the input 𝑥 is passed through a multi-head attention
+layer of which the output is added to the initial input such that. Passing on both the initial input
+and the output ensures that information in the initial sequence is not lost. Layer normalization is
+applied to the output to allow for faster training and small regularization i.e. 𝑥 = LayerNorm(𝑥+
+MHA(𝑥)). The output is then passed to a feed-forward network to allow for more model
+complexity. The output is again added to the original input and layer normalization is applied i.e.
+𝑥 = LayerNorm(𝑥 + FFN(𝑥)). The output of the final block (i.e., the output of each transformer
+encoder in the diagram) is passed through an adaptive max pooling layer to reduce the output
+dimensions. The output of two separate transformer encoders are then concatenated before
+feeding into a MLP classifier for five categories prediction. The five categories probabilities are
+obtained from the final output softmax layer.
+4. Factify Dataset
+4.1. Dataset Description
+The Factify 2 dataset created and supplied by the organisers covers a train, validation, and test
+set. The train set contains 35000 data pairs, while the validation and test sets each contain 7500
+data pairs. Each data pair consists of a claim and a document, each of which comprises an image,
+a text, and OCR text extracted from the image. The data pairs are annotated with one label from
+5 categories including Support_Multimodal, Support_Text, Refute, Insufficient_Multimodal, or
+Insufficient_Text.
+4.2. Text Length Distribution
+The training set text and OCR text length distributions are represented in Figures 2 and 3. The
+text length distribution varies between the claim and document text, with the document text
+that tends to be much longer. This is expected as it is to be used to verify the claim. From Figure
+2 (a), we can can see that claim text is much shorter and less varied for the Refute category
+
+than for the rest of the categories, which all have similar claim text length distributions. Figure
+2 (b) shows that the Support_Multimodal and Support_Text categories have the larger spread
+of document text lengths and also the longest document text lengths. The two Insufficient
+categories have a smaller document text length distribution, and Refute has the smallest variance
+and maximum length in document text length.
+Considering the claim OCR length we see from Figure 3a that the Refute category has a much
+larger claim OCR length distribution and maximum length than any other category. The second
+largest claim OCR length distributions are the Support_Text and the Insufficient_Text categories,
+which then leaves the two Multimodal categories with the shortest claim OCR text lengths. The
+document OCR length distribution is very similar to that of the claim OCR, from Figure 3b we
+see the only real difference is that the two Text categories have a smaller document OCR length
+distrubution than that of the claim OCR.
+(a) Claim Text Length
+(b) Document Text Length
+Figure 2: Boxplot of Text Length of all Categories
+4.3. Image Similarity Distribution
+An image similarity investigation was conducted in order to gain an intuition of the similarity
+between the claim and document images for each category. Using image pairwise CLIP embed-
+dings we calculate a similarity score and analyse it per category. Figures 4a and 4b illustrate
+that the similarity between the claim and document image is comparatively higher within
+the categories for Support_Multimodal and Insufficient_Multimodal than the other categories.
+The label correlation with similarity of image pairs has been largely increased compared to
+
+350
+300
+Length of Text
+250
+200
+150
+100
+50
+0
+Refute
+Support_Multi
+Support_Text
+Multi
+Text
+Insuf_
+Insuf_
+Categories40000
+30000
+Length of Text 
+20000
+10000
+Refute
+Multi
+_Text
+_Multi
+Insuf_Text,
+R
+ Support_
+Insuf_
+S
+Categories(a) Claim OCR Text Length
+(b) Document OCR Text Length
+Figure 3: Boxplot of OCR Text Length of all Categories
+factity 1 dataset [3] last year. This further indicates that there is explicit correlation multimodal
+categories which can be leveraged to learn and verify multimodal entailment categories.
+4.4. Multimodal Similarity Distribution
+The multimodal CLIP similarity between claim-text pairs is explored to investigate our hypothe-
+sis that Image should contain content that is related to claim in order to entail either support or
+refute veracity decision. Figures 5a and 5b depict the cosine similarity scores between the claim
+text and document image. From the figures, there is no clear indicator of the entailment between
+doc image and claim text. However, it is noticeable that “Support_Multimodal” presents the
+highest pairwise similarity correlation between label and claim-evidence pair. “Insufficient text”
+have the lowest pairwise similarity correlation, although our initial hypothesis was that “Insuf-
+ficient_Multimodal” should have the lowest value. This analysis suggests that differentiating
+between the different categories based on the claim text and document image correlation could
+be challenging.
+In terms of correlation between the claim image and document text, due to the maximum
+text sequence constraints with CLIP, text access maximum length is truncated. Consequently,
+longer context of document text is not incorporated in this analysis. As shown in Figure 6a and
+6c, there is low degree of similarity correlation across 5 categories, among which the "Refute"
+category shows highest similarity correlation.
+Lastly, Figure 6b and Figure 6d about the similarity correlation between the claim image and
+the claim text show no significant deviation in similarity scores of different categories when
+
+800
+Length of Text
+600
+400
+200
+工
+工
+0
+Refute
+ Support_Text
+Multi -
+Insuf_Text
+Insuf_
+Categories800
+Length of Text
+600
+400
+200
+工
+0
+Refute
+Support_Text
+Multi -
+Insuf_Text
+Insuf_
+Categories(a) Claim Image and Document Image Similarity Score Histogram
+(b) Claim Image and Document Image Similarity Boxplot
+Figure 4: Claim Image and Document Image Similarity Scores
+the claim image and claim text are compared to each other. For the purpose of this task and this
+dataset, we hypothesis that the claim image provides supplementary information to the claim
+text.
+
+Occurence
+150
+Refute
+Support Multi
+100
+Support_Text
+Insuf Multi
+Insuf Text
+50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Similaritv Score1.0
+0.8
+Score
+0.6
+S
+0.4
+0.2
+Refute
+Multi
+Text
+Insuf Multi
+Insuf_Text -
+Support_
+Categories(a) Claim Text and Document Image Similarity Score Histogram
+(b) Claim Text and Document Image Similarity Boxplot
+Figure 5: Claim Image and Document Image Similarity Scores
+5. Experiments
+5.1. Model settings
+To validate and optimal the effect of evidence retrieval, We attempt to experiment our model
+with 1) with or without evidence selection; 2) vary length of evidence doc text sorted by evidence
+retriever; 3) passage ranking at paragraph level versus sentence level; 4) text-to-text alignment
+with SBERT versus cross-modal alignment with CLIP. Both SBERT and CLIP is used to rank
+evidence doc with paragraph and sentence level; 5) if SBERT model trained on QA dataset
+perform better than general purpose SBERT model. Note that ranking at paragraph level on
+
+Occurence
+Refute
+150
+Support_Multi
+Support_Text
+100
+of
+Insuf Multi
+Frequency
+Insuf Text
+50
+0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.400.40
+0.35
+0.30
+0.25
+0.20
+0.15
+0.10
+Refute
+Support_Multi
+Support_Text
+Insuf_Multi
+Insuf_Text 
+Categories(a) Claim Image and Document Text Similarity Score His-
+togram
+(b) Claim Image and Claim Text Similarity Score His-
+togram
+(c) Claim Image and Document Text Similarity Score
+Boxplot
+(d) Claim Image and Claim Text Similarity Score Box-
+plot
+Figure 6: Similarity Scores
+top <5 or sentence level on top <5 is only option to CLIP due to its maximum allowed length
+restriction.
+For two transformer encoders, we choose an empirical setting of four heads in two MHAs.
+The number of sequential MHA and feed-forward network blocks per embedding input is
+𝑁𝑏𝑙𝑜𝑐𝑘𝑠 = 2. All our experiments are trained on 3-layered MLP and number of nodes per layer
+are set to 3072, 1024 and 5 respectively. A dropout of 0.5 and ReLU activations are applied
+between the MLP layers.
+Preliminary experiments conducted in this work are elaborated in details as follows:
+• "model_w/o_ER": to validate the effectiveness with evidence retrieval, we remove evidence
+retrieval in our system and provide original document text to "Cross-modal veracity
+prediction model".
+
+Occurence
+150
+Refute
+Support Multi
+100
+Support Text
+of
+Insuf Multi
+Frequency
+Insuf Text
+50
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+Similarity ScoreOccurence
+200
+Refute
+150
+Support Multi
+Support Text
+a
+ 100
+Insuf Multi
+Frequency
+Insuf Text
+50
+0
+0.0
+0.1
+0.2
+0.3
+0.4
+Similarity Score0.5
+0.4
+Similarity Score
+0.3
+0.2
+0.1
+0.0
+Refute
+Support_Multi
+Support_Text
+Insuf_Multi
+Insuf_Text
+Categories0.5
+0.4
+Score
+0.3
+Similarity s
+0.2
+0.1
+0.0
+Refute
+Support_Multi
+Support_Text
+Insuf_Multi
+Insuf_Text
+Categories• "SBERT_sentence_ER_top5": One of the "top" 4 performing general purpose SBERT model
+("all-MiniLM-L6-v2") is chosen in our experiment. This is a all-round model tuned for
+many use-cases and 5 times faster while offering good quality compared to best all-round
+model "all-mpnet-base-v2". The model is trained on a large and diverse dataset of over
+1 billion training pairs and also fine-tuned for dot-product score function suitable for
+cosine similarity. The use of all-round model allows us to evaluate the value of adopting
+QA fine-tuned counterpart that we hypothesis as optimal solution. Top 5 sorted sentence
+sorted by all-round SBERT model is configured in this setting.
+• "SBERT_sentence_ER_top10": Top 10 sorted sentence sorted by all-round SBERT model
+is configured in this setting.
+• "SBERT_sentence_ER_top15": Top 15 sorted sentence sorted by all-round SBERT model
+is configured in this setting.
+• "SBERT-QA_paragraph_ER_top5": SBERT QA dataset fine-tuned model (as described
+in 3.3) is adopted in this setting to obtain top 5 paragraphs as evidentiary passages for
+veracity inference in this setting.
+• "SBERT-QA_sentence_ER_top5": Top 5 sentences sorted by SBERT QA model and selected
+as evidentiary passages in this setting.
+• "BigBird_w/o_ER": To evaluate the value of evidence selection against the long context
+modeling solution, the Google’s BigBird pre-trained model fine-tuned on Factity dataset
+from last year [3] is used in replace of Word2Vec model in "Text Embedding layer" with this
+setting. This BigBird model allows a maximum 1396 tokens and contextual representation
+of text is adopted in this setting.
+5.2. Training and validation
+For our experiment, the model was trained up to 80 epochs with early stopping on minimum
+validation loss by minimizing the cross-entropy loss function, the adaptive AdamW optimizer
+[42] with initial learning rate 𝛾 = 1e−4 and epsilon 𝜖 = 1e−8 with batch size 𝑁𝑏𝑎𝑡𝑐ℎ = 16.
+Early stopping patience is set to 5. A linear decreasing learning rate scheduler was used including
+𝑁𝑠𝑡𝑒𝑝𝑠 = 438 warming up training steps during which the learning rate increased linearly to
+the chosen learning rate.
+We have found that data scraping error leads to invalid doc text content in the development
+dateset provided by organiser with 463 and 114 invalid samples in train and val set respectively.
+There also are 112 invalid samples in test set. This results in document text containing only
+"We’ve detected that JavaScript is disabled in this browser ...". The invalid samples are removed
+from our training data.
+6. Results and Discussion
+The best model results in preliminary experiments described in section 5 are presented in Table
+1, Table 2 and Table 3 respectively.
+4The best performing general purpose model is selected with a sorted list of model performances and use cases
+recommended provided by SBERT, accessible via https://www.sbert.net/docs/pretrained_models.html
+
+Table 1
+5-way Classification Results of experiments without ER on val set
+Categories
+model_w/o_ER
+BigBird_w/o_ER
+P
+R
+F1
+P
+R
+F1
+Support_Multimodal
+0.73
+0.79
+0.76
+0.73
+0.81
+0.77
+Support_Text
+0.71
+0.61
+0.66
+0.77
+0.59
+0.67
+Insufficient_Multimodal
+0.66
+0.66
+0.66
+0.64
+0.70
+0.67
+Insufficient_Text
+0.71
+0.75
+0.73
+0.73
+0.75
+0.74
+Refute
+0.99
+0.98
+0.98
+0.98
+0.98
+0.98
+Weighted Avg.
+0.76
+0.76
+0.76
+0.77
+0.77
+0.77
+Table 2
+5-way Classification Results of experiments with all-round SBERT + ER on val set
+Categories
+SBERT_sentence_ER_top5
+SBERT_sentence_ER_top10
+SBERT_sentence_ER_top15
+P
+R
+F1
+P
+R
+F1
+P
+R
+F1
+Support_Multimodal
+0.72
+0.85
+0.78
+0.74
+0.78
+0.76
+0.75
+0.77
+0.76
+Support_Text
+0.63
+0.73
+0.68
+0.71
+0.61
+0.66
+0.71
+0.62
+0.66
+Insufficient_Multimodal
+0.70
+0.64
+0.67
+0.66
+0.67
+0.66
+0.65
+0.67
+0.66
+Insufficient_Text
+0.80
+0.58
+0.67
+0.70
+0.77
+0.74
+0.71
+0.76
+0.73
+Refute
+0.96
+0.99
+0.97
+0.96
+0.99
+0.97
+0.98
+0.98
+0.98
+Weighted Avg.
+0.76
+0.76
+0.75
+0.76
+0.76
+0.76
+0.76
+0.76
+0.76
+Firstly, the Table 1 shows that our veracity model without ER exhibits a reasonably good
+performance and utilising long sequence model (BigBird) for text embedding improves the
+base model with a small margin, by 1% for all categories except "Refute". As comparison,
+further experiments with ER are conducted in Table 2 and Table 3. The results in Table 2
+indicates that all-round SBERT based evidence selection do not provide obvious performance
+improvement based on current preliminary exploration covering three top K sentences settings
+(K=5, 10, 15). In contrast, SERT-QA based model achieves big marginal improvement at both
+paragraph and sentence level. Our experiments covers both top 5 paragraphs and sentences,
+which improves best base model (without ER) by 1% and 2% respectively. Final results across 7
+different experiment setup shows that combining SBERT-QA at top K sentence-level evidence
+passage retrieval achieves optimal performance compared to the base model without ER and
+the use of all-round SBERT model. The best model "SBERT-QA_sentence_ER_top5" obtains 0.79
+weighted avg. F1 with 20th epochs.
+6.1. Competition Result
+Final test set results and competition leaderboard are presented in Table 4. The results shows
+that top 3 participating systems achieves similar performance and our system is ranked at 3rd
+place with a small margin (by 0.028) to the top performing system. Please refer to [43] for the
+
+Table 3
+5-way Classification Results of experiments with SBERT-QA + ER on val set
+Categories
+SBERT-QA_paragraph_ER_top5
+SBERT-QA_sentence_ER_top5
+P
+R
+F1
+P
+R
+F1
+Support_Multimodal
+0.80
+0.77
+0.78
+0.79
+0.83
+0.81
+Support_Text
+0.70
+0.68
+0.69
+0.70
+0.69
+0.70
+Insufficient_Multimodal
+0.66
+0.72
+0.69
+0.71
+0.72
+0.73
+Insufficient_Text
+0.76
+0.72
+0.74
+0.74
+0.72
+0.73
+Refute
+0.96
+1.00
+0.98
+0.99
+0.98
+0.98
+Weighted Avg.
+0.78
+0.78
+0.78
+0.79
+0.79
+0.79
+Table 4
+Factify Official Leaderboard
+Rank
+Team
+Support_Text
+Support_Multi.
+Insufficient_Text
+Insufficient_Multi.
+Refute
+Final
+1
+Triple-Check
+0.828
+0.914
+0.852
+0.892
+1.0
+0.818
+2
+INO
+0.812
+0.9
+0.888
+0.852
+0.999
+0.808
+3
+Logically
+0.804
+0.905
+0.844
+0.856
+0.985
+0.79
+4
+Zhang
+0.766
+0.879
+0.816
+0.879
+0.999
+0.774
+5
+gzw
+0.785
+0.863
+0.814
+0.833
+1.0
+0.761
+6
+coco
+0.773
+0.865
+0.815
+0.83
+1.0
+0.757
+7
+Noir
+0.771
+0.873
+0.785
+0.816
+0.997
+0.745
+8
+Yet
+0.707
+0.826
+0.786
+0.719
+1.0
+0.691
+9
+TeamX
+0.582
+0.709
+0.537
+0.556
+0.698
+0.456
+-
+BASELINE
+0.5
+0.827
+0.802
+0.759
+0.988
+0.65
+competition details.
+7. Conclusion
+In this research, we present our multimodal fact checking system that is submitted to the De-
+Factify 2023 competition. The system consists of various components, including a multimodal
+fact checking dataset, a QA-enhanced evidence passage retrieval component, and a Transformer-
+based cross-modal sequence-to-sequence veracity prediction model. Our findings from the
+De-Factify 2023 competition show that recent advances in pre-trained cross-modal models, such
+as CLIP, have strong zero-shot or few-shot capabilities and can be effectively transferred to a
+variety of downstream tasks, including multimodal fact checking. However, there is still a need
+for more effective techniques for multimodal modeling and explainability, particularly in regards
+to learning finer-grained cross-modal representations by jointly modeling intra- and inter-
+modality relationships and aligning vision regions with sentence words or entities. Additionally,
+more focus should be placed on real-world challenges that involve handling large amounts of
+textual and multimodal information from multiple sources and domains for claim verification.
+There is also a need for techniques that can effectively handle more complex and nuanced
+real-world scenarios, such as those involving sarcasm, irony, and misleading context. The
+
+difficulties in creating large and high-quality multimodal fact checking datasets that accurately
+reflect real-world scenarios, as identified in our last year work, remain a significant challenge.
+References
+[1] D. Wadden, K. Lo, L. Wang, A. Cohan, I. Beltagy, H. Hajishirzi, Multivers: Improving
+scientific claim verification with weak supervision and full-document context, in: Findings
+of the Association for Computational Linguistics: NAACL 2022, 2022, pp. 61–76.
+[2] D. Stammbach, Evidence selection as a token-level prediction task, in: Proceedings of the
+Fourth Workshop on Fact Extraction and VERification (FEVER), EMNLP, Association for
+Computational Linguistics (ACL), 2021.
+[3] J. Gao, H.-F. Hoffmann, S. Oikonomou, D. Kiskovski, A. Bandhakavi, Logically at factify
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+doi:10.48550/ARXIV.2112.09253.
+[4] M. Zaheer, G. Guruganesh, K. A. Dubey, J. Ainslie, C. Alberti, S. Ontanon, P. Pham,
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+modeling, arXiv preprint arXiv:2208.04933 (2022).
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