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label1,label2]Stefan Muff label1,label2]Mauro Fanciulli label1,label2]Andrew P. Weber label2]Nicolas Pilet label2,label3]Zoran Ristić label2,label4]Zhiming Wang label2]Nicholas C. Plumb label2,label5]Milan Radović label1,label2]J. Hugo Dil[label1]Institut de Physique, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland [label2]Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland [label3]Vinča Institute of Nuclear Science, University of Belgrade, 11001 Belgrade, Serbia[label4]Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland [label5]SwissFel, Paul Scherrer Institut, 5232 Villigen, SwitzerlandThe two-dimensional electron gas at the surface of titanates gathered attention due to its potential to replace conventional silicon based semiconductors in the future. In this study, we investigated films of the parent perovskite CaTiO_3, grown by pulsed laser deposition, by means of angular-resolved photoelectron spectroscopy. The films show a c(4×2) surface reconstruction after the growth that is reduced to a p(2×2) reconstruction under UV-light. At the CaTiO_3 film surface, a two-dimensional electron gas (2DEG) is found with an occupied band width of 400 meV. With our findings CaTiO_3 is added to the group of oxides with a 2DEG at their surface. Our study widens the phase space to investigate strontium and barium doped CaTiO_3 and the interplay of ferroelectric properties with the 2DEG at oxide surfaces. This could open up new paths to tailor two-dimensional transport properties of these systems towards possible applications. Calciumtitanate CaTiO_3 surface states two-dimensional electron gas electronic structure ARPES PLD 68.35.B- 68.47.Gh 71.10.Ca§ INTRODUCTIONThe discovery of a two-dimensional electronic state at the interface of LaAlO_3 and SrTiO_3 <cit.> triggered research on other oxide interfaces where similar states were found <cit.>. These two-dimensional states at interfaces of complex oxides give rise to different phenomena such as superconductivity <cit.>, metal-insulator transitions <cit.> or magnetism <cit.>.More recently, a two-dimensional electron gas (2DEG) was also found on clean SrTiO_3 and KTaO_3 (001) surfaces <cit.>. These states at the vacuum interface can, in contrast to the burried interface states, be more easily probed by angular-resolved photoelectron spectroscopy (ARPES) in the UV-range, revealing their band structure in reciprocal space. It was shown by spin-resolved ARPES, that the 2DEG at the surface of SrTiO_3 exhibits a Rashba-like spin splitting of approximately 100 meV, likely enhanced due to the presence of (anti)ferroelectricity and magnetic order at the sample surface <cit.>.The strong electron-phonon coupling of the TiO_2 surface <cit.>, which depends on carrier density, is most likely responsible for a drastic rise of the superconducting transition temperature of a monolayer FeSe deposited on top <cit.>. The variety of observed properties makes these oxide-based two-dimensional states an ideal platform to explore new functionalities and possible ways towards device application in the future.CaTiO_3 is the very first discovered perovskite of the transition metal oxide (TMO) family and is thus closely related to the members recent studies focus on. Like SrTiO_3, KTaO_3 and TiO_2 (all compounds shown to host a 2DEG at their surface) CaTiO_3 is classified as an incipient ferroelectric or quantum paraelectric material, meaning that it is very close to a ferroelectric phase <cit.>. Intermixtures of SrTiO_3, BaTiO_3 and CaTiO_3 form a rich phase diagram, especially regarding the ferroelectric properties, exhibiting para-, ferro- and antiferro-electric phases <cit.>. Pure, crystalline CaTiO_3 undergoes two phase transitions at elevated temperatures; from orthorhombic to tetragonal at 1512 K and from tetragonal to cubic at 1635 K <cit.>. According to band structure calculations for the orthorhombic and cubic crystal lattice the band gap is 2.43 eV or 2.0 eV, respectively <cit.>. In today's electronics, CaTiO_3 is widely used as a ceramic and as rare-earth doped phosphor with excellent luminescence properties.In this work, films of 20 unit cells CaTiO_3 grown by pulsed laser deposition (PLD) on Nb:SrTiO_3 substrates were studied by UV-ARPES and X-ray photoelectron spectroscopy (XPS). Our low-energy electron diffraction (LEED) measurements show that the surface of the CaTiO_3 films reconstruct while XPS indicates a TiO_2 terminated surface. In addition, observed surface plasmon loss features in the region of the Ti 2p core levels suggest the presence of metallic states at the surface of the films. Using ARPES, we found that these metallic states show a purely two-dimensional dispersion with a band width of ≈400 meV. Folded bands are visible as an effect of the surface reconstruction. In contrast to SrTiO_3 where the mixture of two- and three-dimensional states is observed <cit.>, this 2DEG is the only metallic state present at the surface. Therefore the CaTiO_3 surface states yield easy access to directly manipulate the two-dimensional transport properties of this system by surface structure or gating.Furthermore, with the ferroelectricity introduced in Sr_xCa_1-xTiO_3 this is a promising material to investigate the influence of ferroelectricity and the connected electric fields on the 2DEG at the surface of perovskites.§ MATERIALS AND EXPERIMENTAL METHODThe CaTiO_3 films of 20 unit cell thickness used for this study where grown by PLD on commercial TiO_2 terminated SrTiO_3 (001) substrates with a niobium doping of 0.5 wt% (Twente Solid State Technology BV). The growth was performed at a substrate temperature of 680^∘ C in partial oxygen pressure of 5×10^-5 mbar. The growth process and film thickness was monitored by reflection high-energy electron diffraction. The prepared films were in-situ transferred to the experimental station at the Surface and Interface Spectroscopy beam line of the Swiss Light Source at the Paul Scherrer Institut under ultra high vacuum (UHV) conditions and measured without further treatment.The sample was held at a temperature of 20 K in pressures better than 8×10^-11 mbar during the measurements. Photoemission spectra (XPS and ARPES) were taken using a Scienta R4000 hemispherical electron analyzer and circular polarized synchrotron light. LEED patterns were obtained at 20 K before the ARPES measurements. The atomic force microscopy (AFM) topography was measured at the NanoXAS beam line of the Swiss Light Source at the Paul Scherrer Institut with the sample at room temperature in UHV environment.The orthorhombic unit cell of bulk crystalline CaTiO_3 has lattice parameters of a=5.367 Å, b=7.644 Åand c=5.444 Å <cit.>. An approximate representation of the orthorhombic unit cell can be made by a pseudo-cubic unit cell as marked in Fig.<ref>a). The lattice parameters of the pseudo-cubic unit cell a/√(2)≈ b/2 ≈ c/√(2)≈ 3.822 Å are similar to cubic SrTiO_3 with a lattice mismatch of approximately 2%. In LEED we can identify the primary diffraction spots corresponding to the pseudo-cubic unit cell. Further we observe spots indicating a c(4×2) surface reconstruction of the pseudo-cubic lattice with domains rotated 90^∘ with respect to each other (see Fig.<ref>b) and c)). The (1×1) TiO_2 terminated surface at the vacuum interface of TMO perovskites might be unstable due to the unshared oxygen atom of the TiO_2 polyhedron sticking out of the surface.Of the surface reconstructions reported for the closely related SrTiO_3 system, c(4×2) reconstruction has also been observed <cit.>.The AFM topography in Fig.<ref>d) shows that the films are of low roughness and follow the substrate steps with a terrace size of approximately 200 nm. However, the AFM measurements do not have the resolution required to observe the surface reconstruction. The observed presence of domain walls is a further indication of the existence of multiple rotated domains corroborating the LEED data.§ RESULTS AND DISCUSSIONThe XPS spectrum of the films in Fig.<ref>, measured with a photon energy of hν = 600 eV, shows clear signatures of the expected calcium, titanium and oxygen core levels with no detectable contamination. Comparing the spectra taken with the sample surface normal to the analyzer to the more surface sensitive measurement taken at an angle of 45^∘ between the sample normal and the analyzer axis (see sketch inset in Fig.<ref>) we can confirm the TiO_2 termination of the grown films. This termination of the film surface is expected due to the TiO_2 termination of the SrTiO_3 substrate <cit.>. When comparing the peak areas (A_i) after background subtraction the ratio A_Ca 2p/A_Ti 2p of 0.75 at normal emission is significantly higher than the ratio of 0.65 measured at an emission angle of 45^∘. All the titanium peaks show a shoulder towards lower binding energy, indicating the existence of titanium atoms with different valency. The increase of the surface located Ti 3^+ shoulder is a light induced effect commonly observed in this class of materials <cit.>. The appearance of Ti 3^+ ions is likely linked to a distortion of the TiO_2 octahedron, for example due to the creation of oxygen vacancies in the surface region and/or a structural rearrangement and buckling of the surface layers.The Ti 2p as well as the Ca 2p core levels show plasmon loss peaks in their shake-up tail with an energy loss of 13.2 eV for titanium and 9 eV for calcium. Plasmon loss peaks with this loss energy of the Ti 2p core levels have been observed in other perovskites. The measured plasmon energy corresponds to surface plasmons present in TiO_2 where the plasmons are trapped at the interface of the metallic surface and the dielectric bulk due to the sudden change in dielectric constant. <cit.>Consequently, we also expect metallic states to be present at the surface of our CaTiO_3 films. Indeed, the ARPES measurements in Fig.<ref> show an electron-like surface state. The scanover a wide range of photon energies in Fig.<ref>a) shows no dispersion of these states with out-of-plane momentum, verifying their two-dimensional nature. In contrast to the well-studied metallic states present at the surface of SrTiO_3 (001) and KTaO_3 (001) <cit.> we have no indication of three-dimensional features, making the 2DEGthe only states contributing to the metallicity. Similar to the other perovskites the spectral intensity of the 2DEG at the CaTiO_3 surface increases under UV-irradiation. This is attributed to light induced surface rearrangements and induced carriers <cit.>. The circular Fermi surface of Γ_( 100 ) is depicted in Fig.<ref>b) and the corresponding clear free-electron-like parabolic band along the high symmetry direction Γ in Fig.<ref>c). The in-plane momentum g≈ 1.57 Å^-1 at the ring center, corresponding to the momentum of Γ_( 100 ), is equal to a lattice parameter of a≈ 4 Å.This is in good agreement with the lattice parameter of the pseudo-cubic unit cell of CaTiO_3 and the SrTiO_3 substrate. Also clearly visiblein Fig.<ref>a) is the intensity at the Fermi energy of an additional, folded parabola between Γ_( 000 ) and Γ_( 100 ) due to the surface reconstruction observed also in LEED as described in section <ref>. Similar band folding has been observed for the (1×4) reconstructed anatase TiO_2 films <cit.>. The band structure of the 2DEG in Fig.<ref>c) and <ref>b) can be fitted with a free-electron-like parabola yielding an effective mass of m^* ≈ 0.39 m_e, a Fermi momentum of k_F ≈ 0.20 Å^-1, a Fermi velocity of v_F ≈ 6.3×10^5 m/s, and a band minimum at a binding energy of E_b≈400 meV. This corresponds to a charge carrier density per parabola of 6.4 × 10^13 cm ^-2 or 0.1 e^-/a^2 with a=3.822 Å. This charge carrier density is similar to SrTiO_3 <cit.> while the band width is significantly higher and the effective mass much lower than for SrTiO_3 and KTaO_3.The ARPES measurements with s- and p-polarized light in Fig.<ref>(c-f) confirm the xy-symmetry of the 2DEG with no indications of bands with xz- or yz-symmetry. The 2DEG thus consists of the Ti 3d_xy bands splitted from d_xz/d_yz by crystal field splitting and partially filled due to surface band bending and light induced carriers. With the absence of the Ti 3d_xz and 3d_yz bands and the two-dimensional Ti 3d_xy bands at relatively high binding energies, the splitting between the d_xy and d_xz/d_yz-bands has to be large, at least of the size of the observed bandwidth of ≈400 meV. This splitting is considerably larger than the 240 meV measured for SrTiO_3 <cit.> but smaller than for TiO_2 anatase where 1 eV is reported <cit.>. For the orthorhombic oxide LaAlO_3 a comparable noncubic crystal field splitting of 120 meV to 300 meV for the t_2g sub shell is reported <cit.>. However, there is no detectable additional splitting of the Ti 3d_xy band as observed for SrTiO_3 <cit.>. Comparing SrTiO_3 to CaTiO_3 the increased rotation of the TiO_3 octahedron in the later due to the orthorhombicity will likely reduce the local electric fields as observed in other perovskites <cit.>. The resulting weak polarization field at the surface could be the reason that the splitting is too small to be observed in our data.The results of the fitting are indicated in Fig.<ref>a) and b) showing the circular Fermi surface composed by parabolic bands for the primary Γ-points as well as for the reconstructed Γ-points. Along the Γ direction, the Fermi surfaces and parabolic bands corresponding to the folded Γ-points, which are present as a result of the reconstruction, are clearly visible in the data. However their intensity is weaker than the signal of the 2DEG at the primary Γ-points. In contrast to the folding along the high-symmetry direction, the Γ-points offset by 1/4· g in k_y direction are not present in the data. A possible reason for this is a change of the reconstruction from c(4×2) either to a combination of (2×1) and c(2×2) or more likely to p(2×2) under irradiation with UV-light. Since we observe an increasing intensity of the 2DEG as well as the described formation of a low binding energy shoulder on the titanium core levels under UV-light, a change of reconstruction under light due to the deposited energy is plausible. § SUMMARYIn conclusion, we have revealed the existence of metallic states at the surface of CaTiO_3 films consisting solely of a 2DEG. The 2DEG has a band width of ≈400 meV, indicating a large splitting between the unoccupied Ti 3d_xz/d_yz bands and the two-dimensional 3d_xy bands. Due to its metallicity, the surface also hosts plasmons visible as loss peaks in the XPS data. The bands are folded according to the surface reconstruction that is likely changed from c(4×2) to p(2×2) under UV irradiation.Due to the lack of higher-dimensional conducting channels and the affinity of the system to adapt to its surface structure, various paths open up to directly manipulate the surface states. This manipulation may give direct access to the transport properties of the system and its coupling to overlayers.With the possibility to induce ferroelectricity into the quantum paraelectric materials CaTiO_3 and SrTiO_3 by mutual doping, the phase space is open to probe the effect of ferroelectricity on the 2DEG hosted by both of these materials. § ACKNOWLEDGMENTSWe acknowledge financial support from the Swiss National Science foundation Project No. PP00P2_144742/1. elsarticle-num00url<#>1urlprefixURL href#1#2#2 #1#1Ohtomo:2004 A. Ohtomo, H. 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"authors": [
"Stefan Muff",
"Mauro Fanciulli",
"Andrew P. Weber",
"Nicolas Pilet",
"Zoran Ristic",
"Zhiming Wang",
"Nicholas C. Plumb",
"Milan Radovic",
"J. Hugo Dil"
],
"categories": [
"cond-mat.mes-hall",
"physics.app-ph"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170526092601",
"title": "Observation of a two-dimensional electron gas at CaTiO$_3$ film surfaces"
} |
Department of Physics, The Ohio State University, Columbus, OH 43210, USA The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy S_topo for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy S_nonlocal (a generalization of S_topo). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type 1 and type 2 fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that S_nonlocal could obtain geometry-dependent values, and S_nonlocal is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed. Deciphering the nonlocal entanglement entropy of fracton topological orders Yuan-Ming Lu December 30, 2023 =========================================================================== § INTRODUCTION Topological order <cit.> is a gapped quantum phase of matter beyond the description of the Landau-Ginzburg theory of symmetry breaking. Many of the early examples of topological orders (which we will refer to as conventional topological orders) share the following properties: robust ground state degeneracy which depends on the topology of the manifold <cit.>, the ground states are locally indistinguishable <cit.>, the existence of integer dimensional condensates and logical operators that can be topologically deformed <cit.>, nontrivial braiding statistics of anyons or other topological excitations (or topologically charged excitations)e.g., excitations which could not be created alone by local operators <cit.>, they are effectively described by topological quantum field theory (TQFT) at low temperatures <cit.>, and they can be used to do fault-tolerant quantum information processing <cit.>. And it is well known that, in 2D, a suitable linear combination of entanglement entropy with local contributions canceled is a topological invariance called the topological entanglement entropy<cit.>. Topological entanglement entropy is a property of the ground state wave function and it has been used to identify quantum spin liquid phases <cit.>. It also contains information about the ground state degeneracy <cit.> and the forms of low-energy excitations <cit.>. Generalizations of topological entanglement entropy into 3D bulk <cit.> and boundary <cit.> are studied.On the other hand, there are recently discussed 3D exotic topological ordered models <cit.> that do not fit very well into the pictures above. These models have recently been classified into fracton topological orders <cit.>. While fracton models have locally indistinguishable ground states when placed on nontrivial manifolds and the ground state degeneracy is robust under local perturbations <cit.>, the ground state degeneracy depends on the system size (geometry) rather than merely the topology of the manifold <cit.>. While fracton models possess topological excitations <cit.>, these topological excitations areconstrained to move in lower dimensional submanifolds rather than the whole system <cit.>. The condensates and logical operators can be fractal dimensional <cit.> instead of integer dimensional. These models are beyond the description of TQFT. There are type 1 and type 2 fracton topological orders. The type 1 fracton models include the Chamon-Bravyi-Leemhuis-Terhal (CBLT) model <cit.>, the Majorana cubic model <cit.>, and the X-cube model <cit.>, etc.; they have integer dimensional condensates and logical operators. The type 2fracton models include Haah's code <cit.> and many of thefractal spin liquid models <cit.> (see Sec.<ref>). Type 2 fracton models possess fractal condensates and logical operators and the excitations are fully immobile <cit.>. For the ground state entanglement properties of fracton models, a relation to the ground state degeneracy is implied in <cit.> and the entanglement renormalizationgroup transformation of Haah's code is studied in <cit.>. In this work, we construct a direct analogy of topological entanglement entropy by doing linear combinations of entanglement entropies of different subsystems in such a way that the local contributions (from each boundary or corner of the subsystems) are canceled, and we call the linear combination S_nonlocal, the nonlocal entanglement entropy. While S_nonlocal is topologically invariant for conventional topological orders, it is geometry-dependent for fracton models.Explicitly, we choose a conditional mutual information form used by Kim and Brown <cit.> and define the nonlocal entanglement entropy S_nonlocal≡ (S_BC+S_CD-S_C-S_D)|_ρ=I(A:C| B)|_ρ. Where ρ=|ψ⟩⟨ψ| with |ψ⟩ being a ground state.The whole system is the union of subsystems A, B, C, and D withA and C separated by distance l≫ξ, the correlation length. One can check that the local contributions from the boundaries are canceled, and this is why we use the name “nonlocal entanglement entropy." This construction can be used in any dimensions and an example in 2D is shown in Fig.<ref>. With several assumptions about the condensates and topological excitations, a lower bound of S_nonlocal is derived.When applied to known conventional Abelian topological orders e.g. the 2D toric code model and the 3D toric code model <cit.>, the lower bound is topologically invariant and it is identical to the exact result, i.e. the lower bound is saturated. When applied to fracton models <cit.>,the lower bound depends on the sizes and relative locations of the subsystems. For fracton models, there exist choices of subsystems for which the lower bound is nonzero and extensive. It is possible to have S_nonlocal>0for subsystem choices, that are expected to have S_nonlocal=0 if TQFT holds.This method observes an intimate relation between S_nonlocal and topological excitations created in D by a unitary operator U stretched out in CD which could be “deformed" into a unitary operator U^def in AD, and since deformable U is intimately related to condensate operator W (for more details of U, U^def and condensate operator W see Sec.<ref>), this method observes an intimate relation between S_nonlocal and ground state condensates as well.This method allows a lower bound of S_nonlocal to be obtained without calculating the entanglement entropy of any individual subsystem. Furthermore, it provides us with a unified viewpoint to understand the topology-dependent S_nonlocal in the conventional topological orders and the geometry-dependent S_nonlocal in fracton topological orders. Also discussed is the stability of the lower bound under local perturbations.Two additional papers appeared after our work which also study the entanglement entropy of fracton phases, using explicit computation<cit.> and tensor network <cit.>.For non-Abelian models, some of our assumptions breakdown, andour original method does not apply. Nevertheless, a variant of our lower bound is applicable to non-Abelian models <cit.>. The structure of the paper is as follows: In Sec.<ref> we provide a derivation of the lower bound from some assumptions about topological excitations and condensates; In Sec.<ref> we apply our lower bound to several exactly solved Abelian stabilizer models of 2D, 3D conventional topological orders and type 1, type 2 fracton topological orders. In Sec.<ref> we discuss the stability of the lower bound under local perturbations. Sec.<ref> is discussion and outlook.§ THE LOWER BOUND §.§ A few notations and definitionsWe will consider a infinite system without boundaries. The system is divided into subsystems A, B, C, D (nonoverlapping regions in real space, the union of which is the whole system). Each subsystem has a size large compared to the correlation length ξ, and the subsystems A and C are separated by a distance much larger than the correlation length. One example is shown in Fig.<ref>, and similar constructions can apply to any dimensions. For all the examples in this paper, we have chosen B, C, D to be local subsystems while A is not, but there exist other possible choices, say A, B, C local. A local subsystem is a subsystem which can be contained in a ball-shaped subsystem of finite radius. ∂ A, ∂ B are the boundaries of the subsystems. We use A̅ to denote the complement of A.We will use ρ, σ to represent density matrices. In this paper we always use ρ for the ground state density matrix, ρ=|ψ⟩⟨ψ|, and |ψ⟩ is the ground state. We use ρ_ABC, σ_BCwhen we want to specify the subsystems. The entanglement entropy is defined in terms of the (reduced) density matrix as usual S=-tr [σlnσ].WeuseS_ABC|_ρ and S_ABC|_σ to distinguish the entanglement entropy on region ABC with different density matrices ρ, σ. Defineconditional mutual informationI(A:C| B)≡ S_AB+S_BC-S_B-S_ABCand we use I(A:C| B)|_ρ when we want to specify a density matrix. It is known that the conditional mutual information is always nonnegative I(A:C| B)≥ 0. We say σ_ABC is conditionally independent if I(A:C| B)|_σ=0.For unitary operatorsU and U' which create excitations in D when acting on the ground state |ψ⟩, we say U∼ U' or U is similar to U' if the states U|ψ⟩ and U'|ψ⟩ have identical reduced density matrices on ABC, i.e. tr_D [Uρ U^†]= tr_D [U'ρ U'^†]. Otherwise, we say U and U' are distinct. §.§ Prepare for the lower boundIf there is a density matrix σ which is related to the ground state density matrix ρ by ρ_AB=σ_AB and ρ_BC=σ_BC, then we have:I(A:C| B)|_ρ≥ S_ABC|_σ-S_ABC|_ρand the “=" happens if and only if I(A:C| B)|_σ= 0. For a proof, observe that I(A:C| B)|_ρ and I(A:C| B)|_σ has only a single different term, and that I(A:C| B)|_σ≥ 0.For a pure state, the entanglement entropy of a subsystem equals the entanglement entropy of its complement, e.g. S_Ω=S_Ω̅ for any subsystem Ω. Therefore:I(A:C| B)|_ρ= (S_BC+S_CD-S_B-S_D )|_ρ. Observe that the local contributions of the entanglement entropy get canceled due to the fact that A and C are separated. Let us define the nonlocal entanglement entropy (of the ground state)S_nonlocal≡ (S_BC+S_CD-S_B-S_D)|_ρ.The nonlocal entanglement entropy is just another way to write down the conditional mutual information, S_nonlocal=I(A:C| B)|_ρ, and therefore S_nonlocal≥ 0. The form in Eq.(<ref>) has the advantage that it involves only local systems B, C, D. When the system is placed on a torus or other nontrivial manifolds instead of a infinite manifold, the system may have several locally indistinguishable ground states, this form of S_nonlocal in terms of local subsystems is more convenient, and even if ρ is a mixed state density matrix of different locally indistinguishable ground states, S_nonlocal still has the same value.§.§ The key idea about the lower boundThe discussion above suggests a way to obtain a lower bound of S_nonlocal. For any σ satisfyingσ_AB=ρ_AB and σ_BC=ρ_BC,S_nonlocal≥ S_ABC|_σ-S_ABC|_ρ.The density matrix σ does not have to be a density matrix of a pure state. If we could find a σ satisfying the above requirement andS_ABC|_σ>S_ABC|_ρ,a nonzero lower bound is obtained, and then the existence of nonzero nonlocal entanglement entropy is established.Now, let us assume that we could find a set of σ_I with I=1,…,N such that σ_I AB=ρ_AB and σ_I BC=ρ_BC. Then we can do superpositions and define σ≡∑_I=1^N p_I σ_I with {p_I } being a probability distribution, i.e. p_I∈[0,1] and ∑_I=1^N p_I=1. The σ is a new density matrix which satisfies σ_AB=ρ_AB and σ_BC=ρ_BC. So we have a whole parameter space of σ to try.If lucky, we may even be able to find a σ^∗_ABC which is conditionally independent (satisfying I(A:C| B)|_σ^∗=0) andwehave an exact resultS_nonlocal= S_ABC|_σ^∗-S_ABC|_ρ.Or, if we find the lower bound is saturated, we know the σ we used to obtain the lower bound is conditionally independent. We note that, in the quantum case (unlike the classical case), it is not always possible to find a conditionally independent σ^∗ such that σ^∗_AB=ρ_AB and σ^∗_BC=ρ_BC for ρ being a general density matrix <cit.>.Therefore, the existence of such σ^∗ in some system might be interesting by itself. On the other hand, the conditional independent state σ^∗ is known to exist for models satisfying simple conditions(1)(2) in <cit.>.§.§ Calculate the lower bound for Abelian models employing assumptionsWe make a fewassumptions about the condensates and operators creating topological excitations in order to develop away to findσ_I and calculate a lower bound of S_nonlocal. These assumptions are applicable to Abelian models with commuting projector Hamiltonians (with each term acting on a few sites localized in real space), including conventional models e.g. the toric code model in various dimensions <cit.>, quantum double models with any Abelian finite group, andfracton models <cit.> of type 1 and type 2which we will be focusing on in this work. In the context of fracton models, Abelian means no protected degeneracy associated with excitations. Our way of employing the operators is inspired by a method by Kim and Brown <cit.>, where an interesting connection between conditional mutual information and deformable operator U is obtained. While the subsystems we choose have only an unimportant difference from <cit.>, our result is different. The result in <cit.> shows that if S_nonlocal=0, there will be no topological excitations, and therefore, S_nonlocal>0 is needed for the existence of topological excitations. The result is very general since very small amount of assumptions was used. Nevertheless, the result was not powerful as a lower bound for S_nonlocal. In this work, on the other hand, we use more detailed properties of topological excitations and condensates to obtain a powerful lower bound of S_nonlocal. Our method shows that the key to have S_nonlocal>0 in these models is the nonlocal nature of the ground state condensates and the operators creating topological excitations. S_nonlocalcan be extensive in the subsystem size and it is not necessarilytopologically invariant. Whether S_nonlocal is topologically invariant or not depends on whether the operators can be deformed topologically. Before rigorously stating the assumptions and deriving the lower bound, here are a few words about the physical picture. The ground states of topological orders condense extended objects. If a unitary operator W (which has an extended support) acting on the ground state |ψ⟩ gives you W|ψ⟩ =e^iφ|ψ⟩,we call the operator W a condensate operator with eigenvalue e^iφ. Whenever confusion can be avoided, we may call W a condensate for short. Let us further assume W to be a tensor product of operators acting on each site. Then, a suitably defined “truncation" of a condensate operator W onto a subsystem Ω gives you a new operator U. U|ψ⟩ is an excited state withtopologically excitations located around ∂Ω∩ W, and U can be deformed in the sense that you could choose a “truncation" of W^† onto Ω̅ and call it U^def, which creates the same topological excitations and satisfies U|ψ⟩= U^def|ψ⟩. One the other hand, if we have unitary operators U and U^def satisfying U|ψ⟩= U^def|ψ⟩, then [U^def]^† U|ψ⟩ =|ψ⟩, and therefore [U^def]^† U is a condensate operator with eigenvalue +1. Intuitively, a condensate operator W can be “truncated" into adeformable operator U which creates topological excitations, and a pair of deformable operators U and U^def can be “glued together" into a condensate operator W. Therefore U and W are closely related. Some examples of condensate operator W and deformable operator U,are shown in Fig.<ref>. Once we have condensate operators W_i and deformable operators U_j, we use U_j to create topological excitations in D which result in some states σ_I which is identical to the ground state ρ on AB and BC. If the excitations created in D are topological excitations, they can not be created by an operator supported on D, andσ_I ABC and ρ_ABC will have some difference. The difference is detected by a change of the eigenvalue of condensate operator W_i supported on ABC.Then, we use σ_I to obtain a lower bound of S_nonlocal.The following are our assumptions 𝐔-1, 𝐔-2, 𝐔-3; 𝐖-1, 𝐖-2:Assumption 𝐔-1: There exists a set of unitary operators {U_i}supported on CD and a set of unitary operators{U_i^def}supported on AD, with i=1,⋯, M. See Fig.<ref> for an example.When acting on the ground state |ψ⟩, U_i can be “deformed" into U_i^def, i.e.:U_i|ψ⟩ =U_i^def|ψ⟩.Assumption 𝐔-2: For any subsystem Ω, the unitary operator U_i can always be written as a direct product of unitary operators U_i Ω and U_i Ω̅ which act on the subsystem Ω, Ω̅ respectively:U_i=U_i Ω⊗ U_i Ω̅. Assumption 𝐔-3: There are integers n_i such that U_i^n_i=1, and when multiple U_i act on a ground state |ψ⟩, we have the following:(∏_i=1^M U_i^k_i)|ψ⟩ =e^iδ({k_i})(∏_i=1^M [U_i^def]^k_i)|ψ⟩,where integer 0≤ k_i≤ n_i-1, and we allow possible phase factors e^iδ({k_i }).Assumption 𝐖-1: There exists a set of unitary operators {W_i} supported on subsystem ABC such thatW_i |ψ⟩= |ψ⟩ i=1,…,M. Assumption 𝐖-2: The following relation between W_i and U_j holds:W_iU_j=U_jW_i e^iθ_ijwithθ_ij=2π/n_iδ_ij,where δ_ij is the Kronecker delta.Comments about the assumptions:1) 𝐔-1 implies that when U_i is acting on the ground state, it can create excitations only in D, but not in ABC.2) We do not assume U_i, W_i to be string operators and not even assume U_i, W_i to be integer dimensional operators. In fact, we will apply this method to fractal operators later.3) In 𝐔-1 we assumed U can be deformed without changing the excitations. Nevertheless, we do not assume U can be deformed into all topologically equivalent configurations, and we do not assume U can be deformed continuously. As we will see below, in fracton models,deformations exist in weird form, U may not be topologically deformed (i.e. deformed continuously into any topologically equivalent configuration), and U can sometimes be deformed in discontinuous ways into topologically inequivalent configurations.4) 𝐔-2 is not true when a local perturbation is added. We will address the stability of the lower bound under local perturbations separately in Sec.<ref>.5) For the non-Abelian case, the entanglement entropy of an excited state could depend on the quantum dimension of the anyon <cit.>, and𝐔-2 does not apply. On the other hand, the idea in Sec.<ref>still holds, a saturated lower bound for non-Abelian models is recently discussed in <cit.>.6) For systems with boundaries, one may choose D being region attached to boundaries, as is done in <cit.>. An alternative way is to identify D witha boundary region ∂Ω; in this case, 𝐔-1 should be understood as: U_i being an operator supported on C and attached to ∂ C∩∂Ω, and U_i^def being an operator supported on A and attached to ∂ A∩∂Ω.7) According to 𝐖-2, |ψ⟩ and U_i|ψ⟩ are eigenstates of W_i with different eigenvalues, where |ψ⟩ is the ground state. Since W_i is supported on ABC, this implies that U_iis distinct from the identity operator. Similarly, U_i and U_j are distinct for i j. We will refer to this change of eigenvalue of W_i as a detection, e.g. U_i is detected by W_i. The requirement θ_ij=2π/n_iδ_ij is not crucial, and it can be replaced by other numbers as long asthe operator set { W_i } can detect the difference among the set of operators {U_i }.8) For a relatively simple class of models, which has the Hamiltonian H=-∑_k h_k, [h_i,h_j]=0 and h_k^2=1, there isan obvious class of operators that satisfy Eq.(<ref>) in 𝐖-1, namely W_i=∏_k∈ℰ_i h_k, where ℰ_i is a subset of the stabilizer generators. For the ground state |ψ⟩, we have h_i|ψ⟩ =|ψ⟩, it follows that W_i|ψ⟩=|ψ⟩. As U_i could not flip stabilizers in ABC, ℰ_i must contain some h_j in D. It turns out that this simple observation applies to all the Abelian stabilizer models we will use as examples in Sec.<ref>. However, we do not provide a general procedure to find the subset ℰ_i for fracton models. On the other hand, our methodworksfor models not in this simple class also, such as quantum double models <cit.> with Abelian finite groups.Define the set of states|{k_i};ψ⟩≡∏_i=1^MU_i^k_i|ψ⟩with 0≤ k_i ≤ n_i-1with k_i being integers. Define σ({k_i})≡|{k_i};ψ⟩⟨{k_i};ψ|.Note the total number of σ({ k_i}) is ∏_i=1^M n_i. Relabel σ({ k_i}) using a new index I=1,⋯, N with N=∏_i=1^M n_i and call them σ_I. One immediately varifies that 1) 𝐔-1, 𝐔-3 ⇒σ_I AB=ρ_AB and σ_I BC=ρ_BC;2) 𝐖-1, 𝐖-2 ⇒ σ_IABC·σ_J ABC=0 for I J;3) 𝐔-2 ⇒ σ_I ABC=V_I ρ_ABCV_I^† and S_ABC|_σ_I=S_ABC|_ρ.Where V_I is some unitary operator acting on subsystem ABC and recall that ρ is the ground state density matrix.Let σ=∑_I=1^N p_I σ_I with probability distribution {p_I },one derives thatS_ABC|_σ-S_ABC|_ρ=-∑_I=1^N p_Iln p_I ≤ln N“=" if and only if p_I=1/N for all I. From Eq.(<ref>) we findS_nonlocal|_ρ≥ln N =∑_i=1^M ln n_i.§.§ When is the lower boundtopologically invariant?It is instructive to think of the conditions under which our lower bound of S_nonlocal is topologically invariant. Consider a chosen set of subsystems A, B, C, D and the operator sets {U_i }, {U_i^def} and {W_i}. Let us do “topological deformations" of the subsystems andthe operator sets. Here, by “topological deformation" of the operatorsets we mean that we can topologically deform the support of each operator to get new operator sets which preserve the algebra in Eq.(<ref>,<ref>,<ref>,<ref>,<ref>). Note that, these deformations generally change the positions of the excitations, which should be contrasted with the type of deformation in 𝐔-1, in which the positions of the excitations never change. When these conditions are satisfied, the lower bound for the two topologically equivalent choices of subsystems are the same. If such conditions are satisfied for each pair of topologically equivalent choices of subsystems, then our lower bound will be topologically invariant.As is shown in the examples below in Sec.<ref>, S_nonlocal can be either topologically invariant or not, and it is instructive to think of how the conditions above are violated in fracton models <cit.> in which S_nonlocal depends on the geometry of subsystems. § APPLICATIONSIn this section, our lower bound is applied to several stabilizer models of Abelian phases: the 2D and 3D conventional topological ordersand type 1, type 2 fracton phases.§.§ The 2D Toric Code ModelFor a 2D topological order, choose the subsystems A, B, C, D of the same topology as is shown in Fig.<ref>. From the well-known results <cit.>, one derives S_nonlocal=2γ where -γ is the topological entanglement entropy. For the 2D toric code model <cit.>: On a square lattice with a qubit on each link,the Hamiltonian isH=-∑_s A_s -∑_p B_p,where A_s is a product of X_r of a “star" or vertex, B_p is a product of Z_r of a plaquette.A_s=∏_r∈ sX_r B_p=∏_r∈ p Z_r,where X_r, Z_r are Pauli operators acting on the qubit on link r.The ground state of toric code model condenses two types of closed string operators, and the corresponding open string operators (which could be regarded as truncations of closed string operators) create topological excitations at the endpoints.We find the following unitary operators U_1, U_2, W_1, W_2 as is shown in Fig.<ref>. U_1 and W_2 are products of X_r; U_2 and W_1 are products of Z_r. Also notice the feature that the closed string operators W_1(W_2) can be written as a product of stabilizers B_p (A_s) on a 2D disk region surrounded by the corresponding closed strings. 𝐔-1, 𝐔-2, 𝐔-3, 𝐖-1, 𝐖-2 can be checked.The operators satisfy:U_1^2=U_2^2=W_1^2=W_2^2=1 W_iU_j=U_jW_i e^iπδ_ij i,j=1,2.Therefore M=2, n_1=n_2=2 andN=n_1n_2=4. Using the result in Eq.(<ref>), one derives S_nonlocal≥ 2ln 2. By comparing with the knownresult γ=ln 2, S_nonlocal=2γ=2ln 2, we find that our lower bound is saturated. A by-product of a saturated lower bound is an explicit construction of a conditionally independent σ^∗. In the toric code case:σ^∗ =1/4(ρ +U_1 ρ U_1^† + U_2 ρ U_2^† + U_1 U_2 ρ U_2^† U_1^†)and σ^∗ satisfies:1) σ^∗_AB=ρ_AB, σ^∗_BC=ρ_BC; 2) I(A:C| B)|_σ^∗=0.Where ρ=|ψ⟩⟨ψ| is the ground state density matrix.The observation in Sec.<ref> explains why the lower bound is topologically invariant in the toric code model: the operators U_i and W_i can be topologically deformed together with the subsystems A, B, C and D, without changing the algebra in Eq.(<ref>,<ref>,<ref>,<ref>,<ref>).This method can be applied to other 2D Abelian topological orders, e.g., quantum double models with Abelian finite groups, and the lower bounds are saturated. For a variant of the method for non-Abelian models, see <cit.>.§.§ The 3D Toric Code model The 3D toric code model <cit.> is defined on a cubic lattice, with one qubit on each link. The Hamiltonian is of exactly the same form as the one of the 2D toric code model:H=-∑_s A_s -∑_p B_p; A_s=∏_r∈ sX_r;B_p=∏_r∈ p Z_r.Here a star s includes the 6 links around a vertex, and a plaquette p is a square consistent of 4 links.The ground state of the 3D toric code model condenses one type of closed string and one type of closed membrane. There is one type of open string operator that creates point-like topological excitations at the endpoints and one type of open membrane operator which creates loop-like topological excitations at the edge of the membrane, see Fig.<ref>.§.§.§ Subsystem types for 3D modelsThe 3D toric code model is the first 3D model we discuss, and it is a good place to introduce subsystem types for 3D models whichwill be discussed for all 3D models. Wefocus on the following three topologically distinctsubsystem types, i.e. the type-α,β,γ shown in Fig.<ref>, although other choices are possible. We will use the notation S^(α)_nonlocal, S^(β)_nonlocal, S^(γ)_nonlocal to distinguish the nonlocal entanglement entropy for the three topological types. Type-α hasD, which consists of two disconnected boxes, while C D is connected, and it can be used to detect open string-like U_i,which is attached to the two boxes of D. Type-β has D of the topology of a solid torus (and therefore D is not simply connected), while C D is simply connected. It can be used to detect open membrane-like U_i supported on C D which create excitations in D as noncontractible loops. Type-γ has C and CD of the same topology, e.g. the topology of a solid torus, and B is simply connected.Type-α and type-β have already been implied in paper by Kim and Brown <cit.>, in which, similar subsystems types are used to study different types of boundaries of 3D models.For type-γ, S_nonlocal^(γ)=0 for models satisfying the assumptions in <cit.>, e.g. the entanglement entropy of a general subsystem Ω (which haslarge size compared to correlation length) can be decomposed into localplus topological parts:S_Ω= S_Ω,local +S_Ω,topological⇒ S_nonlocal^(γ)=0.Therefore, a model with S_nonlocal^(γ)>0is a model beyond the description of<cit.>. Fractal models do have S^(γ)_nonlocal>0 for some choices of the subsystems and the value can be extensive.Furthermore, the contribution of S^(α)_nonlocal (S^(β)_nonlocal) is not necessarily from open string-like (open membrane-like) U.§.§.§ The 3D Toric Code model has saturated lower bounds for each subsystem typeLet us go back to the 3D toric code model.For type-α,we find M=1, n_1=2 where the operator U_1 is an open string operator which creates a point-like topological excitation in each box of D and W_1 is a closed membrane operator. Therefore N=2 and S_nonlocal^(α)≥ln 2.For type-β, we find M=1, n_1=2 where the operator U_1 is an open membrane operator which creates a loop-like topological excitation at the edge of the membrane (the loop could not continuously shrink within D into a point), and W_1 is a closed string operator. Therefore N=2 and S_nonlocal^(β)≥ln 2.For type-γ, operators supported on CD, which create excitations in D, could always be deformed into D. This is because, for 3D toric code, the operators that create topological excitationscan be topologically deformed keeping the excitations fixed. Therefore we obtain a lower boundS_nonlocal^(γ)≥ 0.Comparing with the known entanglement properties <cit.> of the 3D toric code model, our lower bounds are identical to the exact results:S^(α)_nonlocal=S^(β)_nonlocal=ln 2; S^(γ)_nonlocal=0.§.§ The X-Cube ModelThe X-cube model is a 3D exactly solved stabilizer model, and it is an example of type 1 fracton phase <cit.>. The model is defined on a cubic lattice with one qubit on each link. The Hamiltonian isH=-∑_c A_c -∑_s (B^(xy)_s + B^(yz)_s +B^(zx)_s),where A_c is a product of Z_r on a cube (which includes 12 links), and B^(xy)_s, B^(yz)_s, and B^(zx)_s are products of X_rof 4 links around a vertex which are parallel to the xy-plane, yz-plane, zx-plane respectively. Here we focus ontype-α and consider thegeometry dependence of S_nonlocal, see Fig.<ref>. We find the following lower bounds:Fig.<ref>a, S^(α)_nonlocal≥ (2l_D +O(1))ln 2;Fig.<ref>b, S^(α)_nonlocal≥ (4l_D +O(1))ln 2;Fig.<ref>c, S^(α)_nonlocal≥ 0.Where O(1) denotesorder one contributions which dependent on the detailed shapes of the subsystems, which is not crucial for our discussion.The types of U_i that contribute to the S^(α)_nonlocal in Fig.<ref>a are illustrated in Fig.<ref>a, each U_i stretches out in directions parallel to the yz-plane and creates a pair of “dimension-2 anyons." The translations of the operators U_iin Fig.<ref>a in (1,0,0) direction give you distinct operators (while translations in (0,1,0) or (0,0,1) directions do not give you distinct operators). This gives M=2l_D +O(1).Translations can produce distinct operators, this indicates a breakdown of topological deformation: in the X-cube model, U_iis deformable but not topologically deformable. Another nice example of the breakdown of topological deformation is shown in Fig.<ref>c, in which U and its translations U_T_1, U_T_2, and U_T_1+T_2 are distinct; nevertheless, by thinking of the condensate in Fig.<ref>b, one can show U∼ U_T_1· U_T_2· U_T_1 +T_2.The result S^(α)_nonlocal≥ (4l_D +O(1))ln 2 for Fig.<ref>b can be understood by thinking of contributions from operators parallel to the yz-plane and the xy-plane, which gives M=4l_D +O(1). The resultS^(α)_nonlocal≥ 0 in Fig.<ref>c comes from the fact that the types of U_i discussed above could not connect the two boxes of D separated by a displacement vector d⃗=(d_x,d_y,d_z) with | d_x|, | d_y|, | d_z| >l_D and the inability to find U_i gives M=0.These results for S^(α)_nonlocal may also be calculated using the method in <cit.> and an independent estimation agrees with our lower bounds up to O(1) contributions.The lower bounds of S^(α)_nonlocal for all the cases in Fig.<ref> depend on the length scale l_D and the displacement vector d⃗ but are not sensitive to other details. This is due to the fact that we have chosen “big enough" A,B,C, so that they do not block any U_i. If we consider another extreme, say the subsystem C has a very narrow neck, then S^(α)_nonlocal will be sensitive to the geometry of the neck which determines how many U_i could pass through.One may also apply the same idea to subsystems oftype-β and type-γ and find extensive values of S^(β)_nonlocal and S^(γ)_nonlocal for certain choices of subsystems.§.§ Fractal spin liquidsFractal spin liquids <cit.> is a generalization of Haah's code <cit.>. A common feature of fractal spin liquid models is the existence of fractal condensates. Fractal structures have discrete scale symmetries, and this results in a more complicated dependence of the ground state degeneracy on the system size <cit.> compared to the type1 fracton models.Some of the fractal models possess “hybrid" condensates W_i having both 1D parts and fractal parts, and the truncations of the condensates give you U_i which can be either a string-like operator or a fractal operator. Note that, they do not fit into the definition of type 1 due to the existence of fractal operators. On the other hand, they do not fit into type 2 because the excitations created by the string-like operator are mobile excitations. Some fractal models have only fractal condensates, and no string-like U_i exists. These models are type 2 fracton models.Discussed in the following areways to detect string-like U_i and fractal U_i using condensates. Then, S_nonlocal is shown to be extensive for certain choices of subsystem geometry.§.§.§ The Sierpinski Prism ModelAs an example, we consider themodel (d) in Yoshida's paper <cit.>. Let us call this model the Sierpinski prism model, named after the shape of the condensate in Fig.<ref>a, which looks like a prism with three legs decorated with Sierpinski triangles. This model lives on a 3D cubic lattice with two qubits (A and B) on each site. The Hamiltonian can be written asH=-∑_i,j,k h^Z_(i,j,k)-∑_i,j,k h^X_(i,j,k),where i,j,k are integers labeling the sites on cubic lattice, and the Hamiltonian involves all the translations of the operator h^Z_(0,0,0) and h^X_(0,0,0). Explicitly, in terms of Pauli operators acting on each A and B qubit on different sites, we haveh^Z_(0,0,0) = Z^A_r=(0,0,0)Z^A_r=(0,1,0)Z^A_r=(1,1,0) × Z^B_r=(0,0,0)Z^B_r=(0,0,1)h^X_(0,0,0) = X^A_r=(0,0,0)X^A_r=(0,0,-1) × X^B_r=(0,0,0)X^B_r=(0,-1,0)X^B_r=(-1,-1,0).It is easy to check that all terms in the Hamiltonian commute and [h_(i,j,k)^Z]^2=[h^X_(i,j,k)]^2=1.Using Yoshida's notation:h^Z_(0,0,0)=Z ([ 1+y+xy;1+z ]) ; h^X_(0,0,0)= X ([1+z̅; 1+y̅+x̅y̅ ]).Where x̅≡ x^-1, y̅≡ y^-1 and z̅≡ z^-1. In terms of polynomials f(x), g(x) with coefficients over 𝐅_2, i.e.the coefficients can take 0 or 1:h^Z_(0,0,0)=Z ([ 1+f(x)y; 1+g(x)z ]) ; h^X_(0,0,0)= X ([ 1+g̅(x)z̅; 1+f̅(x)y̅ ]).Here f(x)=1+x and g(x)=1 for the Sierpinski prism model. The polynomials with 𝐅_2 coefficients indicate the locations and the numbers of Pauli Z or X operators in the product; the upper row is forA qubits and the lower row is for B qubits.Choosing other polynomials f(x), g(x) or changing𝐅_2 into 𝐅_p (p>2 prime number) will generally give you other fractal models. The Sierpinski prism model possesses hybrid condensates which consist of 1D parts and fractal parts, see Fig.<ref>. The condensates in Fig.<ref>a and Fig.<ref>b can be constructed as a product of h^Z_(i,j,k) and the condensate in Fig.<ref>c can be constructed as a product of h^X_(i,j,k). As is suggested by the discrete scaling symmetry of fractal structure and the continuous scaling symmetry of a 1D line: the upper and lower surfaces can be separated by an arbitrary distance in z-direction (without changing the size of upper/lower surfaces), and under a rescaling l→ 2^m l the condensates look similar. Under other rescaling factors, the condensates look different but they could be constructed using a product of condensates that looks similar to the ones in Fig.<ref>.While this model does not have any logical qubits under periodic boundary conditions on an L_x× L_y× L_z lattice, i.e. x^L_x=y^L_y=z^L_z=1, it does have logical qubits under some “twisted" boundary conditions (say x^L_x=1, y^L_y=x, z^L_z=1 with L_x=2^m+1, L_y=2^m, ∀L_z and integer m), or under open boundary conditions.Despite the fact that the Sierpinski prism model is one of the simplest fractal models, it nicely illustrates all the important ingredients needed in order to understand how our method works in fractal models. To be specific, it illustrates the following three types of detections:1) The detection of a string-like U using a fractalW.2) The detection of a fractal U using a string-likeW.3) The detection of a fractal U using a fractal W. §.§.§ The detection of sting-like Uby fractal WAs is shown in Fig.<ref>, we have condensates with string-like parts and fractal parts. String-like U_i can be obtained from a truncation of the condensates. In Fig.<ref>a, translations of a string-like U, e.g. U_T_1 and U_T_2 (with T⃗_1=(-2^m,0,0), T⃗_2=(0,2^m,0) and an integer m)are distinct from U, but U∼ U_T_1· U_T_2. It is different from what happens in conventional topological orders but similar to what happens in type 1 fracton models, see Fig.<ref>. The distinctness of a string-like U and its translations indicates a lower bound of S_nonlocal extensive in the subsystem size, and indeed we can use the fractal part of W_i to detect different string-like U_i, see Fig.<ref>b, and get an extensive lower bound. §.§.§ The detection of fractal U by string-like W Very similarly, fractal U can be detected by string-like parts of W, see Fig.<ref>a. It is clear that a translation of a fractal U will be a distinct operator if it anticommutes with a different W. Therefore, by suitably choosing the geometrical shapes of subsystems A, B, C, D, it is possible to get an extensive lower bound of S_nonlocal^(α). S_nonlocal^(β) and S_nonlocal^(γ).§.§.§ The detection of fractal U by fractal WAnother way to detect fractal U is to use a fractal part of a condensate W, see Fig.<ref>b, and it is the only way to detect U for those fractal models without string operators i.e., when the condensates contain only a fractal structure without string-like parts (i.e.,type 2). Therefore, it is important to understand this case.The key features, which can be observed in Fig.<ref>b are the following.1) The fractal condensates aresupported on 2D surfaces with “holes" of different (discrete) length scales, in other words it isless than 2D.2) Fractal U and fractal (part of) W lie in distinct intersecting surfaces, and it is possible to make the operators intersecting at a point. Certain translation of U has a non-overlapping support with W and therefore it commutes with W. This implies that translations of U can give you distinct operators.After some thought, one finds it ispossible to get an extensive lower bound of S_nonlocal^(α), S_nonlocal^(β) and S_nonlocal^(γ) by suitably choosing the geometrical shapes of the subsystems A, B, C, D. A choice of D for type-β or type-γ is shown inyellow color in Fig.<ref>b. §.§.§ Further comments 1) One may consider other subsystem types, for example: type-δ with D consists of three disconnected boxes, andsuitably chosen A,B,C. When putting the three boxes on the positions of the three excitations of a U in Fig.<ref>,S^(δ)_nonlocal>0.2) When rescaling the subsystem sizes according to the discrete scale symmetries (for the Sierpinski prism model, it is L→ 2^m L), the change of S_nonlocal could be investigated using entanglement renormalization group transformation <cit.>. For a rescaling by a factor which is not in the discrete scaling group, S_nonlocal may change in more complicated way.3) It is possible to have a fractal model with a unique ground state on a T^3, in which case, there still exists nonzero S_nonlocal. This indicates that S_nonlocal is in some sense more universal than the ground state degeneracy.4) For models with only fractal condensates (i.e. type 2), although it is possible to find S_nonlocal^(α)> 0 for some choices of A, B, C, D, we get a lower bound 0 when, for example, the two boxes of D of length l_D× l_D× l_D are separated by a displacement vector d⃗ satisfying |d⃗|>λ l_D with some constant λ depending on the model. It might be a general exact result thatS^(α)_nonlocal=0 when |d⃗|>λ l_D no matter how you choose A,B,C but we do not have a proof. The case for Haah's code is a conjecture by Kim [I. H. Kim, private communication. ]. If the conjectured results are true, this may be used to make a clear distinction between type 1 and type 2 fracton models.§ PERTURBATIONSThe stability of quantities under local perturbations is an extremely important topic. If some property of an exactly solved model is totally changed when a tiny local perturbation is added,this property could never be observed in real systems.The ground state degeneracy of topological orders is robust (stable) to arbitrary local perturbations. It is known that the toric code model is stable under arbitrary local perturbations <cit.>. The stability of ground state degeneracy is proved <cit.> for a very general class of models which satisfy assumptions TQO-1 and TQO-2. In the proof, Osborne's modification <cit.> of the quasi-adiabatic continuation <cit.> is employed. This proof is applicable to both conventional and fracton topological orders.We would like to understand the stability of S_nonlocal under local perturbations. It turns out that the stability of S_nonlocal is a trickier problem compared to the stability of the ground state degeneracy. The corresponding problem for the conventional topological orders, e.g. the stability of topological entanglement entropy isnot solved completelywithout additional assumptions. It is known from Bravyi's counterexample (see <cit.> for a published reference) that the argumentsprovided in the original works <cit.>about the invariance of the topological entanglement entropy under perturbation are not complete. Kim obtained a bound of the change of topological entanglement entropy with a 1st order perturbation <cit.> assuming the conditionally independence of certain subsystems.Here we study the stability of our lower bound of S_nonlocal under a finite depth quantum circuit for simplicity, since it is known from the viewpoint of quasi-adiabatic evolution <cit.> that local perturbations for gapped systems can be approximated by finite depth quantum circuit <cit.>.Assumption 𝐒: For subsystems A, B, C as is shown in Fig.<ref>. δ Q is a unitary operator which has support intersecting with A and C. B is separated from δ Q by a distance d. σ is a density matrix of a state with correlation length ξ and replica correlation length <cit.> ξ_α and σ'≡δ Qσδ Q^†. And σ is a density matrix such that(S_AB-S_A )|_σ'≃(S_AB-S_A )|_σ,where “≃" means there is a correction that is negligible when d is large compared to ξ and ξ_α.The assumption 𝐒 should be understood as an assumption about the density matrix σ. It is trivial to check that “≃" can be replaced by “=" when σ_AB=σ_A⊗σ_B, and it may seem intuitive that the difference between the left-hand side and the right-hand side of Eq.(<ref>) should decay as e^-d/ξ. Nevertheless, the original suggestion <cit.> that 𝐒 is true for ξ≪ d is violated in Bravyi's counterexample. It is observed in<cit.> that this is due to the fact that the replica correlation length ξ_α is infinity for the cluster state in Bravyi's counterexample. When the cluster state is deformed, ξ_α becomes finite. For generic local perturbations without symmetry requirement, it is fine-tuned to have ξ_α=+∞ but ξ_α can be arbitrarily large compared to ξ. Judging from a recent conjecture <cit.> , ξ_α≪ d may be the condition required for 𝐒 to be true. In the following, we discuss the stability of our lower bound of S_nonlocal under a depth-R quantum circuit Q which creates a perturbed ground state ρ̅ satisfying 𝐒. We take R∼ξ, and assume ξ and ξ_α much smaller than the length scales of the subsystems. This analysis does not cover all possible local perturbations (especially those with ξ_α→ +∞), but we believe it covers a large class of interesting local perturbations.Let Q be the depth-R quantum circuit (Q Q^†=1) which is responsible for the local perturbation. In other words, we assume the following objects in the perturbated model are related to the corresponding objects in the unperturbed model by1) The new Hamiltonian: H̅=QHQ^†;2) The new (dressed) operators: U̅_i=Q U_i Q^†, U̅_i^def=Q U̅_i^defQ^†, and W̅_i = Q W_i Q^†;3) The new ground state: |ψ̅⟩ = Q |ψ⟩;4) The new density matrices: σ̅_I=Q σ_I Q^† and ρ̅=Qρ Q^†. The dressed operators typically have a “fatter" support than the corresponding operators in the unperturbed stabilizer model, see Fig.(<ref>) for an illustration. It is possible that the support of some U̅_i will overlap with AB, the support of some U̅_i^def will overlap with BC and the support of someW_i will overlap with D. For those operators, we need to throw them away, and supply with other operators if possible. We label the remaining operators using i̅, i̅=1,⋯M̅, where M̅≤ M, and we call the remaining density matrices σ̅_I̅, with I̅=1,⋯, N̅, where N̅=∏_i̅=1^m̅n̅_i̅.With 𝐔-1, 𝐔-2, 𝐔-3, 𝐖-1, 𝐖-2satisfied for the unperturbed system, we supply with 𝐒 in order to complete a result about the stability of the lower bound. With these assumptions, one could verify the following results:1') 𝐔-1, 𝐔-3 ⇒ σ̅_I̅AB=ρ̅_AB and σ̅_I̅BC=ρ̅_BC;2') 𝐖-1, 𝐖-2 ⇒ σ̅_I̅ABC·σ̅_J̅ ABC=0 for I̅J̅;3') 𝐔-1, 𝐔-2, 𝐔-3, 𝐒 ⇒ S_ABC|_σ̅_I̅≃ S_ABC| _ρ̅.The derivation of 1') and 2') are parallel to what is done in Sec.<ref>. Thederivation of 3') follows. There exists a unitary operator δ Q supported on a region within a distance R around∂ C∩∂ D, See Fig.<ref>, such thatρ̅=δ Q ρ̅' δ Q^†, σ̅_I̅=δ Q σ̅'_I̅δ Q^† and σ̅'_I̅ ABC=V̅_I̅ρ̅'_ABCV̅^†_I̅. V̅_I̅ is some unitary operator supported on ABC. It is always possible to find such δ Q and V̅_I̅ given that𝐔-2 is satisfied for the unperturbed case. Therefore, S_ABC|_σ̅'_I̅=S_ABC|_ρ̅'. Then, by applying𝐔-1, 𝐔-3, and 𝐒 we find that𝐔-1, 𝐔-3⇒S_BC|_σ̅'_I̅=S_BC|_ρ̅'S_BC|_σ̅_I̅=S_BC|_ρ̅ ; 𝐒 ⇒ (S_ABC-S_BC)|_ρ̅≃ (S_ABC-S_BC)|_ρ̅' (S_ABC-S_BC)|_σ̅_I̅≃ (S_ABC-S_BC)|_σ̅'_I̅ .After simple algebra one arrives at the result 3') i.e. S_ABC|_σ̅_I̅≃ S_ABC| _ρ̅.Assuming the error caused by “≃" could be neglected, one could apply the same method as Sec.<ref> to arrive at a lower boundS_nonlocal|_ρ̅≥lnN̅ =∑_i̅=1^M̅lnn̅_i̅. For models with topologically deformable U_i operators, like the 2D, 3D toric code models, our lower bound isinvariant under perturbation. Because we can always move the excitations deep inside D, such that the distance from any excitation to the boundary d_e≫ξ∼ R. For large subsystems,we would have M=M' and we do not lose any U_i, U_i^def and W_i.For models with U_i not topologically deformable, e.g. the X-cube model andfractal spin liquids. There is usually some excitation that could not be moved deep inside subsystem D. Therefore, after adding perturbations, we typically lose a few U_i and U_i^def , such that M̅<M and N̅<N. But for large subsystems which possess extensive S_nonlocal before perturbation is added, this modification is small comparing to the leading contribution.To summarize, for local perturbations satisfying assumption 𝐒, and subsystem sizes much larger than ξ and ξ_α we expect:S_nonlocal(perturbed)≃ S_nonlocal(unperturbed)-μξ.For conventional topological orders μ=0, and for fracton topological orders μ>0 being a number depends on the model and subsystem geometry. § DISCUSSION AND OUTLOOKIn this paper, we have obtained a lower bound of the nonlocal entanglement entropy S_nonlocal from assumptions about the topological excitations and the ground state condensates of Abelian topological orders and applied our method to several examples. For conventional topological orders, e.g. the 2D toric code model and the 3D toric code model, our lower bounds are saturated and topologically invariant. Whenever the lower bound is saturated, we get an explicit construction of a conditionally independent density matrix σ^∗. For fracton topological orders <cit.>, e.g. the X-cube model and the Sierpinski prism model, our lower bounddepends on the geometry of the subsystems and S_nonlocal is extensive for certain subsystem choices.This method observes an intimate relation between S_nonlocal and the topological excitations and the ground state condensates, and it obtains a lower bound of S_nonlocal without calculating the entanglement entropy of any subsystem. A nonzero lower bound ofS_nonlocal is a result of the nonlocal nature of topological excitations, i.e. the fact that topological excitations could not be created alone by local operators. This nonlocal nature of topological excitations does not guarantee the operators which create the topological excitations to be topologically deformable and S_nonlocal is not necessarily a topological invariance. Geometry-dependent S_nonlocal is what appears in fracton models.It is beyond an established paradigm, i.e., the topological entanglement entropy, and should be treated as its generalization. The stability of the lower bound is discussed for local perturbations satisfying assumption 𝐒, which should cover a large class of interesting local perturbations.The different behaviors of S_nonlocal may be used to distinguish fracton topological orders from conventional topological orders. Together with othermethods being developed so far <cit.>, our result providesa better understanding of the entanglement properties offractal models. Furthermore, the lower bound suggests (but not prove) different behaviors of S_nonlocalbetween type 1 and type 2 fracton models. These different behaviors may be proven or disproven by later works.Some of the assumptions in our method do not apply to non-Abelian models, a variant of our lower bound of S_nonlocal for non-Abelian models is presented in <cit.>. Also, it might be interesting to investigate possible implications of our method on relations among topological order, topological entanglement entropy and quantum black holes <cit.>. § ACKNOWLEDGEMENTB.S. would like to thank Fuyan Lu for a comment on one assumption, Jeongwan Haah for providing a reference about Bravyi's counterexample, Isaac H. Kim for a discussion and sharing one conjecture, and Michael Levin for a discussion about the entanglement in non-Abelian phases. This work is supported by the startup funds at OSU and the National Science Foundation under Grant No. NSF DMR-1653769 (BS,YML). apsrev | http://arxiv.org/abs/1705.09300v2 | {
"authors": [
"Bowen Shi",
"Yuan-Ming Lu"
],
"categories": [
"cond-mat.str-el",
"hep-th",
"quant-ph"
],
"primary_category": "cond-mat.str-el",
"published": "20170525180008",
"title": "Deciphering the nonlocal entanglement entropy of fracton topological orders"
} |
firstpage–lastpage BRITE-Constellation: Data processing and photometryBased on data collected by the BRITE Constellation satellite mission, designed, built, launched, operated and supported by the Austrian Research Promotion Agency (FFG), the University of Vienna, the Technical University of Graz, the Canadian Space Agency (CSA), the University of Toronto Institute for Aerospace Studies (UTIAS), the Foundation for Polish Science & Technology (FNiTP MNiSW), and National Science Centre (NCN). A. Popowicz<ref> A. Pigulski<ref> K. Bernacki<ref> R. Kuschnig<ref>,<ref> H. Pablo<ref>,<ref> T. Ramiaramanantsoa<ref>,<ref> E. Zocłońska<ref> D. Baade<ref> G. Handler<ref> A. F. J. Moffat<ref>,<ref> G. A. Wade<ref> C. Neiner<ref> S. M. Rucinski<ref> W. W. Weiss<ref> O. Koudelka<ref> P. Orleański<ref> A. Schwarzenberg-Czerny<ref> K. Zwintz<ref> Received; accepted ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We present the detection of 89 low surface brightness (LSB), and thus low stellar density galaxy candidates in the Perseus cluster core, of the kind named `ultra-diffuse galaxies', with mean effective V-band surface brightnesses 24.8–27.1 mag arcsec^-2, total V-band magnitudes -11.8 to -15.5 mag, and half-light radii 0.7–4.1 kpc. The candidates have been identified in a deep mosaic covering 0.3 deg^2, based on wide-field imaging data obtained with the William Herschel Telescope. We find that the LSB galaxy population is depleted in the cluster centre and only very few LSB candidates have half-light radii larger than 3 kpc. This appears consistent with an estimate of their tidal radius, which does not reach beyond the stellar extent even if we assume a high dark matter content (M/L=100). In fact, three of our candidates seem to be associated with tidal streams, which points to their current disruption. Given that published data on faint LSB candidates in the Coma cluster – with its comparable central density to Perseus – show the same dearth of large objects in the core region, we conclude that these cannot survive the strong tides in the centres of massive clusters.galaxies: clusters: individual: Perseus – galaxies: dwarf – galaxies: evolution – galaxies: fundamental parameters – galaxies: photometry. § INTRODUCTIONGalaxies of low surface brightness, once considered a rare part of the overall galaxy population <cit.>, now are recognized to exist in all galaxy mass ranges with a wide variety of properties <cit.>. In addition, improved techniques have led to the detection of increasing numbers of low surface brightness, and thus low stellar density, galaxies <cit.>. These are particularly numerous among the less luminous members of galaxy clusters <cit.>.Galaxy clusters have been and are being surveyed for increasingly faint galaxies, leading to the detection of low-mass dwarf galaxies in the surface brightness regime of Local Group dwarf spheroidals (dSphs) with mean effective surface brightnesses ⟨μ_V⟩_50 > 24 mag arcsec^-2, and even ultra-faint dwarfs <cit.>. With this increasing coverage of the parameter space of magnitude, half-light radius and surface brightness, we therefore consider it necessary to distinguish between a regular – even though faint – dwarf galaxy, and a low surface brightness (LSB) galaxy in the sense of having a surface brightness clearly lower than average at its luminosity. For example, while the Virgo Cluster Catalogue of <cit.> contains hundreds of newly identified dwarf galaxies, many of them being faint in magnitude and surface brightness, their catalogue also includes a handful of LSB objects that seemed to form `a new type of very large diameter (10 000 pc), low central surface brightness (≥ 25 B mag arcsec^-2) galaxy, that comes in both early (i.e., dE) and late (i.e., Im V) types' <cit.>. Further Virgo cluster galaxies of dwarf stellar mass but with unusually large size and faint surface brightness were described by <cit.>, and some similar objects were discovered in the Fornax cluster by <cit.> and <cit.>. Three decades later, galaxies in the same general parameter range were dubbed `ultra-diffuse galaxies' by <cit.>.In the Coma cluster, a large number of over 700 very faint candidate member galaxies with total magnitudes M_B > -13 mag, half-light radii 0.2 < r_50 < 0.7 kpc and central surface brightnesses as low as μ_B,0 = 27 mag arcsec^-2 were identified by <cit.>. In the brighter and overlapping magnitude range -11 ≳ M_g ≳ -16 mag <cit.> and <cit.> reported numerous LSB candidates with μ_g,0≥ 24 mag arcsec^-2 and half-light radii up to 5 kpc in Coma, of which five large objects withr_50≳ 3 kpc are spectroscopically confirmed cluster members <cit.>. The Virgo cluster study of <cit.> revealed four LSB candidates with even lower central surface brightnesses of μ_V,0∼ 27 mag arcsec^-2 and half-light radii as large as 10 kpc. In the Fornax cluster an abundant population of faint LSB galaxies with μ_r,0≥ 23 mag arcsec^-2 were catalogued by <cit.> and <cit.>, of which a few have r_50 > 3 kpc <cit.>. Several such objects in different environments were also reported by <cit.>.Although LSB galaxies have now been detected in large numbers, their origin remains a puzzle. Especially the abundant existence of LSB galaxies of dwarf stellar mass in galaxy clusters raised the question how these low stellar density systems could survive in the tidal field of such dense environments. For example, <cit.> did not report any signs of distortions for the faint LSB candidates identified in the Coma cluster. Other cluster LSB galaxies of dwarf luminosity harbour surprisingly large and intact globular cluster (GC) systems <cit.>. One explanation could be that these galaxies are characterized by a very high dark matter content that prevents disruption of their stellar component. A similar interpretation was given by <cit.> for a population of remarkably round and undistorted dSphs in the Perseus cluster core. Dynamical analyses of two faint LSB galaxies in the Coma and Virgo cluster indeed revealed very high mass-to-light ratios on the order of M/L = 50–100 within one half-light radius <cit.>. Similar or even higher M/L ratios are also characteristic for Local Group dSphs with M_V > -10 mag or ⟨μ_V⟩_50 > 25 mag arcsec^-2 <cit.>. On the other hand, <cit.> suggested that within the MOND theory high M/L ratios could also be explained if the LSB galaxies would contain yet undetected cluster baryonic dark matter.However, apparently the above does not apply to all faint cluster LSB galaxies. For example, two LSB galaxy candidates of very low stellar density in the Virgo cluster show possible signs of disruption <cit.>. One large LSB candidate of dwarf luminosity with a very elongated shape and truncated light profile was also reported in Fornax <cit.>, and several further elongated large LSB candidates were described by <cit.>. In the Hydra I galaxy cluster, <cit.> identified a faint LSB galaxy with S-shaped morphology, indicative of its ongoing tidal disruption. Also <cit.>, who studied populations of faint LSB candidates with r_50≥ 1.5 kpc in eight clusters with redshifts z = 0.044–0.063, reported a depletion of LSB galaxy candidates in the cluster cores, based on number counts. Similarly, the numerical simulations of <cit.> predict the disruption of LSB galaxies that are on orbits with very close clustercentric passages.In this study, we aim to investigate the faint LSB galaxy population of the Perseus cluster core. Perseus is a rich galaxy cluster at a redshift of z = 0.0179 <cit.>. While its mass is in between the lower mass Virgo and the higher mass Coma cluster, its core reaches a density comparable to that of the Coma cluster. There are indications that Perseus is possibly more relaxed and evolved than Coma <cit.>. For example Perseus only has a single cD galaxy in its centre, while the core of Coma harbours two large galaxies. On the other hand, <cit.> interpreted the `non-uniform distribution of morphological types' in Perseus as an indication that this cluster is not yet virialized and instead dynamically young. This may be supported by the observation that on large scales Perseus is not a spherically symmetric cluster like Coma, but shows a projected chain of bright galaxies extending in east–west direction that is offset from the symmetric X-ray distribution.While a significant number of regular dwarf galaxies has already been identified in a smaller field of the cluster core by <cit.>, we focus on galaxies in the same luminosity range with M_V > -16 mag (corresponding to stellar masses of M_* ≲ 10^8 M_⊙) but of fainter surface brightness and thus lower stellar density. This is made possible by our deep wide-field imaging data obtained with the 4.2 m William Herschel Telescope (WHT) Prime Focus Imaging Platform (PFIP), reaching a 5σ V-band depth of about 27 mag arcsec^-2. In this paper, we concentrate on LSB galaxies with ⟨μ_V⟩_50≥ 24.8 mag arcsec^-2, which corresponds to the currently often adopted surface brightness limit of μ_g,0≥ 24 mag arcsec^-2 for the so-called `ultra-diffuse galaxies'. While the definition of the latter refers to objects with r_50 > 1.5 kpc <cit.>, we will not apply any size criterion in this study and generally speak of `faint LSB galaxies', or `LSB galaxies of dwarf stellar mass'. Previous work on the low-mass galaxy population in Perseus includes also the 29 dwarf galaxies studied by <cit.> and <cit.> in Hubble Space Telescope (HST) imaging data, of which six fall within our considered surface brightness range.This paper is organized as follows: in Section <ref>, we describe the observations, data reduction and our final mosaic. We outline the detection of the LSB sources in Section <ref>, and specify their photometry in Section <ref>. We present our results in Section <ref>, where we define our sample of LSB candidates, examine their spatial distribution in the cluster, discuss peculiar candidates and characterize their magnitude–size–surface brightness distribution in comparison to LSB candidates in the Coma cluster. We discuss our results in Section <ref>, followed by our conclusions in Section <ref>. Throughout the paper, we assume a distance of 72.3 Mpc to the Perseus cluster with a scale of 20.32 kpc arcmin^-1 <cit.>. § THE DATAWe acquired deep V-band imaging data of the Perseus cluster core with PFIP at the WHT through the Opticon programme 2012B/045 (PI T. Lisker).The PFIP is an optical wide-field imaging camera with a field of view of 16 × 16 arcmin^2, corresponding to 325 × 325 kpc^2 at the distance of Perseus. The observations were carried out 2012 November 12 and 13. We performed dithered observations on three pointings across the cluster core, with individual exposure times of 120 s. In total, 187 science exposures contribute to the final mosaic.We reduced the data mainly with the image reduction pipeline [, version 2.6.2] <cit.>, which is especially designed to process wide-field imaging data. For the data reduction each exposure was spatially split into two frames, corresponding to the two detectors of the instrument. All frames were overscan- and bias-corrected, as well as flat fielded using twilight flats. To correct for remaining large-scale intensity gradients that may still be imprinted in the data after flat fielding, a master background, containing only signal from the sky, was created. For the latter the sources in all frames were masked, then the frames were normalized and stacked. Assuming the background inhomogeneities are of additive nature, the master background was subsequently subtracted from all frames. Since applying one common master background was not sufficient to remove the large-scale background variations from all frames, individual background models were created in a next step. The individual models are based on object-masked frames, where the masked areas were interpolated based on values from neighbouring unmasked pixels. The resulting images were convolved with a Gaussian kernel with a full width at half-maximum (FWHM) of 512 pixels. The individual background models were subtracted from each frame. We note that the applied filter kernel is large with respect to the extent of our targets, which have typical half-light radii on the order of 20–60 pixels. Then all frames were calibrated astrometrically and distortion corrected, using the Sloan Digital Sky Survey Data Release 9 (SDSS-DR9) <cit.> as a reference catalogue. Finally the frames were resampled and combined to a mosaic, where each frame was weighted according to the square of its inverse sky noise.In a second iteration of the reduction we improved the individual background models of the frames that were contaminated through the extended haloes of the two brightest cluster galaxies. This optimization was done outside thepipeline, mainly using .[ is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.] Manually extending the masks would have resulted in a very high fraction of masked pixels on the single frames. To avoid this, we modelled the light distribution of both galaxies in the first iteration mosaic, usingellipse and bmodel. We then subtracted the galaxy models from the distortion corrected frames before generating new individual background models with . The new background models were then subtracted from the original science frames, and combined to the second mosaic.Lastly we corrected our mosaic for spatial zero-point variations, again outside thepipeline. After selecting suitable stars in our mosaic using SExtractor <cit.>, we measured their magnitudes with thetask photometry on the individual flat fielded frames, before any background model was subtracted. We calculated the zero-point of each frame as median magnitude offset with respect to the SDSS-DR9 catalogue, using the transformation equations from <cit.>. The zero-point variations are then given as the deviation of the magnitude offset of individual stars from the zero-point of the respective frame. We rejected stars that deviate by more than 0.2 mag from the zero-point of the respective frame and only considered stars with small magnitude errors in both the SDSS-DR9 catalogue and the measurements withphotometry, requiring √(Δ mag^2_phot + Δ mag^2_SDSS) < 0.05 mag. We then established a two-dimensional map yielding the zero-point variations across the detector by fitting a two-dimensional surface to the zero-point variations obtained for all frames. Finally, we divided each frame by this map, and repeated the above described reduction steps leading to the final mosaic. The zero-point of the final mosaic is 26 mag, with a mean variation of 0.02 mag with respect to the SDSS-DR9 catalogue.Fig. <ref> (left-hand panel) shows our final deep mosaic of the Perseus cluster core (also Figs <ref> and <ref>). It is not centred directly on the brightest cluster galaxy NGC 1275, but on a region including the chain of luminous galaxies that are distributed to the west of it. The mosaic covers an area of ∼ 0.27 deg^2 (=̂ 0.41 Mpc^2), and extends to a clustercentric distance of 0.57(=̂ 0.70 Mpc^2) from NGC 1275. This corresponds to 29 per cent of the Perseus cluster virial radius for R_vir = 2.44 Mpc <cit.>, or 39 per cent when adopting R_vir = 1.79 Mpc <cit.>. The mosaic reaches an image depth of 27 mag arcsec^-2 in the V-band at a signal-to-noise ratio of S/N = 1 per pixel, with a pixel scale of 0.237 arcsec pixel^-1. The corresponding 1σ and 5σ depths are 28.6 and 26.8 mag arcsec^-2, respectively. The image depth varies across the mosaic, as can be seen in the weight image (Fig. <ref>, right-hand panel). The average seeing FWHM is 0.9 arcsec.For the subsequent detection and photometry of low surface brightness sources we created one copy of the mosaic where we removed most of the sources with bright extended haloes, including the largest cluster galaxies and the haloes of foreground stars. We fitted the light profiles withellipse, generated models withbmodel and subtracted them from the mosaic.§ DETECTION Motivated by the detection of faint LSB galaxy candidates in the Virgo and Coma galaxy clusters by <cit.> and <cit.>, we inserted LSB galaxy models in the same parameter range into our mosaic and then searched systematically for similarly looking objects in Perseus. We decided to search for LSB sources by eye, since automatic detection algorithms often fail in reliably detecting sources with very low S/N. We realized the models with a one component Sérsic profile of Sérsic index n=0.7–1.2 that were convolved with a Gaussian kernel, adopting our average seeing FWHM.We generated a first set of 27 models in the parameter range 24.6 ≤⟨μ_V⟩_50≤ 27.8 mag arcsec^-2, -14 ≥ M_V ≥ -16.6 mag, and 2.1 ≤ r_50≤ 9.7 kpc, assuming an average foreground extinction of A_V = 0.5 mag at the location of Perseus. Among them are nine model types with different magnitudes and half-light radii. For each model type we generated two additional variants with altered position angle and ellipticity, which results in slightly different surface brightnesses. We created a second set of seven nearly round (ellipticity = 0.1) models with ⟨μ_V⟩_50≤ 26.0 mag arcsec^-2 that extend the parameter range to smaller half-light radii of 1.5 kpc and fainter magnitudes of -13.5 mag.From the first model set, we always inserted 30–40 models of one type, i.e. with the same magnitude and half-light radius but varying ellipticity, into one copy of the mosaic. We generated two additional mosaic copies where we inserted the models from the second model set. We used these copies only at a later stage to focus the detection especially on smaller and fainter LSB sources that turned out to be quite numerous based on the search using the first model set. In total we inserted 305 models from the first model set into nine different mosaic copies, and 56 models from the second set into two further copies.To facilitate the visual detection of LSB sources, we used the mosaic variant where we previously fitted and subtracted the light distribution of most of the extended sources (see Section <ref>). To remove the remaining bright sources on each copy of the mosaic, we ran SExtractor to detect all sources with more than 10 pixels above a detection threshold of 1.5 σ, and replaced the pixels above this threshold with zero values, corresponding to the background level of our mosaic. We then convolved the data with a circular Gaussian kernel with σ=1 pixel, and demagnified each copy by a factor of 1.5. We further divided each mosaic copy into four smaller regions of different image depth according to the weight image (see Fig.<ref>, right-hand panel). Finally two of us independently searched visually for diffuse sources in each copy, thereby detecting simultaneously the inserted models and real LSB candidates, without knowing where the former had been inserted. After removing sources that we identified more than once in different copies of the same region, this resulted in a preliminary sample of 214 LSB sources that were identified by at least one of us, and for which we carried out photometry (see Section <ref>).We used the visually identified models from the first model set to get a rough estimate on our detection rate (see Fig. <ref>). We estimated the detection rate for each model type as fraction of the total number of inserted models that were visually identified. We find that the detection rate generally drops with surface brightness. We detected more than 90 per cent of all models with ⟨μ_V⟩_50 < 25.5 mag arcsec^-2, between 70 and 90 per cent of all models with 25.5 ≤⟨μ_V⟩_50 < 27.0 mag arcsec^-2, and about 50 per cent of all models with ⟨μ_V⟩_50 > 27.0 mag arcsec^-2.[The given surface brightnesses refer to the average surface brightness of the three model variants with different ellipticity, and thus surface brightness, that exist per model type.]The models with ⟨μ_V⟩_50 < 27.0 mag arcsec^-2 are in general clearly visible in our data and the main reason for missing some of them seems to be related to overlap with brighter sources. We estimated the area occupied by remaining bright extended sources in our object-subtracted mosaic to be 12 per cent[This accounts for all sources that were detected with SExtractor with more than 1000 connected pixels above a detection threshold of 1.5 σ.], which compares to an average detection rate of 90 per cent of all models with ⟨μ_V⟩_50 < 27.0 mag arcsec^-2. Scatter in the trend of decreasing detection fraction with surface brightness can both be caused by our approach of visual source detection, as well as by the different overlap fractions of the inserted models with brighter sources.[We note that the fraction of models whose centre overlaps with one of the SExtractor-detected sources above 1.5 σ does not exceed 12 per cent per model type.] The detection rate of models with ⟨μ_V⟩_50 < 27.0 mag arcsec^-2 is similar in all regions of our mosaic, even in the shallowest region (Region 1; see Fig.<ref>, right-hand panel). For models with ⟨μ_V⟩_50 > 27.0 mag arcsec^-2 we find, however, a lower detection rate in Region 1 and Region 2, compared to the other two regions. While Region 1 is the shallowest region, the lower detection rate in Region 2 might be related to the higher galaxy density compared to the other regions. § PHOTOMETRYPhotometry of LSB sources is challenging and the measurements suffer in general from higher uncertainties compared to sources of brighter surface brightness. One reason for this is that the radial flux profile of the former is characterized by a larger fraction of flux at large radii, where the S/N is typically very low. This also implies that contamination from close neighbour sources and the presence of background gradients is more severe for these objects. We quantify the arising uncertainties in our data using inserted LSB galaxy models (see Section <ref>).We derived magnitudes and sizes from growth curves through iterative ellipse fitting withellipse, rather than from fits to analytical models. The first step was to obtain a first guess of the centre, ellipticity and position angle of all sources. We used SExtractor to measure the parameters of 131 objects that were detected with a detection threshold of 1 σ (128 objects) or 0.8 σ (3 objects). For 83 objects that were not detected with SExtractor or that had obviously wrong parameters we estimated their centre and shape visually based on the Gaussian smoothed and demagnified mosaic. Then we ran ellipse with fixed parameters, adopting the previously measured or estimated centres, ellipticities and position angles. We chose a linear step-size of 5 pixels for consecutive isophotes. We used the first ellipse fit results to generate two-dimensional brightness models withbmodel that we subtracted from the fitted source.The residual images served as a basis to create masks of neighbouring sources from SExtractor segmentation images. We ran SExtractor in two passes, one with a minimum number of 28 connected pixels above a detection threshold of 1 σ, the other with a lower detection threshold of 0.6 σ and requiring a minimum number of 1000 connected pixels. In both passes, we used SExtractor with the built-in filtering prior to detection. We combined both segmentation images and extended the masked areas by smoothing with a Gaussian kernel. We ran ellipse in a second pass with the masks to exclude that flux from neighbouring sources contributes to the ellipse fits. From the second iteration residual images we created improved masks where the masked regions are somewhat larger. We unmasked the centre of nucleated candidates and ellipse fit residuals when necessary.The next step was to determine the background level from the third pass ellipse fit results using the improved masks. Getting the background level right is a very subtle task and the major source of the uncertainties in the magnitude and size measurements. Therefore, we determined the background level for each of our detected LSB objects individually. We first measured the radial flux profiles out to large radii (350 pixels) for each object. We then manually adjusted the radius and width of the background annulus, whose median flux we adopted as the background level. The inner radius of the background annulus was set at the first break in the flux profile where the intensity gradient significantly changes and the flux profile levels out. We set the width of the annulus to 50 pixels. Its shape follows the ellipticity and position angle of the measured object.Although all neighbour sources were carefully masked, still some flux profiles show signs of contamination. Especially at larger radii where faint flux levels are reached, the flux of the LSB source can be comparable to the flux of a neighbour source that still extends beyond the masked area (e.g. some very extended haloes of foreground stars or bright cluster galaxies). Also background inhomogeneities remaining in the data after the reduction can contaminate the flux profiles. Possible contamination can become apparent in a flux profile when, for example, the profile continues to decline after the first break instead of levelling out to zero. In this case we nevertheless set the inner radius of the background annulus to the first break in the profile, and eventually decrease its width to make sure that the flux profile is flat in this region.Even though we might truncate a galaxy at too high intensity, resulting in a systematically fainter magnitude and a smaller half-light radius, restricting the analysis to the uncontaminated inner profile helps to preserve the true surface brightnesses (see the right-hand panels in Fig. <ref> and Section <ref>). After subtracting the background offset, we then obtained a first estimate of the magnitudes and sizes by running ellipse in a fourth pass on the background corrected images and taking into account the masked sources. We determined the total flux from the cumulative flux profile[We adopted the median of the cumulative fluxesfrom the ellipse fit tables, namely of the five isophotes between the inner radius of the background annulus and 20 pixels further, as an estimate of the total flux. Since ellipse does not account for masked regions when calculating the total flux within an isophote, we replaced the masked regions with values from the 2-D model created withbmodel from the radial flux profile.] and derived the half-light radius along the semimajor axis, as well as the mean effective surface brightness within one half-light radius.In the final iteration we measured the centre, ellipticity and position angle of our LSB sources more accurately, using our first guess parameters as input values. We usedimcentroid to derive the centre, and calculated the ellipticity and position angle from the image moments within a circular area defined by our first-guess half-light radius. We also further improved the masks by manually enlarging the masks of extended neighbour sources with faint haloes.[Using<cit.> regions andmskregions.] After that we ran ellipse in a fifth pass with the new parameters and masks to adjust the inner radius of the background annulus. We adopted the new background level and derived the final magnitudes, half-light radii and mean effective surface brightnesses in a last pass of ellipse fitting. We corrected the derived magnitudes for extinction, using the IRSA Galactic Reddening and Extinction Calculator, with reddening maps from <cit.>. The average foreground extinction of our measured sources is A_V = 0.5 mag. § FAINT LSB GALAXIES IN THE PERSEUS CLUSTER CORE§.§ SampleWe define our sample of LSB galaxy candidates to include all objects with ⟨μ_V⟩_50≥ 24.8 mag arcsec^-2. This corresponds to the currently often adopted surface brightness limit of μ_g,0≥ 24.0 mag arcsec^-2 for `ultra-diffuse galaxies' <cit.>, when assuming an exponential profile with Sérsic n=1 <cit.>, g-r = 0.6 and using the transformation equations from <cit.>. Of our preliminary sample, 133 objects fall into this parameter range. We carefully examined all of them, both on the original as well as on the smoothed and demagnified mosaic. We also compared them to an independent data set of the Perseus cluster, obtained with WIYN/ODI in the g,r and i filters (programme 15B-0808/5, PI: J. S. Gallagher). Since the single-band images are shallower than our data, we used the stacked g,r,i images for the comparison.Based on a more detailed visual examination of their morphology, we classified 82 of our candidates as likely galaxies. They are characterized by a smooth morphology and are confirmed in the independent data set. We classified seven further candidates as possible galaxies (all of them are shown in Fig. <ref> in the bottom row). Three of them (candidates 26, 31 and 44) are clearly visible in our data, but their morphology does not appear very regular. Since these objects are also visible in the WIYN/ODI data, we rule out that they are image artefacts. However a confusion with cirrus cannot be excluded (see Section <ref>). The four other candidates (candidates 27, 49, 57 and 81) are classified as possible galaxies since they are only barely visible in our data, due to their low surface brightness or low S/N, and are not confirmed in the shallower independent data set. We rejected 44 LSB sources from our sample, since we cannot exclude that these are remaining background inhomogeneities from the reduction, or residuals from ellipse fitting of the brighter galaxies. Most of them are of very diffuse nature (80 per cent have ⟨μ_V⟩_50≥ 26.5 mag arcsec^-2) and often do not have a smooth morphology.Our final sample includes 89 LSB galaxy candidates in the Perseus cluster core. We show our sample in Fig. <ref> and provide the photometric parameters in Table <ref>. We also compare our sample to overlapping HST/ACS images, in order to investigate whether some of our objects would classify as background sources, based on possible substructure in the form of, e.g., spiral arms. Seven of our LSB candidates fall on HST/ACS pointings, and none of them shows signs of substructure. We therefore expect that the overall contamination through background galaxies is low in our sample, based on the morphological appearance in the HST as well as in the WHT images and due to the location of our sample in the core region rather than in the cluster outskirts. Certain cluster membership can, however, only be established through measurements of radial velocities. The six brightest candidates in the HST/ACS images with 24.8 ≤⟨μ_V⟩_50≤ 25.4 mag arcsec^-2, as measured in our data,were previously identified in <cit.> (candidates 62, 64, 69, 70, 73 and 87). One of them (candidate 62) was first catalogued by <cit.>. The faintest candidate, with ⟨μ_V⟩_50 = 26.5 mag arcsec^-2 (candidate 82), is only barely visible in the HST/ACS images and was not published previously. §.§ PropertiesFig. <ref> shows the spatial distribution of our sample of 89 faint LSB galaxy candidates in the Perseus cluster core. The sample spans a range of 47 ≤ d ≤ 678 kpc in projected clustercentric distance, with respect to the cluster's X-ray centre[The X-ray centroid almost coincides with the optical location of NGC 1275.] <cit.>. This corresponds to 0.02-0.28 R_vir when assuming a virial radius of R_vir=2.44 Mpc <cit.>. About half of our sample is located closer than 330 kpc to the cluster centre.We find three LSB candidates that appear to be associated with structures resembling tidal streams (see Fig. <ref>, right-hand panels). Candidate 44 seems to be embedded in diffuse filaments, candidates 26 and 31 appear connected via an arc-shaped stream. We find one further galaxy with tidal tails (see Fig. <ref>, bottom left panel), which has a slightly brighter surface brightness of ⟨μ_V⟩_50 = 24.4 mag arcsec^-2 and therefore was not included in our sample. We will analyse faint cluster galaxies with brighter surface brightnesses in a future paper. It is noticeable that all four objects are confined within one region to the south–west of the cluster centre, within a clustercentric distance range of about 300 - 400 kpc. Also the peculiar more luminous galaxy SA 0426-002 <cit.> falls on our mosaic, which shows a disturbed morphology with extended low surface brightness lobes (see Fig. <ref>, top left panel).We show the radial projected number density distribution of our sample in Fig. <ref>. It was derived by dividing the number of galaxies in radial bins of a width of 100 kpc by the area of the respective bin that falls on our mosaic. The bins are centred on the Perseus X-ray centre. We find that the number density is nearly constant for clustercentric distances r ≥ 100 kpc, but drops in the very centre at r < 100 kpc,[Only two galaxies are contained in the central bin with r < 100 kpc.] with a statistical significance of 2.8 σ with respect to the average number density at larger radii. For comparison, a preliminary analysis showed that the distribution of bright cluster members is consistent with the expectation of being much more centrally concentrated.Fig. <ref> shows the magnitude–size and magnitude–surface brightness distribution of our Perseus cluster LSB galaxy sample. We include the Coma cluster LSB galaxies and candidates from <cit.> and the three very low surface brightness galaxy candidates in Virgo from <cit.>. For comparison, we also show Virgo cluster early- and late-type galaxies (compilation of ; based on the Virgo Cluster Catalogue (VCC), ), Virgo cluster dSphs <cit.>, as well as dSphs from the Local Group <cit.>.Our sample spans a parameter range of 24.8 ≤⟨μ_V⟩_50≤ 27.1 mag arcsec^-2, -11.8 ≥ M_V ≥ -15.5 mag and 0.7 ≤ r_50≤ 4.1 kpc. The surface brightness range of our sample is comparable to the LSB galaxy sample from <cit.> and approaches the surface brightness of the two brighter Virgo LSB candidates from <cit.>. With regard to magnitudes and sizes our sample includes smaller and fainter LSB candidates than the sample from <cit.>, which is likely due to their resolution limit. At faint magnitudes, our samples overlaps with the parameter range of cluster and Local Group dSphs. We note that the apparent relation between magnitude and size of our sample is created artificially. The bright surface brightness limit arises due to our definition of including only sources fainter than ⟨μ_V⟩_50 = 24.8 mag arcsec^-2 in our sample. The faint limit is due to our detection limit. At brighter magnitudes M_V ≤ -14 mag, the LSB candidates of our sample are systematically smaller at a given magnitude than the LSB candidates identified in the Coma cluster, with all but one LSB candidate having r_50 < 3 kpc. However, <cit.> cover a much larger area of the Coma cluster, while we only surveyed the core region of Perseus.[According to tests with the inserted model galaxies (see Section <ref>) sources in the surface brightness range of the LSB galaxy sample from <cit.> can easily be detected in our data.] Our total observed area corresponds to 0.41 Mpc^2. This translates to a circular equivalent area with a radius of R = 0.15 R_vir,Perseus, when assuming a virial radius for Perseus of R_vir,Perseus = 2.44 Mpc <cit.>.[We note that our field is not centred directly on the cluster centre, but extends to the west of it.] When selecting all LSB candidates from the <cit.> sample that are located in the core of Coma, within a circular area with clustercentric distances smaller than R = 0.15 R_vir,Coma, where R_vir,Coma = 2.8 Mpc <cit.>, seven LSB candidates remain. These are marked with black squares in Fig. <ref>. One can see that also only two of them reach sizes of r_50 > 3 kpc. Since the sample of <cit.> has a brighter magnitude and larger size limit than our study, we restrict the comparison to objects with M_V ≤ -14 mag and r_50≥ 2 kpc, which should well have been detected by <cit.>. Five LSB candidates in the Coma cluster core are in this parameter range, whereas in Perseus we find seven. A similar result is obtained when comparing to the independent sample of Coma cluster LSB galaxy candidates from <cit.>. When selecting LSB candidates of the Coma core region in the same surface brightness range as our sample and with M_V ≤ -14 mag and r_50≥ 2 kpc, we find 10 LSB candidates in this parameter range, where three LSB candidates have r_50≥ 3 kpc. While it seems that the Virgo cluster galaxies shown in Fig. <ref> are also rare in this parameter range, we note that the catalogue we used is not complete at magnitudes fainter than M_r = -15.2 mag.Thus, in summary, we find that first, the core regions of the Perseus and the Coma cluster harbour a similar number of faint LSB galaxy candidates in the same parameter range of M_V ≤ -14 mag and r_50≥ 2 kpc, and secondly, that large LSB candidates with r_50≥ 3 kpc seem to be very rare in both cluster cores.§.§ UncertaintiesIn Fig. <ref>, we try to include realistic photometric uncertainties for our sample. Our major source of uncertainty in the measured total fluxes, which translate to uncertainties in half-light radii and surface brightnesses, lies in the adopted background level (see Section <ref>). To test how large the resulting uncertainties are, we probed this using inserted LSB galaxy models that were generated similarly to those described in Section <ref>. We created eight model types that span the parameter range of our sample. Four model types have ⟨μ_V⟩_50 = 25.5 mag arcsec^-2, the other four have ⟨μ_V⟩_50 = 26.5 mag arcsec^-2, with varying magnitudes M_V = -12.5 to -15.5 mag and sizes 0.8 ≤ r_50≤ 4.9 kpc. The models have one component Sérsic profiles with n=1, are nearly round (ellipticity = 0.1) and were convolved to our average seeing FWHM. We inserted 10 models of each type into one copy of our mosaic, respectively. We then measured M_V, r_50 and ⟨μ_V⟩_50 similarly to our sample of real LSB candidates. We calculated the average offset between true and measured parameters for each model type, as well as the scatter of the measured parameters. We indicate the average parameter offsets with arrows in the right-hand panels of Fig. <ref>. The arrow tips point to the true values, with M_V being systematically estimated as too faint by on average 0.4 mag, and r_50 being underestimated by on average 0.5 kpc. We largely preserved the true surface brightness, which results from our approach of considering the uncontaminated part of the flux profile only (see Section <ref>). The offsets in ⟨μ_V⟩_50 are small, and do not exceed 0.1 mag arcsec^-2. In general the parameter offsets are more severe for model types with the largest size and faintest surface brightness, and negligible for model types with the smallest size and brightest surface brightness. The error bars in Fig. <ref> give the standard deviation of the measured M_V, r_50 and ⟨μ_V⟩_50 values for each model type, with average standard deviations of Δ M_V = ± 0.3 mag, Δ r_50 = ± 0.3 kpc and⟨μ_V⟩_50 = ± 0.1 mag arcsec^-2.We also tested the implications of our estimated uncertainties on our results from Section <ref>, and applied the average systematic offsets in M_V, r_50 and ⟨μ_V⟩_50 between the models and the measured parameters of our LSB galaxy sample. In this case the number of LSB candidates in the considered parameter range of M_V ≤ -14 mag and r_50≥ 2 kpc would increase to 25 candidates in the Perseus cluster core, but still only two LSB candidates would have sizes larger than r_50≥ 3 kpc. Thus, while the number of LSB candidates would now be significantly higher in Perseus compared to the number of LSB candidates in the same parameter range in the Coma cluster core, the conclusion of only finding very few large LSB galaxy candidates in the cluster core would remain unchanged.Since the core regions of massive clusters are characterized by a particularly high density of galaxies, one possible concern is that this may have influenced our ability of detecting large LSB galaxy candidates with r_50≥ 3 kpc. Our tests with the inserted LSB galaxy models indicate, however, that we are in principle able to detect objects with r_50 > 3 kpc in the surface brightness range ⟨μ_V⟩_50 < 27 mag arcsec^-2 in our data, if these were present (see Section <ref>). Nevertheless we might have missed objects in close vicinity to bright cluster galaxies or foreground stars, although we modelled and subtracted the light profile of the latter in most cases. The apparent absence of LSB candidates in regions around bright sources in Fig. <ref> might therefore not be a real effect. Due to the location of the Perseus cluster at low Galactic latitude (l = 13) we cannot exclude the presence of diffuse emission from Galactic cirrus in our data. Cirrus is often visible in deep wide-field imaging data, and the resulting structures can be very similar in appearance to stellar tidal streams <cit.>. We therefore compared our candidates with possible streams to the WISE[Wide-field Infrared Survey Explorer <cit.>] 12μm data that trace Galactic cirrus, in order to search for possible counterparts in the 12μm emission. Fig. <ref> shows our data in comparison to both the original WISE data with 6 arcsec resolution, as well as to the reprocessed data from <cit.> with 15 arcsec resolution that were cleaned from point sources. We clearly see diffuse emission in the 12μm data at the position of Perseus. However, we are not able to identify obvious structures in the WISE maps that would match to the candidates with possible streams we observe in our data, due to the insufficient resolution of the latter. Therefore, we neither can confirm nor exclude that the nature of these structures may be cirrus emission rather than LSB galaxy candidates with tidal streams. § DISCUSSIONWe detected a large number of 89 faint LSB galaxy candidates with ⟨μ_V⟩_50≥ 24.8 mag arcsec^-2 in the Perseus cluster core. It is interesting to note that all but one candidate have r_50 < 3 kpc. We thus speculate that LSB galaxies with larger sizes cannot survive the strong tidal forces in the core region and possibly have lost already a considerable amount of their dark matter content. This observation is consistent with the study of <cit.> who found a decreasing number density of faint LSB galaxy candidates in the cores of galaxy clusters. Also, the numerical simulations of <cit.> predicted the disruption of LSB galaxies orbiting close to the cluster centre.The effect of tides on LSB galaxies in galaxy clusters is possibly also reflected in the radial number density distribution we observe for our sample. The nearly constant projected number density for clustercentric distances r ≥ 100 kpc implies that the three-dimensional distribution should actually increase with distance from the cluster centre. This may be a further argument that LSB galaxies are depleted in the cluster core region due to tidal disruption. Very close to the cluster centre, for clustercentric distances r < 100 kpc, the number density drops, with only two LSB candidates from our sample being located in this region. Here tidal effects from the central cluster galaxy NGC 1275 may become apparent <cit.>. For example, the slightly more compact peculiar galaxy SA 0426-002 (M_B = -16.3 mag, r_50 = 2.1 kpc), being located only ∼ 30 kpc from the cluster centre, shows signs of being tidally disturbed (see Fig. <ref>, top left panel). Also, in the Fornax cluster core a drop in the number density profile of faint LSB candidates is seen within 180 kpc of the cluster centre <cit.>.We can use the observed limit in r_50 as a rough constraint on the dark matter content of the LSB candidates in the cluster centre <cit.>. The tidal radius R_tidal is given byR_tidal = R_peri( M_obj/M_cl(R_peri) (3 + e))^1/3,with the pericentric distance R_peri, the total object mass M_obj, the cluster mass M_cl(R_peri) within R_peri and the eccentricity of the orbit e <cit.>. We find about 50 per cent of our sample (44 objects) at projected clustercentric distances below 330 kpc. Assuming that this is representative of the orbital pericentre for at least a fraction of the population,[While on the one hand, most objects are likely to be situated somewhat further away from the centre than the projected value suggests, on the other hand, it is also likely that their orbital pericentre is located further inwards from their current location.] we estimate R_tidal for a typical LSB candidate of our sample with M_V = -14 mag and R_peri = 330 kpc, assuming an eccentric orbit with e = 0.5. We adopt the cluster mass profile from <cit.>, where M_cl(330 kpc) = 1.3 × 10^14 M_⊙. Assuming a galaxy without dark matter, and adopting a mass-to-light ratio of M/L_V = 2 for an old stellar population with subsolar metallicity <cit.>, the mass of an object with M_V = -14 mag would be M_obj = 7 × 10^7 M_⊙ accordingly, resulting in a tidal radius of 1.8 kpc. This compares to a range of observed r_50≃ 1.0 - 2.5 kpc for LSB candidates from our sample with M_V ≃ -14 mag. We note that we can generally probe our objects out to more than one half-light radius in our data, thus the tidal radius would be within the observed stellar extent. However, since most objects from our sample do not show obvious signs of current disruption, we suspect that they may contain additional mass in order to prevent tidal disruption.If we assume a higher mass-to-light ratio of M/L_V = 10, the tidal radius of the same object would increase to 2.9 kpc. For M/L_V = 100 the tidal radius would be R_tidal = 6.2 kpc, and for M/L_V = 1000 we derive R_tidal = 13.3 kpc. For M/L_V close to 1000 the tidal radius is significantly larger than the observed range of half-light radii. If such a high mass-to-light ratio would be reached within the tidal radius, we might expect to find a higher number of galaxies with r_50≳ 3 kpc in the cluster core. However, for M/L_V ≲ 100, the tidal radius would be on the order of 1–2 r_50, which is also consistent with the mass-to-light ratios derived from dynamical measurements of similar galaxies. For example, <cit.> found a mass-to-light ratio of ∼ 50 within one half-light radius for one LSB galaxy in the Coma cluster (M_V = -16.1 mag, r_50 = 4.3 kpc),[Based on stellar dynamics of the galaxy.] and <cit.> derived a mass-to-light ratio of ∼ 100 within one half-light radius for one LSB galaxy in Virgo (M_g = -13.3 mag, r_50 = 2.8 kpc).[Based on GC system dynamics of the galaxy.] We note that based on similar analytical arguments as described above <cit.> also estimated a dark matter fraction of ≳ 100 per cent within an assumed tidal radius of 6 kpc for a sample of faint LSB candidates within the core region of the Coma cluster.While the above approach gives an estimate of the radius beyond which material is likely going to be stripped, another approach to estimate the effect of tides on galaxies in clusters is to compare the density of the tidal field to the density of the orbiting galaxy <cit.>. The density of the tidal field ρ_tidal is given by Poisson's equation, ρ_tidal = F_tidal / (4 π G), where F_tidal is the trace of the tidal tensor. We consider the extended mass distribution of the cluster[Unlike in the first approach, where a point-mass approximation was used.] and approximate the strength of the tidal force at a given clustercentric distance r_0 as F_tidal = | dg(r)/dr|_r_0, where g(r) is the gravitational acceleration exerted by the mass of the cluster. For g(r) we adopt the gravitational acceleration due to the Perseus cluster potential given by <cit.>, where we only consider the contribution of the NFW-profile, which is the dominant component at clustercentric distances r ≳ 10 kpc. We approximate the average density of the orbiting galaxy, assuming spherical symmetry, as ρ_gal = M_gal(R) / (4 π R^3 / 3), where M_gal(R) is the total mass of the galaxy within a radius R. Requiring that the density of the galaxy is larger than the tidal density to prevent its disruption, the limiting radius R_lim is given asR_lim≥√(3 G M_gal(R)/| dg(r)/dr|_r_0)Considering again a typical galaxy from our sample, with M_V = -14 mag at a clustercentric distance r_0 = 330 kpc, we find R_lim = 0.8 kpc for M/L_V = 2, R_lim = 1.3 kpc for M/L_V = 10, R_lim = 2.8 kpc for M/L_V = 100 and R_lim = 6.1 kpc for M/L_V = 1000. Thus, in comparison to the tidal radius derived with the first approach, the limiting radius obtained with the second approach is a factor of two smaller. If we assume that M/L_V = 100 would be characteristic for a considerable fraction of our sample, then the limiting radius would be on the order of only 1 r_50.Does this imply that a few of the largest LSB candidates in the Perseus cluster core should be in process of tidal disruption right now? – We do identify three LSB candidates in Perseus that show possible signs of disruption (see panels on the right-hand side in Fig. <ref>). Candidate 44 appears to be embedded in stream like filaments. It is, however, unclear whether we see here still a bound galaxy or rather a remnant core of a stream. Candidates 26 and 31 seem to be connected via an arc-like tidal stream. This could point to a low-velocity interaction between those two candidates, since such interactions produce the most severe mass-loss. The convex shape of the stream with respect to the cluster centre might suggest that these two objects are not in orbit around the cluster centre, but instead still bound to a possibly recently accreted subgroup of galaxies. The association with a subgroup could be supported by the observation that these three candidates, together with the candidate of brighter surface brightness with tidal tails (see Fig. <ref>, lower left panel), are located closely together in a region south–west of the cluster centre, within a clustercentric distance range of 300–400 kpc. It is also interesting to note that <cit.> found a generally more complex and distorted morphology for LSB candidates in galaxy groups than in galaxy clusters, indicating that the group environment may play an important role in shaping galaxies of low stellar density.The comparison to the LSB galaxy samples in Coma <cit.> showed that both cluster cores hold a similar number of faint LSB candidates with r_50≥ 2 kpc and M_V ≤ -14 mag. Based on the 1.5 times lower cluster mass of Perseus[Assuming M_vir,Coma = 1.3×10^15 M_⊙ <cit.> and M_vir,Perseus = 8.5×10^14 M_⊙ <cit.>.], we would expect a somewhat lower number of all galaxy types in Perseus. However, with regard to the density in the cluster core, both clusters reach a comparable galaxy surface number density within 0.5 Mpc <cit.>, thus causing comparable disruptive forces in both cluster cores. Therefore, according to the cluster mass and density, we would expect a similar or even lower number of LSB galaxies of such large size in Perseus, which is in agreement with our observations. One important question to investigate would be whether there exists a possible evolutionary link between LSB galaxies that are red and quiescent and those that are blue and star-forming. The cosmological simulations of <cit.> suggest that faint LSB galaxies with large sizes may form as initially gas-rich star-forming systems in low-density environments. In this context, the quenching of star formation should be related to external processes, like, e.g., ram pressure stripping. <cit.> examined a sample of faint LSB candidates in group environments. Since they found the red LSB candidates closer to the respective group's centre than the blue systems this could imply that the group environment was efficient in removing the gas that fuels star formation. This is also seen among the dwarf galaxies of the Local Group, which show a pronounced morphology – gas content – distance relation <cit.>. However, a few quiescent and gas-poor LSB galaxies of dwarf luminosity are also observed in isolation <cit.>, which would not fit into this scenario. An essential aspect would be to understand whether the physical processes governing the formation and evolution of LSB galaxies are controlled by stellar density or by stellar mass. The latter could possibly explain the observed wide variety of LSB galaxy properties from low-mass dSphs to massive LSB disc galaxies.§ SUMMARY AND CONCLUSIONSWe obtained deep V-band imaging data under good seeing conditions of the central regions of Perseus with PFIP at the WHT that we used to search for faint LSB galaxies in the surface brightness range of the so-called `ultra-diffuse galaxies'. We detected an abundant population of 89 faint LSB galaxy candidates for which we performed photometry and derived basic structural parameters. Our sample is characterized by mean effective surface brightnesses 24.8 ≤⟨μ_V⟩_50≤ 27.1 mag arcsec^-2, total magnitudes -11.8 ≥ M_V ≥ -15.5 mag and half-light radii 0.7 ≤ r_50≤ 4.1 kpc. A comparison to overlapping HST/ACS imaging data indicates that the sample is relatively uncontaminated by background objects.We find good evidence for tidal disruption leading to a deficiency of LSB galaxy candidates in the central regions of the cluster. This is indicated by a constant observed number density beyond clustercentric distances of 100 kpc and the lack of very large LSB candidates with r_50≥ 3 kpc except for one object. However, only a few candidates show structural evidence of ongoing tidal disruption. If LSB systems are to remain gravitationally bound in the cluster core, the density limits set by the Perseus cluster tidal field require that they have high M/L values of about 100, assuming a standard model for gravity.In comparison to the Coma cluster – with its comparable central density to Perseus – we find that our sample statistically resembles the LSB galaxy population in the central regions of Coma. Given the same dearth of large objects with r_50≥ 3 kpc in both cluster cores we conclude that these cannot survive the strong tides in the centres of massive clusters. § ACKNOWLEDGEMENTSThe William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias (programme 2012B/045). We thank Simone Weinmann and Stefan Lieder for useful comments when preparing the WHT observing proposal. CW is a member of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD). RK gratefully acknowledges financial support from the National Science Foundation under grant no. AST-1664362. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.mnras | http://arxiv.org/abs/1705.09697v2 | {
"authors": [
"Carolin Wittmann",
"Thorsten Lisker",
"Liyualem Ambachew Tilahun",
"Eva K. Grebel",
"Christopher J. Conselice",
"Samantha Penny",
"Joachim Janz",
"John S. Gallagher III",
"Ralf Kotulla",
"James McCormac"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170526195133",
"title": "A population of faint low surface brightness galaxies in the Perseus cluster core"
} |
[email protected] ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Victoria, Australia [email protected] Instituto de Física,Universidade de São Paulo, São Paulo – SP, Brazil [email protected]ó Catalana de Recerca i Estudis Avançats (ICREA), Departament d'Estructura i Constituents de la Matèria,Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, SpainC.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA We analyze the scattering of fermions, Higgs and electroweak gauge bosons in order to obtain the partial–wave unitarity bounds on dimension–six effective operators, including those involving fermions.We also quantify whether, at the LHC energies, the dimension–six operators lead to unitarity violation after taking into account the presently available constraints on their Wilson coefficients.Our results show that for most dimension–six operators relevant for the LHC physics there is no unitarity violation at the LHC energies, and consequently, there is no need for the introduction of form factors in the experimental and phenomenological analyses, making them model independent. We also identify two operators for which unitarity violation is still an issue at the LHC Run–II. 14.70-e, 14.80.BnUnitarity Constraints on Dimension-six Operators II: Including Fermionic Operators M. C. Gonzalez–Garcia==================================================================================== § INTRODUCTION The discovery of a scalar state with properties in agreement with those of the Standard Model (SM) Higgs boson at the Large Hadron Collider (LHC) set the final stone in establishing the validity of the model.Presently there are no high energy data that are in significant conflict with the SM predictions.In this framework, with no other new state yet observed, one can parametrize generic departures from the SM by an effective Lagrangian constructed with the SM states and respecting the SM symmetries, abandoning only the renormalizabilty condition which constrains the dimension of the operators to be of dimension four or less. In particular the established existence of a particle resembling the SM Higgs boson implies that the SU(3)_C ⊗ SU(2)_L ⊗ U(1)_Y symmetry can be realized linearly in the effective theory, an assumption under which we will work in this paper.In this framework the first departures from the SM at the LHC which also respect its global symmetries appear at dimension–six. When using such an effective field theory to quantify possible deviations from the SM predictions, one must be sure of its validity in the energy range probed by the experiments.As is well known, the higher–dimensional operators included in the effective Lagrangian can lead to perturbative partial–wave unitarity violation at high energies, signaling a maximum value of the center–of–mass energy for its applicability.If there is unitarity violation we must either modify the effective field theory, e.g.by adding form factors that take into account higher order terms <cit.>, or it should be replaced by an ultraviolet (UV) complete model. In Ref. <cit.> we presented a general study of unitarity violation in electroweak and/or Higgs boson pair production in boson and/or fermion collisions associated with the presence of dimension–six operators involving bosons, concentrating on those which are blind to low energy bounds.The rationale behind this choice was that not–blind operators were expected to be too constrained by electroweak precision data to be relevant.In this work we revisit this assumption and extend the study to introduce the effects of these operators sensitive to low energy observables, and in particular those involving the coupling of fermions to electroweak bosons.This is timely since the LHC has already started to be able to probe triple electroweak gauge boson couplings with a precision comparable to, or even better than, LEP <cit.>. With such a precision the LHC experiments are already sensitive to deviations of the couplings of electroweak gauge bosons to fermions that are of the order of the limits obtained using the electroweak precision data (EWPD) <cit.>. In this paper we evaluate the unitarity bounds on bosonic and fermionic dimension–six operators from boson pair production amplitudes.As in Ref. <cit.> we take into account all coupled channels in both elastic and inelastic scattering and all possible helicity amplitudes. Moreover, we consistently work at fixed order in the effective Lagrangian expansion[Other studies in the literature have been performed either considering only one non–vanishing coupling at a time, and/or they did not take into account coupled channels, or they worked in the framework of effective vertices <cit.> ].We also study the variation of the constraints under the assumption of the flavour dependence of the fermionic operators.With these results we can address whether, at the LHC energies, the dimension–six operators can indeed lead to unitarity violation after taking into account the presently available constraints on the anomalous couplings. This is accomplished by substituting the present limits of the Wilson coefficients in our partial–wave unitarity bounds to extract the center–of–mass energy at which perturbative unitarity is violated.In order to do so we consistently derive the EWPD constraints on the coefficients of the non-blind operators.Our results show that for all dimension–six operators relevant for the LHC physics, except for just two (O_Φ,2 and O^(1)_Φ d), there is no unitarity violation at the LHC energies, and consequently, we can safely neglect the introduction of form factors in the experimental and phenomenological analyses, making them cleaner and free of ad-hoc parameters. In the case of the operator O_Φ,2 there is no unitarity violation up to subprocess center–of–mass energies of the order of 2.1 TeV, meaning that we have to be more careful in analyzing the high energy tail of processes where the Higgs boson can participate.On the other hand for O^(1)_Φ d perturbative unitarity holds for diboson (VV) subprocess center–of–mass energy less than 3.5 TeV. This paper is organized as follows. Section <ref> contains the dimension–six operators relevant for our analyses, while we present in Section <ref> the unitarity bounds for bosonic and fermionic operators (listing the unitarity violating amplitudes in Appendix <ref>).We discuss the consequences of these results in Section <ref> taking into account the existing constraints on the Wilson coefficients of the dimension–six operators.We present the details of our fit to the EWPD in Appendix <ref>, while for completeness we summarize in Appendix <ref> the unitarity constraints on fermion dipole operators. § EFFECTIVE LAGRANGIANWe parametrize deviations from the Standard Model (SM) in terms of higher dimension operators asL_ eff =L_ SM + ∑_n>4,jf_n,j/Λ^n-4 O_n,j . The first operators that impact the LHC physics are of n=6, or dimension–six.Their basis contains 59 independent operators, up to flavor and hermitian conjugation, where we impose the SM gauge symmetry, as well as baryon and lepton number conservation <cit.>. Of those, 49 can be chosen to be C and P conserving and do not involve gluons. Since the S-matrix elements are unchanged by the use of the equations of motion (EOM), there is a freedom in the choice of basis <cit.>. Here we work in that of Hagiwara, Ishihara, Szalapski, and Zeppenfeld <cit.>.§.§ Bosonic OperatorsAssuming C and P conservation there are ninedimension–six operators in our basis involving only bosons that take part at tree level in two–to–two scattering of gauge and Higgs bosons after we employ the EOM to eliminate redundant operators <cit.>.We group these operators according to their field content. In the first class there is just one operator that contains exclusively gauge bosons. 𝒪_WWW =Tr[W_μ^νW_ν^ρW_ρ^μ] . In the next group there are six operators that include Higgs and electroweak gauge fields. [ 𝒪_WW = Φ^†W_μνW^μνΦ, 𝒪_BB = Φ^†B_μνB^μνΦ,; 𝒪_BW = Φ^†B_μνW^μνΦ, 𝒪_Φ,1 = (D_μΦ)^†ΦΦ^†(D^μΦ) ,;𝒪_W = (D_μΦ)^†W^μν(D_νΦ),𝒪_B = (D_μΦ)^†B^μν(D_νΦ). ] The final class contains two operators expressed solely in terms of Higgs fields 𝒪_Φ,2 = 1/2∂^μ(Φ^†Φ)∂_μ(Φ^†Φ) and 𝒪_Φ,3 =1/3(Φ^†Φ)^3. Here Φ stands for the Higgs doublet and we have adopted the notation B_μν≡ i(g^'/2)B_μν, W_μν≡ i(g/2)σ^aW^a_μν, with g and g^' being the SU(2)_L and U(1)_Y gauge couplings respectively, and σ^a the Pauli matrices. The dimension–six operators given in Eqs. (<ref>)–(<ref>) affect the scatterings VV → VV and f f̅→ VV, where V stands for the electroweak gauge bosons or the Higgs, through modifications of triple and quartic gauge boson couplings, Higgs couplings to fermions and gauge bosons, interactions of gauge bosons with fermion pairs, and the Higgs self-couplings; see Table <ref>.Moreover, these anomalous couplings enter in the analyses of Higgs physics, as well as, triple gauge couplings of electroweak gauge bosons and were analyzed in <cit.>. §.§ Operators with fermions After requiring that the dimension–six operators containing fermions conserve C, P and baryon number, we are left with 40 independent operators (barring flavour indexes) in our basis which do not involve gluon fields. We classify them in four groups. In the first group there are three dimension–six operators that modify the Yukawa couplings of the Higgs boson, and therefore do not contribute to the processes that we study at high energies. The second class possesses 25 four–fermion contact interactions that again do not take part in our analyses. The third group includes the operators that lead to anomalous couplings of the gauge bosons with the fermions that exhibit the same Lorentz structures as the SM vertices and are relevant for our analyses. This class contains eight dimension–six operators [ O^(1)_Φ L,ij=Φ^† (i _μΦ)(L̅_iγ^μ L_j) , O^(3)_Φ L,ij =Φ^† (i _μΦ)(L̅_iγ^μ T_a L_j) ,;O^(1)_Φ Q,ij=Φ^† (i _μΦ) (Q̅_iγ^μ Q_j) ,O^(3)_Φ Q,ij=Φ^† (i _μΦ)(Q̅_iγ^μ T_a Q_j) ,; O^(1)_Φ e,ij=Φ^† (i_μΦ)(e̅_R_iγ^μ e_R_j), ; O^(1)_Φ u,ij=Φ^† (i _μΦ)(u̅_R_iγ^μ u_R_j) , ; O^(1)_Φ d,ij=Φ^† (i _μΦ)(d̅_R_iγ^μ d_R_j) , ;O^(1)_Φ ud,ij=Φ̃^† (i _μΦ)(u̅_R_iγ^μ d_R_j + h.c.) ,] where we defined Φ̃=i σ_2Φ^*, Φ^†_μΦ= Φ^† D_μΦ-(D_μΦ)^†Φ and Φ^†_μΦ= Φ^† T^a D_μΦ-(D_μΦ)^† T^a Φ where T^a=σ^a/2.We have also used the notation of L for the lepton doublet, Q for the quark doublet and f_R for the SU(2)_L singlet fermions, where i, j are family indices. The set of operators in Eq. (<ref>) is redundant as two can be removed by the use of the EOM of the electroweak gauge bosons. We chose to remove from the basisthe following combinations of fermionic operators <cit.>[This is a different choice with respect to the basis in Ref. <cit.>, there these two fermionic operators are kept in exchange of the bosonic operators O_W and O_B.] ∑_iO^(1)_Φ L,ii , and ∑_iO^(3)_Φ L,ii . We notice that the operators in the third group not only contribute to VV → VV and f f̅→ VV processes, but they can also be bounded by the EWPD, in particular from Z–pole and W–pole observables; see Section <ref> and Appendix <ref>. By using the EOM to remove the combinations in Eq. (<ref>) we have selected the operator basis in such a way that there are no blind directions in the analysis of the EWPD data <cit.>.To avoid the generation of too large flavor violation, in what follows we assume no generation mixing in these operators, that is, for any operator O_ij= O_iiδ_ij. Finally we notice that the complete basis of dimension–six operators also contains a fourth group of dipole fermionic operators (i.e. with tensor Lorentz structure) and that can participate in two–to–two scatterings of fermions into gauge and Higgs bosons but that do not modify the Z–pole and W–pole physics at tree level, since their contributions do not interfere with the SM ones. They are [ O_eW,ij = i L̅_̅i̅σ^μνℓ_R,jW_μνΦ , O_eB,ij = i L̅_̅i̅σ^μνℓ_R,jB_μνΦ ,; O_uW,ij = i Q̅_̅i̅σ^μν u_R,jW_μνΦ̃ , O_uB,ij = i Q̅_̅i̅σ^μν u_R,jB_μνΦ̃ ,; O_dW,ij =i Q̅_̅i̅σ^μν d_R,jW_μνΦ , O_dB,ij =i Q̅_̅i̅σ^μν u_R,jB_μνΦ , ] where i,j are family indices. These operators lead to partial–wave unitarity violation in different channels from the operators in Eq. (<ref>), and therefore can be bounded independently. For completeness we present the corresponding unitarity violating amplitudes and bounds in Appendix <ref>.§ CONSTRAINTS FROM UNITARITY VIOLATION IN TWO–TO–TWO PROCESSES Let us start by studying the unitarity violating amplitudes associated with the bosonic operators listed in Eqs. (<ref>)–(<ref>) of which all but O_Φ,3 lead to amplitudes which grow with s in the two–to–two scattering of electroweak gauge bosons and Higgs V_1_λ_1V_2 _λ_2→V_3_λ_3V_4_λ_4. Thehelicity amplitude of these processes is then expanded in partial waves in the center–of–mass system, following the conventions of <cit.> ℳ (V_1_λ_1V_2 _λ_2→V_3_λ_3V_4_λ_4)=16 π∑_J( J+1/2) √(1+δ_V_1_λ_1^V_2_λ_2)√(1+δ_V_3_λ_3^V_4_λ_4) d_λμ^J(θ) e^i M φ T^J(V_1_λ_1V_2 _λ_2→V_3_λ_3V_4_λ_4), where d is the usual Wigner rotation matrix and λ=λ_1-λ_2, μ=λ_3-λ_4, M = λ_1 - λ_2 - λ_3 + λ_4, and θ (φ) is the polar (azimuth) scattering angle. In the case one of the vector bosons is replaced by the Higgs we use this expression by setting the correspondent λ to zero. The helicity scattering amplitudes for the operatorsand O_BW are presented in the Appendix <ref>, while the corresponding amplitudes for the other bosonic operators can be found in Ref. <cit.>.Notice that the contributions of the bosonic operators to VV → VV scattering amplitudes grow with s since the gauge invariance leads to the cancellation of potential terms growing as s^2 <cit.>.Moreover, all bosonic operators contribute to the J=0 and J=1 partial waves, however, O_WWW also leads to the growth of J ≥ 2 amplitudes. Nevertheless, the most stringent bounds come from the J=0 and 1 partial waves, therefore, we restrict our attention to these channels. Furthermore unitarity violating amplitudes arise for the three possible charge channels Q=0,1,2; see Ref. <cit.> for notation and a list of all the states contribution to each (Q,J) channel. In order to obtain the strongest bounds on the coefficients of the eight operators, we diagonalize the six matrices containing the T^J_Q amplitudes for each of the (Q,J) channels and impose that all their eigenvalues (a total of 59) satisfy the constraint |T^J(V_1_λ_1V_2 _λ_2→V_1_λ_1V_2_λ_2)| ≤ 2 . Initially we obtain the unitarity bounds on the eight bosonic operators assuming that only one Wilson coefficient differs from zero at a time.This is a conservative scenario, i.e. leads to stringent bounds, since we do not take into account that more than one operator contributing to a given channel could lead to cancellations and therefore looser limits. For this case we obtain: [ |f_ϕ,2/Λ^2s|≤33, |f_ϕ,1/Λ^2s|≤50, |f_W/Λ^2s| ≤ 87,;;|f_B/Λ^2s| ≤ 617,|f_WW/Λ^2s| ≤ 99, |f_BB/Λ^2s| ≤ 603,;; |f_BW/Λ^2s| ≤ 456, |f_WWW/Λ^2s| ≤ 85. ]Next we study the constraints on the full set of eight bosonic operators when they are all allowed to vary. In order to find closed ranges in the eight–dimensional parameter space we need to consider also the constraints from fermion annihilation into electroweak gauge bosons <cit.>.To do so we obtain the helicity amplitudes of all processes f_1σ_1f̅_2σ_2→ V_3λ_3 V_4λ_4, and then perform the expansion in partial waves of the center–of–mass system; for further details and conventions see Ref. <cit.>. We present in the Appendix <ref> the leading order terms of the scattering amplitudes that give rise to unitarity violation at high energies taking into account the dimension–six operators in Eqs. (<ref>)–(<ref>).These amplitudes are proportional to δ_σ_1,-σ_2 since we neglect the fermion masses in the high energy limit.It is interesting to notice that the dimension–six operators O_WWW, O_W and O_B modify the triple electroweak gauge boson couplings (TGC), therefore, as expected, their presence would affect the SM cancellations that cut off the growth of the f f̅→ VV amplitudes. On the other hand, the operator O_BW also modifies the TGC, however, its effects on the Z/γ wave function renormalizations cancel the growth with the center–of–mass energy due to the anomalous TGC. Similar cancellations occur for O_Φ,1 which contributes to triple gauge vertices, as well as the coupling of gauge bosons to fermions. In order to obtain more stringent bounds and to separate the contributions of the different operators, we consider the processes <cit.> X → V_3λ_3 V_4λ_4, where X is a linear combination of fermionic initial states: | X ⟩ = ∑_f_i σ_i x_f_1 σ_1, f_2σ_2 | f_1σ_1f̅_2σ_2⟩ , with the normalization ∑_f_i σ_i | x_f_1 σ_1, f_2σ_2 |^2 =1. The corresponding bounds read <cit.> ∑_V_3_λ_3,V_4_λ_4|T^J(X→V_3_λ_3V_4_λ_4)|^2 ≤ 1 . In particular using the linear combinations as displayed in the first three lines of Table <ref> we are able to impose independent bounds in each of the Wilson coefficients of the three bosonic operators participating in the ff̅→ VV scattering amplitudes. Combining those with the conditions from partial–wave unitarity of the 59 eigenvalues of the elastic boson scattering amplitudes discussed above we find the most general constraints in the eight–dimensional parameter space. In summary, allowing all coefficients to be nonzero, and searching for the largest allowed values for each operator coefficient while varying over the other coefficients, yields: [ |f_ϕ,2/Λ^2s| ≤ 209, |f_ϕ,1/Λ^2s| ≤ 151, |f_W/Λ^2s| ≤ 436,;|f_B/Λ^2s| ≤ 1460,|f_WW/Λ^2s| ≤ 319, |f_BB/Λ^2s| ≤ 1340,; |f_BW/Λ^2s| ≤ 1386,|f_WWW/Λ^2s| ≤85 . ] As expected these bounds extend the region of validity of the effective theory with respect to the case where just one Wilson coefficient is allowed to be non–vanishing. This is the most optimistic scenario because it implicitly assumes that the values of the Wilson coefficients are tuned to have the largest energy region where the effective theory is valid.It is important to stress that both the one–dimensional bounds in Eq. (<ref>) and the general bounds in Eq. (<ref>) hold independently of the values of the coefficients of the fermionic operators due to the choice of initial states in Table <ref>.§.§ Bounds on Generation Independent OperatorsNow we focus our attention on the operators involving fermions, what require assumptions concerning their flavour structure as we discuss next.Initially we assume that the new physics giving rise to the dimension–six operators is generation blind.In this case we can drop the generation index in all coefficients.Therefore, the constraint on the operators in Eq. (<ref>) implies that the operators O_Φ L^(1) and O_Φ L^(3) are redundant. As for the case of the bosonic operators, in order to obtain more stringent bounds and to separate the contributions of the different operators we consider specific initial states X. In particular choosing the linear combinations as displayed in Table <ref> we are able to impose independent bounds in each of the fermionic Wilson coefficients participating in the ff̅→ VV scattering amplitudes. Starting from the states defined in Table <ref> and the scattering amplitudes given in Appendix <ref> we obtain the partial–wave helicity amplitudes also listed in Table <ref>.The corresponding constraints on the Wilson coefficients of the fermionic operators are [|f^(1)_Φ e/Λ^2 s| <53 ,|f^(1)_Φ u/Λ^2s | < 34, |f^(1)_Φ d/Λ^2s| < 23,;; |f^(1)_Φ Q/Λ^2s| < 14, |f^(3)_Φ Q/Λ^2s| < 123 , |f^(1)_Φ ud/Λ^2s | < 13. ] §.§ Bounds on Generation Dependent Operators In this case first we need to eliminate the redundant operators in Eq. (<ref>). In order to do so we define four independent combinations of the six leptonic operators O^(3)_Φ L,ii and O^(1)_Φ L,ii which are not removed by the EOM's. They are [O^(1)_Φ L,22-11= O^(1)_Φ L,22- O^(1)_Φ L,11 ,O^(1)_Φ L,33-11= O^(1)_Φ L,33-O^(1)_Φ L,11 ,;;O^(3)_Φ L,22-11= O^(3)_Φ L,22- O^(3)_Φ L,11 ,O^(3)_Φ L,33-11= O^(3)_Φ L,33- O^(3)_Φ L,11 ,;] and we denote the corresponding Wilson coefficients as f^(1)_Φ L,22-11, f^(1)_Φ L,33-11, f^(3)_Φ L,22-11, and f^(3)_Φ L,33-11 respectively.It is interesting to notice that the sum over the three generations for the Q=0 leptonic +-00 amplitudes cancel for the left–handed operators because this is the combination removed by the EOM.With this in mind we define the initial states in Table <ref> to impose bounds on each of the fermionic operators. Using these initial states and the corresponding helicity amplitudes, the bounds coming from f f̅→ VV on each of the Wilson coefficients of the fermionic operators read [|f^(1)_Φ e,jj/Λ^2s |<69,|f^(1)_Φ u,jj/Λ^2s |<42, |f^(1)_Φ d,jj/Λ^2s |<34 ,;;|f^(1)_Φ Q,jj/Λ^2s |< 23 , |f^(3)_Φ Q,jj/Λ^2s|<174 ,|f^(1)_Φ ud,jj/Λ^2s |<22 ,;;|f^(1)_Φ L,jj-11/Λ^2s |<53 , |f^(3)_Φ L,jj-11/Λ^2s |<213 , ]with j=2,3 in the last two inequalities. As expected, the above limits are weaker than the ones displayed in Eq. (<ref>) for the generation independent operators.§ DISCUSSION Let us start by noticing that even in the most general case, allowing for all operators to have non–vanishing coefficients, we have obtained bounds that are closed ranges. This means that there is a bounded region of the parameter space for whichthe effective theory is perturbatively valid. In other words there is no combination of Wilson coefficients that can extend indefinitely the energy domain where there is no partial–wave unitarity violation.Second we want to address whether, within that region of coefficients, violation of unitarity can be an issue at the Run II LHC energies. Our procedure to quantitatively answer this question is to determine the maximum center–of–mass energy for which the unitarity limits are not violated given our present knowledge on the Wilson coefficients of the dimension–six operators from lower energy data.For definiteness we considered the maximum allowed value of these coefficients at the 95% confidence level in our analysis. Clearly the results depend on this hypothesis and the energy range where perturbative unitarity holds is extended if we consider these coefficients at 68% confidence level.In this respect EWPD gathered at the Z–pole and W–pole lead to stringent constraints on operators contributing at linear order to these observables and these results are model independent. These are the fermionic operators leading to Z and W couplings to fermions with the same Lorentz structure as the SM, most of the fermionic operators in Eq. (<ref>), togetherwith the bosonic operators O_BW and O_Φ,1. The 95% CL allowed range for their coefficients obtained from a global analysis performed in the full multi–dimensional parameter space are presented in Table <ref>; further details of the analysis are given in Appendix <ref>. With these results we have quantified the maximum center–of–mass energy for which partial–wave unitarity holds for each operator in two scenarios. In the first we do not allow for cancellations among the contributions of the bosonic operators in the s-growing terms in VV→ VV scattering, and therefore, we use the constraints obtained with just one non-vanishing Wilson coefficient; see Eq. (<ref>). In addition to that we considered generation independent fermion operators, Eq. (<ref>), and the corresponding bounds on the Wilson coefficients in the central column in Table <ref>.In the second scenario we use the unitarity constraints on the bosonic operators allowing for cancellations in the VV→ VV scattering amplitudes, as in Eq. (<ref>), together with the assumption of generation–dependent fermion operators, Eq. (<ref>), and the corresponding bounds on the Wilson coefficients in the last column in Table <ref>.The maximum center–of–mass energy for which partial–wave unitarity holds in both scenarios is: [ O_Φ,1√(s)_ max =18TeV ,O_BW√(s)_ max = 16.TeV ,; O^(1)_Φ e√(s)_ max =21TeV , O^(1)_Φ u √(s)_ max = 9.2 TeV ,; O^(1)_Φ d √(s)_ max =3.5TeV , O^(1)_Φ Q√(s)_ max = 8.3TeV ,; O^(3)_Φ Q√(s)_ max =14TeV ,; O^(1)_Φ L,22-11√(s)_ max =11TeV , O^(1)_Φ L,33-11 √(s)_ max =9.2TeV ,; O^(3)_Φ L,22-11√(s)_ max =12TeV , O^(3)_Φ L,33-11√(s)_ max=9.2TeV .; ]Notice that the fermionic operator O^(1)_Φ ud does not contribute to the observables used in the Z–pole and W–pole data analysis at the linear level as it gives a right–handed W coupling which does not interfere with the SM amplitude.It does however, contribute linearly to observables which depend on specific entries of the CKM matrix via a finite renormalization of the quark mixing, in particular to deep inelastic scattering of neutrinos off nucleons, as well as, measurements of the CKM matrix elements in hadronic decays <cit.>. The derivation of the bounds on its coefficient from this data involves additional assumptions about its flavour structure and the presence of further four–fermion operators which also contribute to these processes, making them more model dependent. Under the assumption of generation independent couplings with no cancellation with the additional four–fermion operators one obtains the constraints in Refs. <cit.>, -0.006≤f^(1)_Φ ud/Λ^2≤ 0.01 which imply √(s)_ max =25. TeV. For the remaining dimension–six operators that we studied, the present bounds on their Wilson coefficients come from global fits to Higgs physics and TGC <cit.> at the LHC Run I and the correspondingmaximum center–of–mass energy for which partial–wave unitarity holds is: [O_B√(s)_ max =7.2TeV ,O_W √(s)_ max=4.7TeV ,; O_BB √(s)_ max = 10.TeV , O_WW√(s)_ max= 5.2TeV ,;O_Φ,2 √(s)_ max=2.1TeV ,O_WWW √(s)_ max = 5.7TeV . ] In order to access the importance of the above results for the LHC analyses we should keep in mind that, presently, the most energetic diboson(VV) events possess a center–of–mass energy of the order of 3 TeV; see for instance <cit.>. As more integrated luminosity is accumulated this maximum energy will grow to 4–5 TeV, so we consider that as long as unitarity violation occurs only above these energies, it will not be an issue within the present LHC runs. This condition, of course, will have to be revisited at higher luminosity runs, but at that point also one will have to take into account the possiblemore stringent bounds on the Wilson coefficients. From the results in Eq. (<ref>)–(<ref>) we read that there is no need of modification of the dimension–six effective theory to perform the LHC analyses for most operators. One exception is the operator O^(1)_Φ d whose relatively lower √(s)_ max = 3.5 TeV, however, originates from the weaker bounds on its coefficients induced by the 2.8σ discrepancy of A_ FB^0,b in the EWPD. Notwithstanding, studies of anomalous triple gauge couplings in diboson production should analyze more carefully the high energy tail of the distributions if they include this coupling. Furthermore, there is one additional exception that is the operator O_Φ,2. Since this operator modifies the production and decay of Higgs bosons, as well as, the VV → VV scattering in vector boson fusion the high energy tails of these processes may also need a special scrutiny.An eventual caveat of the above conclusions is that the UV completion might be strongly interacting and the lowest center–of–mass energy exhibiting perturbative unitarity violation then marks the onset of the strongly interacting region. If this were the case at the LHC we should observe new states, which has not yet been the case yet. § ACKNOWLEDGMENTSWe thank J. Gonzalez–Fraile for his valuable contributions to this work. O.J.P.E. is supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); MCG-Gis supported by USA-NSF grant PHY-1620628, by EU Networks FP10 ITN ELUSIVES (H2020-MSCA-ITN-2015-674896) and INVISIBLES-PLUS (H2020-MSCA-RISE-2015-690575), by MINECO grant FPA2016-76005-C2-1-P and by Maria de Maetzu program grant MDM-2014-0367 of ICCUB. T.C. is supported by the Australian Research Council.§ HELICITY AMPLITUDES We present in this appendix the list of unitarity violating amplitudes considered in this work that must be complemented by those in Ref. <cit.>.§ CONSTRAINTS FROM EWPDWe briefly summarize here the details of our analysis of EWPD. Similar analyses for different choices of operator basis can be found in <cit.>. We work on the Z–scheme where the input parameters are chosen to be α_s, G_F, α_em, M_Z <cit.>, and M_h <cit.>.In addition to these quantities we also consider the fermion masses as input parameters. All the other quantities appearing in the Lagrangian are implicitly expressed as combinations of experimental inputs. In our analyses we evaluated the dimension–six contributions to the observables keeping both SM contribution and the linear terms in the anomalous couplings, i.e. the we considered only the interference between the SM and the anomalous contributions.The predictions for the shift in the observables of the Z and W pole physics with respect to their SM values are ΔΓ_Z=2 Γ_Z, SM( ∑_f (g_L^fΔ g_L^f+g_R^fΔ g_R^f)N_C^f/∑_f(|g_L^f|^2+|g_R^f|^2)N_C^f) ,Δσ_h^0= 2σ_h, SM^0( (g_L^eΔ g_L^e+g_R^eΔ g_R^e)/|g_L^e|^2+|g_R^e|^2 +∑_q (g_L^qΔ g_L^q+g_R^qΔ g_R^q)/∑_q(|g_L^q|^2+|g_R^q|^2) -ΔΓ_Z/Γ_Z, SM) ,Δ R_l^0≡Δ(Γ_Z^ had/Γ_Z^l)= 2 R_l, SM^0 (∑_q (g_L^qΔ g_L^q+g_R^qΔ g_R^q)/∑_q(|g_L^q|^2+|g_R^q|^2) - (g_L^lΔ g_L^l+g_R^lΔ g_R^l)/|g_L^l|^2+|g_R^l|^2) ,Δ R_q^0≡Δ(Γ_Z^q/Γ_Z^ had)= 2R_q, SM^0( (g_L^qΔ g_L^q+g_R^qΔ g_R^q)/|g_L^q|^2+|g_R^q|^2 - ∑_q' (g_L^q'Δ g_L^q'+ g_R^q'Δ g_R^q')/∑_q'(|g_L^q'|^2+|g_R^q'|^2)) , Δ𝒜_f=4 𝒜_f, SM g_L^fg_R^f/|g_L^f|^4-|g_R^f|^4(g_R^fΔ g_L^f-g_L^fΔ g_R^f) ,Δ P_τ^ pol=Δ𝒜_l ,Δ A_FB^0,f=A_FB,SM^0,f( Δ𝒜_l/𝒜_l+ Δ𝒜_f/𝒜_f) ,ΔΓ_W=Γ_W, SM(4/3Δ g_WL^ud+2/3Δg_WL^eν+Δ M_W) ,Δ Br_W^eν=Br_W, SM^eν(-4/3Δ g_WL^ud+4/3Δ g_WL^eν) ,where we write the induced corrections to the SM fermion couplings of the Z boson (g^f_L(R)) as Δ g_L,R^f=g_L,R^fΔ g_1+Q^fΔ g_2+Δg̃^f_L,R . The universal shifts of the fermion couplings in Eq. (<ref>) due to dimension–six operators are Δ g_1=1/2(αT-δ G_F/G_F), Δ g_2=^2/(^2 (αT-δ G_F/G_F) - 1/4^2αS) .where we denoted the sine (cosine) of the weak mixing angle by(). The cosine and sine of twice θ_W are then denotedandrespectively. The tree level contributions of the dimension–six operators to the oblique parameters <cit.> are [α S=-e^2v^2/Λ^2 , αT= -1/2v^2/Λ^2,αU=0, δ G_F/G_F=-2 f_LLLLv^2/Λ^2 + (f^(3)_Φ L,11+f^(3)_ΦL,22)vˆ2/Λ^2 ] where for completeness we also included the effect of the dimension–six four–fermion operator contributing with a finite renormalization to the Fermi constant O_LLLL=(L̅γ^μ L)(L̅γ^μ L). The coupling modifications that depend on the fermion flavor are given by [ Δg̃^u_L= -v^2/8 Λ^2 (4f^(1)_Φ Q- f^(3)_Φ Q) , Δg̃^u_R= -v^2/2 Λ^2f^(1)_Φ u,; Δg̃^d_L= -v^2/8 Λ^2 (4f^(1)_Φ Q+ f^(3)_Φ Q) ,Δg̃^d_R= -v^2/2 Λ^2f^(1)_Φ d ,; Δg̃^ν_L=-v^2/8 Λ^2 (4f^(1)_Φ L- f^(3)_Φ L),Δg̃^ν_R= 0 ,;Δg̃^e_L=-v^2/8 Λ^2 (4f^(1)_Φ L+ f^(3)_Φ L) , Δg̃^e_R=-v^2/2 Λ^2f^(1)_Φ e . ] As for the couplings of the W to fermions,in the SM we normalize the left (right)–handed couplingsas 1 (0)and the corresponding shifts on these couplings due dimension–six operators are Δ g_WL^ff'=Δ g_W+ Δg̃_WL^ff',Δ g_WR^ff'=Δg̃_WR^ff', with the universal shift given by Δ g_W=Δ M_W/M_W-1/2δ G_F/G_F ,where the correction to the W mass coming from the dimension–six operators reads Δ M_W/M_W=^2/2αT -1/4αS +1/8^2αU -^2/2δ G_F/G_F. The fermion dependent contributions of the dimension–six operators to the W-couplings are Δg̃_WL^ud=v^2/4 Λ^2f^(3)_Φ Q , Δg̃_WR^ud=v^2/Λ^2f^(1)_Φ ud , Δg̃_WL^eν=v^2/4 Λ^2f^(3)_Φ L ,Δg̃_WR^eν=0 . We notice that, as we are including the effect of the operators in the observables at linear order, the operator O^(1)_Φ ud,ij does not contribute since it leads to a right-handed CC current which does not interfere with the corresponding SM amplitude. We perform two different fits which differ on the assumptions on the generation dependence of the fermionic operators.§.§ Fit withGeneration Independent Operators In the first case we assume that the fermionic operators are generation independent. In this case, as discussed above, we can drop the generation index in all coefficients. Furthermore removing the operators in Eq. (<ref>) implies that those two operators do not appear in the fit to the EWPD. We have then 8 coefficients to be determined {f_BW/Λ^2,f_Φ, 1/Λ^2,f_LLLL/Λ^2,f^(1)_Φ Q/Λ^2,f^(3)_Φ Q/Λ^2,f^(1)_Φ u/Λ^2,f^(1)_Φ d/Λ^2, f^(1)_Φ e/Λ^2}. In our analyses we fitted 15 observables of which 12 are Z observables <cit.>: Γ_Z ,σ_h^0, A_ℓ(τ^ pol) , R^0_ℓ, A_ℓ( SLD) , A_ FB^0,l, R^0_c ,R^0_b ,A_c, A_b, A_ FB^0,c, and A_ FB^0,b (SLD/LEP-I), supplemented by three W observablesM_W ,Γ_W ,andBr( W→ℓν) that are, respectively, its average mass from <cit.>, its width from LEP-II/Tevatron <cit.>, and the leptonic W branching ratio for which the average in Ref. <cit.> is taken.The correlations among the inputs can be found in Ref. <cit.> and have been taken into consideration in the analyses. The SM prediction and its uncertainty due to variations of the SM parameters are taken from <cit.>. When performing the fit within the context of the SM the result is χ^2_ EWPD,SM=18.0, while including the 8 new parameters it gets reduced to χ^2_ EWPD,min=5.3. The results of the analysis are shown in Table <ref> where we quote the 95% C.L. allowed ranges for each parameter in the center column. The range for parameter x is obtained accounting for all possible cancellations in the multiparameter space by imposing the condition Δχ^2_ EWPD, marg(x) <4 where by Δχ^2_ EWPD, marg(x) we denote the value of Δχ^2_ EWPD minimized with respect to the other seven parameters for each value of the parameter x.We notice that the only operator coefficient not compatible with zero at 2σ is f^(1)_Φ d, a result driven by the 2.7σ discrepancy between the observed A_ FB^0,b and the SM expectation.§.§ Fit with Generation Dependent Operators Lifting the assumption of generation independent operators we are left with seven independent leptonic operators. These are, three O^(1)_Φ e,ii plus four combinations of O^(1)_Φ L,ii and O^(3)_Φ L,ii defined in Eq. (<ref>). On the other hand, for operators involving quarks there is not enough information in the observables considered to resolve the contributions from the two first generations. Consequentlywe make the simplifying assumption that operators for the first and second generations have the same Wilson coefficients and only those from the third generation are allowed to be different. Furthermore, for the third generation of quarks only O^(1)_Φ Q,33 and the linear combination f^(1)_Φ Q,33 +1/4 f^(3)_Φ Q,33 contribute independently tothe Z and W observables; see Eq. (<ref>).Altogether there area total of 16 coefficients to be determined from the fit to the Z and W observables:[ f_BW/Λ^2, f_Φ 1/Λ^2 ,f_LLLL/Λ^2 , f^(1)_Φ Q,11/Λ^2=f^(1)_Φ Q,22/Λ^2 ,;; f^(1)_Φ Q,33/Λ^2+1/4 f^(3)_Φ Q,33/Λ^2 ,f^(3)_Φ Q,11 /Λ^2=f^(3)_Φ Q,22/Λ^2 ,f^(1)_Φ u,11/Λ^2= f^(1)_Φ u,22/Λ^2 ,f^(1)_Φ d,11/Λ^2= f^(1)_Φ d,22/Λ^2 ,;;f^(1)_Φ d,33/Λ^2 ,f^(1)_Φ e,11/Λ^2 ,f^(1)_Φ e,22/Λ^2 ,f^(1)_Φ e,33/Λ^2 ,;; f^(1)_Φ L,22-11/Λ^2 , f^(1)_Φ L,33-11/Λ^2 , f^(3)_Φ L,22-11/Λ^2 , f^(3)_Φ L,33-11/Λ^2 . ] In order to obtain the corresponding constraints on these 16 parameters a fit including 24 experimental data points is performed. These are19Z observables <cit.>: [Γ_Z , σ_h^0,R^0_e , R^0_μ, R^0_τ,;A_FB^0,e,A_FB^0,μ,A_FB^0,τ,𝒜_e(τ^ pol) ,𝒜_μ(τ^ pol) ,;𝒜_e( SLD) ,𝒜_μ( SLD) ,𝒜_τ( SLD) ,R^0_c ,R^0_b ,; 𝒜_c, 𝒜_b, A_FB^0,c,andA_FB^0,b, ] plus five W observables: M_W ,Γ_W , Br (W →eν) ,Br(W →μν) ,and Br(W →τν) ,where the three leptonic W branching ratios were taken from Ref. <cit.>. The correlations among the inputs can be found in Refs. <cit.> and were considered in the analysis. As in the previous analysis,the SMprediction for these observables and its uncertainty due to variations of the SM parameters are taken from <cit.>. The fit within the context of the SM leads to χ^2_ EWPD,SM=29, while with the inclusion of the 16 new parameters the minimum gets reduced to χ^2_ EWPD,min=8.2. The 95% allowed ranges for each of the 16 parameters are shown in the last column in Table <ref>. As in the previous case, for each coupling the range is obtained after marginalization over the other 15 couplings.As we can see from Table <ref>, removing the generation independence hypothesis leads to looser constraints, as could be anticipated. Moreover, flavor independent Wilson coefficients and the ones related to the first two families agree with the SM at the 2σ level, with the exception of f^(1)_Φ L,22-11/Λ^2. On the other hand, we can see clearly the effect of the observable A_ FB^0,b on almost all thethird generation Wilson coefficients whose 2σ allowed ranges do not contain the SM.§ DIPOLE OPERATORSThe leading high energy contributions of the dipole fermionic operators in Eq. (<ref>) to the f f̅→ VV scattering is given in Table <ref>. Neglecting fermion masses the dipole fermionic operators contribute to different helicity states to those from operators in Eq. (<ref>) as can be seen from Tables <ref> and <ref>, due to the presence of σ^μν in Eq. (<ref>).Assuming that the Wilson coefficients of the dipole operators are generation independent, we can obtain, using Table <ref>, the following unitarity bounds: 1/√()|∑_i=1^ e^-_-,i e^+_-,i⟩→ W^+_0 W^-_+⇒ |f_eW/Λ^2 s |≤ 491/√()|∑_i=1^ e^-_+,i ν̅_+,i⟩→ W^+_0 A_- ⇒ |(f_eW+f_eB) /Λ^2 s | ≤ 144 ⇒|f_eB/Λ^2 s |≤ 193 1/√() |∑_i=1^∑_a=1^ u^a_-,i u̅^a_-,i⟩→ W^+_+ W^-_0⇒ |f_uW/Λ^2 s |≤ 281/√() |∑_i=1^∑_a=1^ d^a_+,i u̅^a_+,i⟩→ W^-_0 A_+ ⇒ |(f_uW-f_uB) /Λ^2 s | ≤ 83⇒|f_uB/Λ^2 s |≤ 1111/√() |∑_i=1^∑_a=1^d^a_-,i d̅^a_-,i⟩→ W^+_0 W^-_+ ⇒ |f_dW/Λ^2 s |≤ 281/√()|∑_a=1^∑_i=1^d^a_+,i u̅^a_+,i⟩→ W^-_0 A_- ⇒ |(f_dW+f_dB) /Λ^2 s | ≤ 83 ⇒|f_dB/Λ^2 s | ≤ 111 Dropping the generation independence hypothesis for the dipole operators,we can use the same set of amplitudes as in Eq. (<ref>) but now without summing over generations. In this case the partial–wave unitarity constraints read: |f_eW,ii/Λ^2 s |≤ 85 , |f_eB,ii/Λ^2 s |≤ 334|f_qW,ii/Λ^2 s |≤ 49 , |f_qB,ii/Λ^2 s |≤ 193where the last line applies for q=u,d. Because they flip the fermion chirality these operators do not interfere at tree–level with the SM amplitudes and also generically they are expected to be suppressed by the fermion Yukawa[Bounds on the CP conserving coefficients of the dipole operators for light fermions can be obtained, in principle, from their tree-levelcontribution to the corresponding anomalous magnetic moment, but for the light quark dipole operators these are hard to extract in a model independent way, therefore, being subject to large uncertainties. For leptons, the current g-2 bounds <cit.> indicate that in the absence of cancellations between the O_eW and O_eB contributions, for operators involving electrons or muons unitarity is guaranteedwell beyond LHC energies.]. In this case only the operators involving the top quark can be sizable. There is an extensive study of the top quark properties at the LHC which includes the operators O_uW and O_uB; see, for instance, Ref. <cit.> and references therein.In particular the measurement of W–boson helicity in top–quark decays <cit.> give us direct access to f_uW/Λ^2. Using the global fit to the top quarks properties in Ref. <cit.> (|f_uW/Λ^2| < 3.8 TeV^-2) we estimate that the operator O_uW does not lead to perturbative unitarity violation in top–quark processes at the LHC for maximum center–of–mass energies up to 2.7 TeV. | http://arxiv.org/abs/1705.09294v1 | {
"authors": [
"Tyler Corbett",
"O. J. P. Éboli",
"M. C. Gonzalez-Garcia"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170525180002",
"title": "Unitarity Constraints on Dimension-six Operators II: Including Fermionic Operators"
} |
firstpage–lastpage 2017Strong lensing and Observables around 5D Myers-Perry black hole spacetimeK D Purohit ^d Accepted . Received ; in original form============================================================================ Asteroid families are valuable source of information to many asteroid-related researches, assuming a reliable list of their members could be obtained.However, as the number of known asteroids increases fast it becomes more and more difficult to obtain robust list of members of an asteroid family. Here we are proposing a new approach to deal with the problem, based on the well known Hierarchical Clustering Method (HCM). An additional step in the whole procedure is introduced in order to reduce a so-called chaining effect. The main idea is to prevent chaining through an already identified interloper. We show that in this way a number of potential interlopers among family members is significantly reduced.Moreover, we developed an automatic on-line based portal to apply this procedure, i.e to generate a list of family members as well as a list ofpotential interlopers. The Asteroid Families Portal (AFP) is freely available to all interested researchers.asteroids: general § INTRODUCTIONAsteroid families are formed when a collision breaks apart a parent body into numeroussmaller fragments <cit.>. As such, they are very important for almost anyasteroid-related research. For example, families may provide us a clue about collisionalhistory of the main asteroid belt <cit.>,asteroid interiors <cit.>, as well as space weathering effects that alterasteroids colour with time <cit.>. Moreover, they may even tell us about the originsof some near-Earth asteroids <cit.>, main-belt comets <cit.>,or help us put constraints on some past phenomena like the Late Heavy Bombardment <cit.>.The impact ejects fragments away from the parent body at velocities similar to the parent's bodyescape speed, that is typically not more than a few tens of meters per second. In the main asteroidbelt these velocities are much slower than the orbital ones, which are about 15 - 20 km/s.Therefore, the impact-produced objects initially keep orbits similar to the orbit of their parent body.This fact allows identification of the so-called dynamical asteroid families, i.e. groups of asteroidsidentified based on similarity of their orbital parameters. Still, from the orbital similarity pointof view, it is important to distinguish young asteroid families from older ones. Very young families (t_age < 1 m.y.) may be still recognized in the space of instantaneous (osculating) orbital elements, because different perturbations have not yet had enough time to disperse orbits of their members <cit.>. On the other hand, older families are usually identified using the so-called proper orbital elements which are nearly constant over time <cit.>. So far a large number of asteroid families have been discovered across the main asteroid belt<cit.>. These families offera wide range of possibilities for further studies, yet an essential prerequisite for all these studies is a reliably established family membership. However, as nowadays the number of known asteroidsincreases fast, it becomes more and more difficult to obtain a robust list of members of anasteroid family.For the purpose of family identification the Hierarchical Clustering Method (HCM), proposed by <cit.>,is most widely used.[Besides the HCM, the Wavelet Analysis Method (WAM)was also successfully applied to identify asteroid families <cit.>.] It connects all objects whose mutual distances in the three-dimensional proper-element space are below a threshold value.The HCM, however, obviously has some limitations in distinguishing between real family members and nearby background asteroids.[This problem is not due the HCM itself, and is not limited to this method only. For instance, the WAM suffers from the same problem <cit.>.]This gives a rise to the well known problem of presence of interlopers among the members of an asteroid family <cit.>.A possible way to deal with this problem is to use additional information to discriminate real family members from interlopers. Asteroids that belong to the same family generallyhave similar mineral composition <cit.>. Thus, data about their spectra/colours oralbedos may help to determine membership of an asteroid family more reliably. These additional information may be used in different ways. One possible method to exploitavailable physical data is to apply the HCM in extended space, i.e. in the space that also includes physical data in addition to the three proper elements. <cit.> applied the HCM in 4-dimensional space, using the Sloan Digital Sky Survey <cit.> colours as the fourth dimension. <cit.> extend this approach to the 5th dimensionusing albedos as an additional dimension, significantly reducing the percentage of known interlopers with respect to other methods.Another possible procedure is to first separate the main belt asteroids into two populations (typically representing C- and S-type objects) according to theircolour and albedo values. The HCM is then applied to each of these populations separately <cit.>. In this way Masiero et al. managed to identify several new families.These methods, despite being in general very efficient, have a serious limitation,that they can only be applied to a reduced set of main belt asteroids for which the colours and albedos are obtained.Despite significant increase of available physical data in the recent years,the number of asteroids for which these data are at our disposal is still several times smaller than the number of objects for which proper elements have been computed. As explained by <cit.>, the dynamical parameters, in this case the proper elements, have a larger information content than the physical observations, because the latter are available either for significantly smaller catalogues, or with lower relative accuracy.For these reasons some authors adopted a bit different approach, so that the so-called dynamical families are first obtained in the space of proper elements, and available physical data is used only posteriori to distinguish possible overlapping families or to identify interlopers among family members <cit.>. All the above described methodologies are useful to some extent. Still, they have some limitations imposed by the HCM itself. A well-known drawback of the HCM based onthe single linkage rule is the so-called chaining phenomenon; that is,first concentrations tend to incorporate nearby groups, forming as a result a kind ofchain which may consists of non-family members <cit.>.In this paper we extend previous works in two directions:* First, we are introducing an additional step in the whole procedure, aiming mainly to reduce the chaining effect. The main idea is that if an interloper is identified as a family member, it may cause more interlopers to be added due to the chaining effect. In this manner we managed to further reduce the number ofpotential interlopers among family members, by preventing chaining through an already identified interloper.* Second, we are presenting here an automatic, free, on-line based tool to apply our procedure. It allows to generate a list of family membersas well as a list of potential interlopers along with criterion for their rejection. This is important because despite numerous papers dealing with family membersidentification and removal of potential interlopers, there is still alack of publicly available information on these.§ METHODThe approach that we are using here is essentially very similar to methods that first identify dynamical families using only the proper orbital elements, and then apply physical data to further refine family membership. The only exception in this respect is an additional stage. The whole procedure could be divided in four main stpdf (Fig. <ref>): * In the first step, the HCM analysis is performed using the whole (initial) catalogue ofproper elements in order to obtain a preliminary list of family members.* In the second step, physical and spectral properties are used in order to identify interlopersamong asteroids initially linked to a family.* In the third step, objects identified as interlopers in the second step, are excluded fromthe initial catalogue of proper elements, producing a modified catalogue* Finally, in the fourth step, the HCM analysis is performed again, but this time using the reduced (modified)catalogue of proper elements, produced in the third step. §.§ STEP 1: Obtaining a preliminary membershipTo obtain a preliminary list of family members we apply the HCM in the space of three proper orbital elements[The catalogue of synthetic proper elements is available at AstDys web page: http://hamilton.dm.unipi.it]: semi-major axis a_p, eccentricity e_p,and sine of inclination sin (i_p). The HCM identifies an asteroid as a part offamily if its distance d from the closest neighbour is smaller than an adopted cut-offdistance d_cut. The common definition of this metric is <cit.>:d = na_p√(C_a (δ a_p/a_p)^2 + C_e (δ e_p)^2 + C_i (δsin(i_p))^2)where na_p is the heliocentric orbital velocity of an asteroid on a circular orbit having the semi-major axes a_p, δ a_p = a_p_1 - a_p_2, δ e_p = e_p_1 - e_p_2 and δsin(i_p) = sin(i_p_1) - sin(i_p_2), where the indexes (1) and (2) denote the two bodies whose mutual distance is calculated.The distance d is expressed in meters per second.The above metric is derived based on the well-known relationship between the components of ejection velocities of the fragments, and the resulting differences in their orbital elements (Gaussian equations). For the constants, the most frequently used values are: C_a = 5/4, C_e = 2 and C_i = 2 <cit.>, but other values yield the similar results <cit.>. Here we applied the HCM around a selected central asteroid. In this case the volume of the proper elements space is not fixed a priory, but it grows around the central object as the velocitycut-off distance increases. In order to define a preliminary list of family members we need to adopt an appropriate value of cut-off distance d_cut. Generally, this is the distance in the space of proper elements that best describes a family.However, for a preliminary definition of the family our aim isslightly different. As we want to identify as many potential interlopers as possible, weused the largest reasonable value of d_cut, rather than the most appropriate one.Thus, in the first step we derive the cut-off value as follows. Starting from the lowestvalue of 10 ms^-1, we increased d_cut by fixed step of 5 ms^-1, until the family mergeswith a local background population of asteroids. For the threshold value we adopt the one two stpdf(10 ms^-1) below the distance at which the family merges with the background. As mentioned above, the cut-off value obtained in this way is often too large to define a nominal family. In some cases this approach caused two (or even more) nearby families to merge into single group(see Sections <ref> and <ref> for discussion on these cases). Still, we use it in order to get as many family members (and consequently also interlopers) as possible. §.§ STEP 2: Interlopers identificationOnce a preliminary list of family members is obtained, the next step is the identification of potential interlopers among them. In this respect, let us recall the well known fact that members of a collisionalfamily typically share similar physical and spectral characteristics <cit.>. Therefore, different spectral and photometric data couldbe used to complement the results of the HCM analysis, in order to obtain a list of potential interlopers. Available observational data is often good enough only to distinguish between C and S classes of asteroids, while other taxonomy classes could not be separated reliably. An obvious exception are spectral data, which are however available for a very limited number of asteroids. The latter may also be true for an advance classification algorithm developed by <cit.>,based on the SDSS data. Still, to stay on the safe side, we decided to resolve here only two aforementioned broadasteroid classes. This analysis isperformed using SDSS colours, geometric albedos from different surveys, and availablespectroscopic data. Then, if it is found that a family is mostly composed of C-type asteroids,all potential members that belong to S-type are defined as interlopers, and vice-versa.§.§.§ The Sloan Digital Sky Survey data The SDSS survey used 5-colour photometric system u, g, r, i and z, with central wavelengths of 0.3543, 0.4770, 0.6231, 0.7625, 0.9134 μm, respectively. The fourth release of the SDSS Moving Object Catalogue (SDSS MOC) contains photometric data for 471,569 moving objects observedup to March, 2007. Among this data, 220,101 entries were successfully linked to 104,449 unique moving objects. It was shown by <cit.> that SDSS photometry is consistent with available spectra of asteroids, meaning that the colours provided by the SDSS could be used to separate at least broad taxonomicclasses, such as C and S. This was also confirmed by <cit.> and <cit.> whoused the third and fourth release of the SDSS MOC, respectively. In both cases authors used aprincipal component analysis approach to link SDSS colours to spectral types. The main difference between the method utilized by <cit.> and <cit.> is that latter authors excluded u-band from their analysis, due to the large noise in u-band presented in the forth version of SDSS MOC.Moreover, <cit.> defined a new classification algorithm based on the SDSS colours that can be used to assign taxonomicclasses to SDSS observations. This scheme is compatiblewith the Bus taxonomy and allows finer distinction between taxonomic classes than methodology used by <cit.> or <cit.>.For our purpose here, we chose to use the fourth release of the SDSS MOC to distinguish between C and S taxonomic complexes. Thus, we adopt <cit.> approach excluding u-band from our analysis, and usinga^* colour defined by <cit.> as:a^* = 0.89 (g - r) + 0.45 (r - i) - 0.57 It is known that asteroids show bimodal distribution in a^*, where C-type objectsare characterized with a^*< 0, while S-type objects typically have a^* > 0.Being interested here in identifying potential interlopers among members of an asteroid family we proceed in the following way. First, for each family, we calculate the mean value a_m^* and its corresponding standard deviation σ (a_m^*), using colours of potential family members. The mean value is then used to define the spectral class of the family. Finally, all potential family members having their individual a^* colour inconsistent with the one derived for the family are classified as interlopers. In order to keep the interloper identification as reliable as possible we introduce a safetymargin between two types, based on the value of standard deviation of a^* colour of an asteroid σ (a^*). For instance, if an asteroid family isclassified as C-type, only objects having a^* < -3σ (a^*) are consider to beinterlopers, and likewise, if a family is S-type then objects with a^* > 3σ (a^*) are interlopers. This methodology is illustrated in Fig <ref> for the case of the Klumpkea family.[It should be noted that the Klumpkea family is listed as the Tirela family (FIN 612) in <cit.> classification.]§.§.§ The geometric albedos - WISE, AKARI and IRASAnother information that could be used to separate C- from S- taxonomic class, and is available for relatively large number of asteroids, is the geometric albedo. The vast majority of these data isobtained by three different infrared surveys. The catalogue of asteroid albedos provided by the Infrared Astronomical Satellite (IRAS) survey was presented by <cit.>, and contains albedos and diameters for 2470 objects. The AKARI survey provides a catalogue with datafor 5208 asteroids <cit.>. Finally, most of currently known asteroid albedos are provided by the Wide-field Infrared Survey Explorer <cit.>. Based on the data provided by this survey,<cit.> published diameters and albedos for more than 100,000 Main-Belt asteroids.Aiming here to use these albedos to identify interlopers among potential family members, we need to define strict criteria. In this respect, first a spectral type of considered asteroid family should be determined, and then it should be checked for each potential member if its own albedo is consistent with the spectral type of the family. In order to determine the dominant taxonomic class of a family, we simply calculate the average albedo p^*_v over all members for which this information is provided by at least one of three available surveys[If the albedo is provided by more than one survey, we always give priority to the most recent data.], and check if the obtained value is consistent with one of the spectral types. For this purpose we assume that an albedo of 0.13 is a border between two types, but, similarly as in the case of the SDSS data, we also introduced a safety margin. Therefore, an asteroid family is assigned C-type if p^*_v < 0.12, while family is definedto be of S-type if p^*_v > 0.14.Most of known families fall in one of two broad spectral types defined above. However, as these two intervals do not overlap, in some cases this causes ambiguity in the determination of family spectral type. Therefore, for families with the average albedo in [0.12, 0.14] interval, we used predefined confidence intervals, suitable only for these families. Such an example is the Eos family (see Section <ref>).Once a spectral type is assigned to a family, all potential members of an inconsistent spectral type are defined as interlopers.As a starting point we used results obtained by <cit.>.These authors calculated averagealbedo p_v, and its corresponding standard deviation σ(p_v),for different classes of asteroids, using data from three mentioned surveys.Assuming that albedos of both spectral types (C and S) do not spread more than3σ from their average values, we calculated the corresponding confidenceintervals. The average albedo and its corresponding standard deviation forC-type objects are p_v=0.06 and σ(p_v)=0.01, respectively;thus, a confidence interval is [0.03,0.09]. Similarly, p_v=0.23 and σ(p_v)=0.02 for S-type, resulting in a confidence interval of [0.17,0.29]. Finally, if a family is of C-type, all potential members with p_v - 3 σ > 0.09 are interlopers. Accordingly, if a family belongs to the S-type, interlopers are all potentialmembers having p_v + 3 σ < 0.17. For asteroid families withpredefined confidence interval,interlopers are those objects that have p_v ± 3 σ outside the corresponding interval.The above defined criteria should be good enough even at small sizes, for which<cit.> found that the S-complex partially overlaps the lowalbedo C-complex.§.§.§ The spectroscopic data The different taxonomy classifications could also be used to exclude interlopers from asteroid families. Unfortunately these data are available only for a very limited number of asteroids. Still, this information should be more reliable than e.g. SDSS colours. Therefore, we use them in our analysis.Before we describe in details how individual interlopers are identified using spectroscopic data, it is important to note that a family spectral class is defined based on the results obtained using colour and albedo information. This is because the number of family members with available spectra is often too small to reliably determine the dominant spectral class of the family.Therefore, if, following the criteria described above, both, colour and albedo data suggest that the family is either C or S class, we proceed with the exclusion of interlopers using spectroscopic data. However, if these two sources of information are not inagreement[This may happen if a family belongs to X-type. It is because the SDSS a^* colour dose not separate C from X type, although these two types are characterized byquite different albedos.], we completely omitted spectral data from our analysis. If applied, this analysis is based on <cit.>, <cit.> and <cit.> classifications.As in the case of albedos, if more classifications provide a spectral type of an asteroid, priority is given to the most recent data.<cit.> used spectrophotometric results from ECAS (Eight-Colour Asteroid Survey) to classify asteroids in different taxonomy groups. The main groups are C, S and X. Dark, carbonaceous asteroids are usually part of the C-group and they could be divided in a few different classes by comparing slope and maximum wavelength of its spectrum. In this respect, members of the C-group are B, F, G and C classes. Metallic asteroids are members of X group that is divided into the E, M and P classes according to the asteroid geometric albedo. Siliceous-stony asteroids are members of the S-group. There are some groups similar to S, but with slightly different spectrum slope and maximum wavelength, such as Q, R, V and A classes. D and T classes can not be easily classified in any of three broad groups defined by <cit.>. If an asteroid family is mostly composed of asteroids belonging to C group, thenan objects is an interloper if according to the Tholen's spectral classification it belongs to one of the following classes: S, E, M, P,Q, R, V, D and T. Analogously, if a family belongs to S group, its potentialmember is an interloper if belongs to one of the following classes: C, B, F, G, D, T or EMP.<cit.> used data from Small Main-Belt Asteroid Spectroscopic Survey (SMASS), which provided continuous spectrum, but covered smaller wavelength range then the ECAS. This classification divided asteroids in 26 taxonomic classes, with most of the classes being similar to those defined by <cit.>, although some Bus & Binzel classes are separated into subclasses. For example, the S group is separated in S_a, S_k, S_l, S_q, S_r and S class, and hence, if a family is part of C group, all asteroids that belong to the S group are identified as interlopers. In this case we exclude as interlopers asteroids of V, O, L, D, T, A, R, Q, K and E classes. Similarly, if a family belongs to S group, interlopers are those asteroids that belong to the C group or B, V, T and D classes. A similar classification logic was applied in <cit.> classification, but these authors used also the infrared part of the spectrum. Therefore, their classification is similar to that of Bus and Binzel, with some small modifications that eliminated subclasses Ld, Sk, and Sl, but added a new Sv subclass within S class.§.§ STEP 3 and STEP 4 §.§.§ Excluding interlopersIn the third step, asteroids identified as interlopers due to their reflectance characteristics are excluded from the initial catalogue of proper elements. This produces a modified catalogue, that is free of known interlopers in the corresponding family. Therefore, the interlopers are excluded from the catalogue of proper elements, rather than from the list of family members, as it is usually done.In the fourth step the HCM is applied again to the modified catalogue of proper elements, produced in the third step. This is the main difference from the methods that have already been used for determining interlopers in asteroid families. The point of this step is to reduce the chaining effect in the HCM. By removing interlopers from the catalogue, we have also removed objects linked to the family through some of these interlopers. An example of the efficiency of this methodology in the Klumpkea family is shown in Fig <ref>. Asteroid (340653) is identified as an interloper based on the albedo data. Once it is removed from the input catalogue, the eighth additional objects are removed.§.§.§ The final membership Finally, in the fourth step, the family members are re-identified using the modified catalogue of proper elements, produced in the third step by removing interlopers identified based on colour, albedo and spectral data. For a purpose of comparison, we run the HCM on the modified catalogueusing again the cut-off distance adopted in the first step. This allows to estimate the fraction of removed interlopers within each family.On the other hand, as discussed above, the cut-off used in the first step is too high to obtain the nominal membership of an asteroid family. A large cut-off value may easily associate many background asteroids to a family. This may even create bridge between different families resulting in a wrong membership. Therefore, the final membership of each family should be obtained using the cut-off that describes a family the best on the modified catalogue of proper elements. Hence, the determination of a nominal cut-off is a very important step in this analysis, and should be carefully performed. The method used here to select the nominal cut-off value is as follows. First, it is checked how the number of family members changes as a function of cut-off values (Fig. <ref>). Then, in order to estimate a nominal cut-off we search for a plateau, i.e. an interval of cut-off values where a number of family members is almost constant. As a general rule, if such a plateau is well defined, the nominal cut-off is adopted to be around the centre of the plateau. Still, in some cases the plateau practically does not exist, and a detailed analysis ofcorresponding family should be performed in order to determine its nominal cut-off value.The latter cases are discussed below, in Section <ref>. § RESULTS In this section we demonstrate how the above described method works, and present the most important results.For this purpose we selected 17 large families from the classification of <cit.>,each of them with more than 1,000 potential members. These families are statisticallyreliable, but at the same time are also expected to have a large number of interlopers.Before discussing these results, let us to recall that they are obtained using unrealistically high velocity distance cut-off values (d_cut). The purpose here is only to show how the method works, and in particular to highlight the importance of our step #4. Realistic results, obtained using the most appropriate values of d_cut, are given below in Section <ref>.The results for the 17 analysed families are summarized in Tables <ref> and <ref>. The total fraction of interlopers found among potential family membersvary from below 1% for families well separated from the background population (e.g. the Hoffmeister family), to more than 20% for cases where an additional family may be present, as for example in the case of the Klumpkea family. Despite these differences, the results are generally in agreement with the expected fraction of interlopers estimated by <cit.>.To better appreciate the importance of step #4, which is the main improvement of our approach with respect to previous works, one should compare the numbers shown in columns 7 and 8 of Table <ref>.The 7th column gives the total number of interlopers identified using available physical and spectral data (# STEP2), while the 8th column provides the number of asteroids linked to the family through some of the identified interlopers (Chaining).The numbers in these two columns are often comparable, and in some cases the number of interlopers removed due to chaining effect is even larger then the number of interlopers identified based on the physical and spectral data. The only exceptions are families with a very small number of interlopers (e.g. Veritas and Hoffmeister), and cases where two or more families overlap (e.g. Eunomia family). Nevertheless, usage of step #4 seems to be important and fully justified.In order to further demonstrate how the method works we selected two families as the case studies, namely the (1040) Klumpkea and (15) Eunomia families. The Klumpkea family was selected due to its location within the main-belt. It is situated in the outer part of the main-belt,where most of the asteroids belong to C spectral type, while Klumpkea belongs to S-type.Therefore, we expect to find a relatively large fraction of the interlopers among members of the Klumpkea family.The Eunomia family is a large group in the middle main-belt, and it is aninteresting example because there may be another family buried inside the Eunomia family. §.§ The case study #1: Klumpkea family Analysing the outputs of the HCM applied to the initial catalogue for different cut-off values,it could be noted that for cut-off below 55 ms^-1, Klumpkea family members are situatedbetween the 11/5 and 21/10 mean motion resonances with Jupiter. Starting from 55 ms^-1 family members are crossing the 21/10 resonance, while at 80 ms^-1 family members cross also the 11/5 resonance. At the cut-off velocity of 90 ms^-1, the family merges with the background population of asteroids. Therefore, the cut-off value of 80 ms^-1 is adopted for the step #1, yielding the initial family membership that contains 2,794 asteroids (see Fig. <ref>).These potential members of Klumpkea family have an average albedo p_v= 0.142 ± 0.044, and an average SDSS colour a^* = 0.112 ± 0.035. Hence, according to the criteria explained in Section <ref> the family belongs to the S-type.In the second step, among the initial family members 452 interlopers are detected (see Table <ref>). Fig. <ref> shows the distribution of these interlopers indifferent planes. It should be noted that most of the potential members located at a < 3.075 au are likely interlopers.After removing the interlopers from the initial catalogue, and applying the HCM to the modified catalogue[Using the same d_cut as in the first step], 2,115 asteroids were identified as family members. Therefore, there are 679 members less than in the initial step, and amongthese 227 asteroids are removed in step #4 due to the chaining effect.In order to determine the nominal cut-off for the Klumpkea family, we study again the HCM results, but this time obtained on the modified catalogue. The situation seems to be different with respect to the results obtained before removal of interlopers. The most important new feature is that family members do not cross the 11/5 resonance, for cut-off values below 85 ms^-1 when the family merges with the local background population (Fig. <ref>). These results are consistent with two recent classification performed by <cit.> and <cit.>, where the authors found that at the inner side of the 11/5 resonance there is another family, namely that of (96) Aegle. Moreover, a low inclination part of the initial family is completely removed (see upper-right panel of Fig. <ref>). Finally, it is interesting to note that in the a-H plane (lower-right panel of Fig. <ref>) most of the objects located below the V-shape are identified as interlopers, and the removal of these asteroids improve the visibility of the family V-shape.[The so-called V-shape is a characteristic shape of the real collisional families when projected on the semi-major axis vs. absolute magnitude (or inverse diameter) plane. It is mainly the result of the action of Yarkovsky thermal force which disperse more quickly smaller objects <cit.>. The shape structure is mostly used to estimate the age of a family <cit.>, but could also be used to search or confirm existence of asteroid families <cit.>.]For the nominal cut-off value of the Klumpkea family we adopt d_nom = 60 ms^-1,that corresponds to the removal of 9.8% of asteroids initially linked to the family.About 18% of interlopers are removed in the fourth step of our procedure, that is introduced to reduce the chaining effect. §.§ The case study #2: Eunomia family To get the initial list of family members we start from the cut-off value of 5 ms^-1 and increase it until the family merges with background asteroids (Fig. <ref>). Based on the defined criteria we adopt a cut-off value of 60 ms^-1, resulting in 11,889 potential members. The above defined Eunomia family has an average albedo p_v= 0.199 ± 0.056, and average SDSS colour a^* = 0.084 ± 0.030, suggesting that this family belongs to the S-type.In the preliminary list of members we found 1,595 interlopers (see Table <ref> for more details), whose distributions are shown in Fig. <ref>. Applying the HCM with the same d_cut as in the first step to the modified catalogue, 9,978 asteroids are identified as the family members (Figs. <ref> and <ref>). Therefore, there are 1,911 (about 16%) asteroids linked to the family less than in the initial step, with 316 of them rejected in step #4.A cut-off value of 55 ms^-1 is adopted as a nominal to define the family, resulting in removal of almost 20% of asteroids linked to the family as interlopers. <cit.> analyzed the Eunomia family and found that a large fraction of interlopers ∼ 20% should indeed be expected in this family, in agreement with the results obtained in our work. A possible explanation for such a large fraction of interlopers may be the presence of another asteroid family. Additional argument in favour of this hypothesis is concentration of interlopers in specific part of the Eunomia family (see Fig. <ref>). Examining the available data we found that C-type Adeona should be this overlapping family, as its location corresponds to the groupings of interlopers visible within the borders of the Eunomia family.Examining the V-shape structure of the Eunomia family shown in Fig. <ref>, we noted that it is a very well defined, and practically all objects positioned outside the V-shape are identified as interlopers. Moreover, even the interlopers form a visible V-shape structure, although cut at outer side in terms of the semi-major axis. This is an additional evidence that most of the interlopers belong to another collisional family, i.e. to the Adeona family.The results of this case study illustrate the efficiency of the method developed here, when dealing with two overlapping families of different spectral types. §.§ Special casesThe special cases presented in this subsection refer to groups composed of two or more overlapping collisional families. Aiming to keep our approach fully automatic and relativelysimple, obviously we do not expect to completely solve these complex cases. Moreover, as we are distinguishing here only between the C- and S- taxonomic complex, we could not resolvemore than two overlapping families. Therefore, the main purpose of this part is to demonstratehow our method works in these cases, rather than to provide a full explanation for them.§.§.§ Minerva clan The asteroids from the regions around (1) Ceres are strongly perturbed by the nearby secular resonances with Ceres <cit.>. The existence of these resonances complicates any attempt to identify asteroid families.Asteroid Minerva was first considered as a part of Ceres dynamical family <cit.>. The later works also recognized this group, but its largest associated object has changed. It was proposed that asteroid Ceres is not a member of this group[For a discussion on the missing Ceres family see <cit.>, <cit.> and <cit.>.] making (93) Minerva the largest member, and consequently the group is named Minerva family. However, later on it was realized that this family mainly consists of S-type asteroids, indicating that C-type Minerva asteroid is actually an interloper. Thus, the family name has changed once again to Gefion family. As <cit.> identified the so-called dynamical families,[Dynamical families are groupings of asteroids identified using purely dynamical characteristics, i.e. proper orbital elements. Therefore, they may not necessarily represent collisional families.] their classification includes the Minerva family. Therefore, we applied our analysis to this group, but note that asteroid (93) Minerva may not be a parent body of this group.In the initial step, applied usingcut-off of 75 ms^-1 and asteroid (93) Minerva as central object, we identified a group of 7,015 members. As this group includes more than one family (see discussion below), we refer to it as Minerva clan <cit.>. In the second step of our procedure applied to the Minerva clan we calculated the average colour: a_* = 0.044±0.032 and albedo: p_v = 0.137 ± 0.04. Both these values are somewhere in between the typical values for S and C type. While according to our criteria colour data suggest the group should be marginally classified as S-type, the albedo value falls exactly between the values used to discriminate between C and S type. This is an indication that the Minerva clan includes asteroids of different composition and possibly more than one single family. This is further supported by the fact that the Minerva clan, as identified here, includes asteroids (1) Ceres, (668) Dora and (1272) Gefion.Due to existence of asteroids of different taxonomic types, we continue in two directions; that is, we first perform analysis assuming that the Minerva clan is S-type, but then repeated this step assuming it is C-type. In the first case, we found 1,497 interlopers, objects not compatible with S-type taxonomy. Among them there are asteroids (1), (93) and (668). After removing the interlopers from the initial catalogue, we obtained the modified catalogue of proper elements. Because asteroid (93) Minerva is not present in the new catalogue, we applied the HCM analysis using (1272) Gefion as a central body. For nominal cut-off value in this case we adopted d_nom=50 ms^-1, and obtained well-defined Gefion family with 2,306 members. For the second case we assume Minerva clan is C-type and, in step #2, identified 985 asteroids as interlopers, including (1272) Gefion. Again, we exclude them from the initial catalogue, and apply the HCM using nominal cut-off of 65 ms^-1 and asteroid Minerva as a central body. To this group of 3,184 asteroids we refer here as Minerva group. Our results show that difference between the initial and the final Minerva group is 1,356 asteroids. This large fraction of interlopers could be explained by the presence of Gefion family members, which we identify as separate family composed of S-type asteroids.As shown in Fig. <ref>, the Minerva group and Gefion family, as defined here, partly overlap in the proper elements space. In order to better understand the real nature of the Minerva group, we analysed the distribution of the family members in orbital space. We note that the largest concentration in the Minerva group coincides with the location of the Gefion family. On the other hand the distribution of the dark asteroids in this region is roughly uniform (Fig. <ref>), except at the location of the Dora family that was included in the membership of the Minerva group. Based on this evidence, we concluded that asteroids belonging to the Minerva group that coincide with the location of the Gefion family, are actually the members of the latter family for which albedo and/or colour data are not available. The above mentioned facts imply that the Minerva dynamically family is not a collisional family, while nearby there is one bright (Gefion) and one dark (Dora) family. Still, caution is needed before any definite conclusion is derived about the families in this region. The secular resonances with Ceres complicate any attempt to identify asteroid families, and instead of the HCM, other techniques should be used to search for potential families, as explained in <cit.>. §.§.§ Hertha/Nysa-Polana clan The Hertha family is one of the largest families that is considered in this paper. Other paper usually referred to it as Nysa-Polana complex of asteroids. This complex was first proposed as a single family by<cit.>, but later data showed that there are two or more overlapping collisional families <cit.>. As this group is known to contain multiple families we will refer to it as Hertha clan. Following our approach, in the first step by using the cut-off value of 45ms^-1 22,851 asteroids are identified as members of the Hertha clan (Figure <ref>). According to the described criteria, the Hertha clan should be classified as S-type. Interestingly, the average colour and albedo do not show any indication that there may exist a family of different spectral type (see Table <ref>). This may be because among the objects belonging to the Hertha clan, the S-type asteroids with available spectral data are more numerous than the C-type ones. Being similar to the Minerva clan, we employed the same approach here in order to define families within the Hertha clan, i.e. we did not use only the spectral type of the clan derived inthe second step, but run the procedure twice, once assuming the family is S-, and once assuming it is C-type. In the first run we identified 1,603 C-type asteroids as interlopers, including asteroid (142) Polana. After excluding C-type interlopers from the initial catalogue of proper elements, the HCM analysis is performed using modified catalogue. For a nominal cut-off of 35ms^-19,815 asteroids are associated to the family, with asteroid (135) Hertha as its the lowest numbered member. Therefore, after excluding C-type asteroids, the Hertha family is obtained.[The second largest asteroid in the Hertha family identified in this work is (878) Mildred. <cit.> excluded asteroid (135) Hertha from its namesake family due to difference in spectrum of (135) Hertha and the rest of family members. Therefore, these authors used asteroid (878) as the parent body of the family which they called Mildred family. As our conservative criteria did not exclude asteroid (135) Hertha from the family membership, we still called it the Hertha family. ] Asteroid (44) Nysa is included in the Hertha family, but at a larger cut-off value of 45ms^-1.Interestingly, <cit.> proposed that the Hertha family consists of two sub-families, namely Hertha 1 and Hertha 2, however our methodology is not able to provide any evidence in this respect.In the second run, 2,537 S-type asteroids are identified as interlopers, and among these there are (20) Massalia, (44) Nysa and (135) Hertha asteroids. Therefore, in order to obtain C-type family within the Hertha clan, these S-type asteroids are excluded as interlopers. After doing this, asteroids (135) Hertha and (44) Nysa no longer belong to the Hertha clan. For this reason, the next largest asteroid (142) Polana is used as a central body in order to define a newfamily. The HCM at the cut-off velocity of 45ms^-1 listed 11,522 asteroids as members of the Polana family. This group seems to be even more complex, and potentially contains more than a single collisional family. Other works refer to this part of the Hertha clan as the Eulalia family <cit.> or even three separate families <cit.>. Distributions of C and S-type interlopers within the Hertha clan is shown in Fig. <ref>. The interlopers of the two spectral types are concentrated in two separate groups, revealing the presence of at least two collisional families. In Fig. <ref> it could be seen that C-type interlopers are mostly located at smaller proper eccentricities, while the S-type interlopers groups at somewhat larger values.[It should be noted that although we are calling these objects interlopers, they are likely members of one of these two overlapping families. Hence, the term interlopers is used to denote that objects for which we know that they do not belong to the C or S-type family.] In conclusion, we successfully distinguished two separate groups within the Hertha clan, namely the Hertha and Polana families. A further distinction among additional overlapping sub-families requires detailed study of these groups, and probably a somewhat different methodology. This is beyond the scope of our automatic approach, that is designed to be as simple as possible.§.§.§ The potential Levin family The existence of an asteroid family around (2076) Levin was first proposed by <cit.>. The Levin family is located in the inner part of the main belt, where both large number density of asteroids and complex dynamical environment make any attempt to identify families more difficult. Moreover, it seems that some families share this part of the orbital elements space, by overlapping each other.Applying our step one, we found that the concentration of asteroids around (2076) Levin merges with background population at 55 ms^-1. Thus, our selection criteria yields to cut-off value of 45 ms^-1. It is important to note here that the Levin family defined at 45 ms^-1, also includes the (298) Baptistina family, although the latter was treated as a separate group in <cit.>.Actually, the membership of the Baptistina family, as defined at PDS[http://sbn.psi.edu/pds/resource/nesvornyfam.html] by <cit.>, exactly matches the membership of our Levin family obtained at 45 ms^-1 m/s. A similar definition of the Baptistina family was also proposed by <cit.>. In this respect, thereis some disagreement whether the Levin and Baptistina are different families, or they belong to a single family. Therefore, it would be interesting to see how removal of potential interlopers would influence the situation.Our search for interlopers among members of the Levin family, performed within stpdf two and three, did not result in two separated families at 45 ms^-1. This might be an indication that the Levin and Baptistina belong to a single collisional family. However, the large number densityof asteroids in this region also complicates a reliable identification of interlopers, thus caution is needed. Still, a close inspection of the family V-shape caused by the Yarkovsky effect suggests that both groups may belong to the same family (see Figure <ref>).It is interesting to note that a part of the V-shape is also visible on the left side, for absolute magnitudes above about 15.5, in agreement with results by <cit.> who found that most of the objects in this magnitude range should be able to cross the 7/2 resonance with Jupiter.In summary, the Levin and Baptistina may belong to the same collisional family. Still, as described by <cit.>, it might be challenging to explain the current shape of this group in the proper elements space starting from fragments produced in a single collisional event. It should be noted however, that this complexity depends on the cut-off distance used to identified the family, and could be avoided by adopting different, typically slightly larger, cut-off distances. Therefore, although there are some reasons to believe that Levin and Baptistina belong to thesame collisional family, this question remains open. In this respect our approach of interloper removal does not help much. §.§ Other familiesIn this Section we briefly discussed some of the results obtained for other families,focusing on the definition of the most appropriate cut-off value, i.e. the so-callednominal cut-off (see Table <ref>).In most cases the nominal cut-off value is determined using the standard approach,i.e. the centre of a plateau (see Figure <ref>). For some families the methodology was somewhat modified, as discussed below.(20) Massalia: The number of asteroids associated to this family is steadily increasing with cut-off distance, until 40 ms^-1 when the family merges with a local background population, and the number of associated objects jumps dramatically (see Figure <ref>). Therefore, the nominal cut-off should be between 20 and 35 ms^-1. We adopt a cut-off of 30 ms^-1 to define the Massalia family, mainly because the V-shape is well defined in this case. The number of identified interlopers is consistent with a number of expected interlopers estimated by <cit.>. (145) Adeona: Similarly as in the case of the Massalia family, there is no clear plateau for the Adeona family in Figure <ref>, although the number of associated asteroids grows slowly from 35 to 50 ms^-1. The absence of a clear plateau for this family may be due to the close encounters with massive asteroids that significantly affected the dynamical evolution of the Adeona family <cit.>.At 55 ms^-1 the number of linked asteroids increases by factor of a few, and this is because the Adeona merges with the larger Eunomia family as discussed above. At this cut-off the family also extends on the outer side of the nearby 8/3 resonance with Jupiter, located at a=2.705AU. Examining the V-shape of Adeona family it seems that some family members should indeed be found on this side of the 8/3 resonance. However, due to proximity of the Eunomia family the nominal cut-off should be selected below 55 ms^-1, and consequently possible members at a>2.705AU are lost. We adopt the membership found at 45 ms^-1 as the nominal definition of the family.(221) Eos: This family is unique from many aspects. The albedos of family members cover a range of values somewhere in between the typical values of C- and S-type, with an average albedo of p_v = 0.13 ± 0.04. The obtained p_v falls in the ambiguity interval (see Section <ref>). Hence, the albedo confidence interval for the Eos family needs to be set manually, and we adopt the 0.1-0.25 range.In the space of proper orbital elements the Eos family is located very close to the Veritas family[According to <cit.> asteroid (490) Veritas might be an interloper in a family named after it. Therefore, the largest member of this family is (1086) Nata <cit.>.],and this complicates determination of its membership. These two families merge atcut-off of 40 ms^-1, meaning that lower distance should be used to separate the two groups. However, at 35 ms^-1 almost a half of the Eos family is missing. This is illustrated in Figure <ref>.Therefore, a kind of artificial distinction between the Eos and Veritas family is necessary in order to reasonably define the membership of both of the groups. Still, it should be noted that a smaller fraction of the Eos family, with H>15 mag and a>3.156 au, overlaps the Veritas family, suggesting that the latter may be contaminated by some members of the former group. Also, as noted by <cit.>,members of the Eos family could be found at a<2.95 au, but a too high cut-off is needed to jump across the 7/3 resonance with Jupiter that limited the family at the inner edge.§ ASTEROID FAMILIES PORTALThe large size and complexity of data acquired in the recent years create a demand for the new tools necessary to analyse this unprecedented flow of data. This is because it is often difficult to gather, manage and analyse all these data. One of the ways scientists are using to cope with these new challenges is the development of open access web-portals. Such portals, typically devoted to specific purposes, allow scientists to easily review and analyse large amount of available data.In this work we developed and launched this kind of a portal devoted to asteroid families. The portal is called Asteroid Families Portal (AFP) and is available at this link: asteroids.matf.bg.ac.rs/fam/ The screen-shot of the AFP home page is shown in Fig. <ref>.The aim of the AFP is to collect different data about asteroid families and make these freely available to all interested researchers around the world. It can be used to quickly assess and visualized data.Still, this should not be the only purpose of this portal, but it should also allow to apply different tools and methods to the provided data. For instance, the well-known Hierarchical Clustering Method (HCM) could be employed on-line to obtained the most recent list of a family members. Similarly, the method presented in this work could be used to produce list of potential interlopers among family members. A general idea of the AFP is to have two levels of the functionality for all the available tools,the automatic and the advance mode. The automatic mode is developed to produce the results using the most widely adopted settings and standards, and is mainly devoted to scientist which are not specialist in families.The advanced mode, on the other hand, offers to adjust many parameters used in the computations, and is primarily devoted to asteroid families specialist.The AFP is foreseen to be continuously updated and upgraded. The current datasets will be updated frequently by adding newly available data. Among the next tools, we foresee for instance to offer a fully automatic estimation of family age, based on the so-called V-shape method.We also aim to develop an algorithm that will provide different basic information about an asteroid family, such is the size of the parent body and its corresponding escape velocity, the slope of the magnitude-frequency distribution, etc.§ CONCLUSIONS In this work we present an automatic approach to exclude interlopers from the asteroid families. This approach combines the available data about spectral reflectance characteristics of asteroids and the iterative application of the Hierarchical Clustering Method. This algorithm shows very promisingcharacteristics, and some advantages with respect to previously used techniques. The two most important improvements with respect to previous methods are the reduction of the chaining effect, the well known draw-back of the HCM, and the fully automatic application of our method, freely available on-line at the Asteroid Families Portal.There are different possibilities to improve the presented method. Basically two directions could be followed: i) to refine the way that we are usingthe existing data or ii) to include new data. Regarding the first direction, an option definitely would be to try discriminate between different spectral types, not only between the C- and S-complex. It is known that SDSS data, as used here, does not allow to distinguish between the C- and X-type, while with the albedo data we cannot separate the S- from the X-type. However, taken together these information would allow to distinguish between the C-, S- and X-type. Another option would be to use SDSS based classification proposed by <cit.>, that makes possible even finer distinction between the asteroid spectral types.There are also other data that may be useful to identify interlopers among the asteroid families. 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"authors": [
"V. Radović",
"B. Novaković",
"V. Carruba",
"D. Marčeta"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170525152646",
"title": "An automatic approach to exclude interlopers from asteroid families"
} |
numbers, compressnatbib F OMSzplmmntheoremTheorem definitionDefinition lemmaLemma propositionProposition remarkRemark exExample assAssumptioncorollaryCorollary arg max arg min* ℙ ℚ ℱ 𝔼 frame=tb, language=Python, aboveskip=3mm, belowskip=3mm, showstringspaces=false, columns=flexible, basicstyle=, numbers=none, numberstyle=, keywordstyle=, commentstyle=, stringstyle=, breaklines=true, breakatwhitespace=true, tabsize=3 Fisher GANYoussef Mroueh^*, Tom Sercu^*, * Equal Contribution AI Foundations, IBM Research AI IBM T.J Watson Research Center December 30, 2023 =====================================================================================================================================================================================================================Generative Adversarial Networks (GANs) are powerful models for learning complex distributions. Stable training of GANs has been addressed in many recent works which explore different metrics between distributions. In this paper we introduce Fisher GAN which fits within the Integral Probability Metrics (IPM) framework fortraining GANs. Fisher GAN defines a critic with a data dependent constraint on its second order moments. We show in this paper that Fisher GAN allows for stable and time efficienttraining that does not compromise the capacity of the critic, and does not need data independent constraints such as weight clipping. We analyze our Fisher IPM theoretically and provide an algorithm based on Augmented Lagrangian for Fisher GAN. We validate our claims on both image sample generation and semi-supervised classification using Fisher GAN.§ INTRODUCTION Generative Adversarial Networks (GANs) <cit.> have recently become a prominent method to learn high-dimensional probability distributions. The basic framework consists of a generator neural network which learns to generate samples which approximate the distribution, while the discriminator measures the distance betweenthe real data distribution, and this learned distribution that is referred to as fake distribution. The generator uses the gradients from the discriminator to minimize the distance with the real data distribution. The distance between these distributionswas the object of study in <cit.>, and highlighted the impact of the distance choice on the stability of the optimization. The original GAN formulation optimizes the Jensen-Shannon divergence, while later work generalized this to optimizef-divergences <cit.>, KL <cit.>,the Least Squares objective <cit.>.Closely related to our work, Wasserstein GAN (WGAN) <cit.> uses the earth mover distance, for which the discriminator function class needs to be constrained to be Lipschitz. To impose this Lipschitz constraint, WGAN proposes to use weight clipping, i.e. a data independent constraint, but this comes at the cost of reducing the capacity of the critic and high sensitivity to the choice of the clipping hyper-parameter. A recent development Improved Wasserstein GAN (WGAN-GP) <cit.> introduced a data dependent constraint namely agradient penalty to enforce the Lipschitz constrainton the critic, which does not compromise the capacity of the critic but comes at a high computational cost.We build in this work on the Integral probability Metrics (IPM) framework for learning GAN of <cit.>. Intuitively the IPM defines a critic function f, that maximally discriminates between the real and fake distributions. We propose a theoretically sound and time efficient data dependent constraint on the critic of Wasserstein GAN, that allows a stable training of GAN and does not compromise the capacity of the critic.Where WGAN-GP uses a penalty on the gradients of the critic, Fisher GAN imposes a constraint on the second order moments of the critic. This extension to the IPM framework is inspired by the Fisher Discriminant Analysis method.The main contributions of our paper are:* We introduce in Section <ref> the Fisher IPM, a scaling invariant distance between distributions. Fisher IPM introduces a data dependent constraint on the second order moments of the critic that discriminates between the two distributions. Such a constraint ensures the boundedness of the metric and the critic. We show in Section <ref> that Fisher IPM when approximated with neural networks, corresponds to a discrepancy between whitened mean feature embeddings of the distributions. In other words a mean feature discrepancy that is measured with a Mahalanobis distance in the space computed by the neural network.* We showin Section <ref> that Fisher IPM corresponds to the Chi-squared distance (χ_2) when the critic has unlimited capacity (the critic belongs to a universal hypothesis function class).Moreover we prove in Theorem 2 that even when the critic is parametrized by a neural network, it approximates the χ_2 distance with a factor which is a inner product between optimal and neural network critic. We finally derive generalization bounds of the learned critic from samples from the two distributions, assessing the statistical error and its convergence to the Chi-squared distance from finite sample size.* We use Fisher IPM as a GAN objective [Code is available at <https://github.com/tomsercu/FisherGAN>] and formulate an algorithm that combines desirable properties (Table <ref>):a stable and meaningful loss between distributions for GAN as in Wasserstein GAN <cit.>,at a low computational cost similar to simple weight clipping,while not compromising thecapacityof the critic via a data dependent constraint but at a much lower computational cost than <cit.>.Fisher GAN achieves strong semi-supervised learning results without need of batch normalization in the critic.§ LEARNING GANS WITH FISHER IPM §.§ Fisher IPM in an arbitrary function space: General framework Integral Probability Metric (IPM).Intuitively an IPM defines a critic function f belonging to a function class ℱ, that maximally discriminates between two distributions. The function class ℱ defines how f is bounded, which is crucial to define the metric. More formally, consider a compact space X in ℝ^d. Let ℱ be a set of measurable, symmetric and bounded real valuedfunctions on X. Let 𝒫(X) be the set of measurable probability distributions on X. Given two probability distributions ℙ,ℚ∈𝒫(X), the IPM indexed by a symmetric function space ℱ is defined as follows <cit.>: -0.35ind_ℱ(ℙ,ℚ)= sup_f ∈ℱ{x∼ℙ𝔼 f(x) -x∼ℚ𝔼f(x)}. -0.1 in It is easy to see that d_ℱ defines a pseudo-metric over 𝒫(X). Note specifically that if ℱ is not bounded, sup_f will scale f to be arbitrarily large. By choosing ℱ appropriately <cit.>, various distances between probability measures can be defined. First formulation: Rayleigh Quotient. In order to define an IPM in the GAN context, <cit.> impose the boundedness of the function space via a data independent constraint. This was achieved via restricting the norms of the weights parametrizing the function space to a ℓ_p ball. Imposing such a data independent constraint makes the training highly dependent on the constraint hyper-parametersand restricts the capacity of the learned network, limiting the usability of the learned critic in asemi-supervised learning task. Here we take a different angle anddesign the IPM to be scaling invariant as a Rayleigh quotient. Instead of measuring the discrepancy between means as in Equation (<ref>), we measure astandardized discrepancy, so that the distance is bounded by construction. Standardizing this discrepancy introduces as we will see a data dependent constraint, that controls the growth of the weights of the critic f and ensures the stability of the training while maintaining the capacity of the critic.Given two distributions ℙ,ℚ∈𝒫(X) the Fisher IPM for a function space ℱ is defined as follows:d_ℱ(ℙ,ℚ)=sup_f ∈ℱx∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)]/√(1/2𝔼_x∼ℙf^2(x)+1/2𝔼_x ∼ℚ f^2(x)). While a standard IPM (Equation (<ref>)) maximizes the discrepancy between the means of a function under twodifferent distributions, Fisher IPM looks for critic f that achieves a tradeoff between maximizingthe discrepancy between the means under the two distributions (between class variance), and reducing the pooled second order moment (an upper bound on the intra-class variance). Standardized discrepancies have a long history in statistics and the so-called two-samples hypothesis testing.For example the classic two samples Student's t- test defines the student statistics as the ratio between means discrepancy and the sum of standard deviations. It is now well established that learning generative models has its roots in the two-samples hypothesis testing problem <cit.>. Non parametric two samples testing and model criticism from the kernel literature lead to the so called maximum kernel mean discrepancy (MMD) <cit.>. The MMD cost function and the mean matching IPM for a general function space has been recently used for training GAN <cit.>. Interestingly Harchaoui et al <cit.> proposed Kernel Fisher Discriminant Analysisfor the two samples hypothesis testing problem, and showed its statistical consistency. The Standard Fisher discrepancy used inLinear Discriminant Analysis (LDA) or Kernel Fisher Discriminant Analysis (KFDA) can be written: sup_f ∈ℱ( x∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)])^2/Var_x∼ℙ(f(x))+Var_x∼ℚ(f(x)) ,where Var_x∼ℙ(f(x))=𝔼_x∼ℙf^2(x)-(𝔼_x∼ℙ(f(x)))^2. Note that in LDA ℱ is restricted to linear functions, in KFDA ℱ is restricted to a Reproducing Kernel Hilbert Space (RKHS). Our Fisher IPM(Eq (<ref>)) deviates fromthe standard Fisher discrepancysince the numerator is not squared, and we use in the denominator the second order moments instead of the variances. Moreover in our definition of Fisher IPM, ℱ can be any symmetric function class.Second formulation: Constrained form. Since the distance is scaling invariant, d_ℱ can be written equivalently in the following constrained form:d_ℱ(ℙ,ℚ)=sup_f ∈ℱ, 1/2𝔼_x∼ℙf^2(x)+1/2𝔼_x ∼ℚ f^2(x)=1 ℰ(f):=x∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)]. -0.10 in Specifying ,: Learning GAN with Fisher IPM. We turn now tothe problem of learning GAN with Fisher IPM.Given a distribution ℙ_r∈𝒫(X), we learn a function g_θ: Z⊂ℝ^n_z→X, such that for z∼ p_z, the distribution of g_θ(z) is close to the real data distribution ℙ_r, where p_z is a fixed distribution on Z (for instance z∼𝒩(0,I_n_z)).Let ℙ_θ be the distribution of g_θ(z),z∼ p_z. Using Fisher IPM (Equation (<ref>)) indexed by a parametric function class ℱ_p,the generator minimizes the IPM: min_g_θ d_ℱ_p(ℙ_r,ℙ_θ). Given samples {x_i ,1… N} from ℙ_r and samples {z_i ,1… M} from p_z we shall solve the following empirical problem: -0.25 inmin_g_θsup_f_p∈ℱ_pℰ̂(f_p,g_θ):=1/N∑_i=1^N f_p(x_i) - 1/M∑_j=1^M f_p(g_θ(z_j)) Subject to Ω̂(f_p,g_θ)=1,-0.15 in where Ω̂(f_p,g_θ)= 1/2 N∑_i=1^N f^2_p(x_i) + 1/2 M∑_j=1^M f^2_p(g_θ(z_j)). For simplicity we will have M=N. §.§ Fisher IPM with Neural NetworksWe will specifically study the case where ℱ is a finite dimensional Hilbert space induced by a neural network Φ_ω (see Figure <ref> for an illustration). In this case, an IPM with data-independent constraint will be equivalent to mean matching <cit.>. We will now show that Fisher IPM will give rise to a whitened mean matching interpretation, or equivalently to mean matching with a Mahalanobis distance.Rayleigh Quotient. Consider the function space ℱ_v,ω, defined as followsℱ_v,ω={f(x)=vΦ_ω(x) | v ∈ℝ^m, Φ_ω: X→ℝ^m },Φ_ω is typically parametrized with a multi-layer neural network. We define the mean and covariance (Gramian) feature embedding of a distribution as in McGan <cit.>:μ_ω(ℙ)=x ∼ℙ𝔼( Φ_ω(x)) and Σ_ω(ℙ)=x∼ℙ𝔼(Φ_ω(x)Φ_ω(x)^⊤), Fisher IPM as defined in Equation (<ref>) on ℱ_v,ω can bewritten as follows:d_ℱ_v,ω(ℙ,ℚ)=max_ωmax_vvμ_ω(ℙ)-μ_ω(ℚ)/√(v^⊤(1/2Σ_ω(ℙ)+ 1/2Σ_ω(ℚ)+γ I_m )v),where we added a regularization term (γ>0) to avoid singularity of the covariances.Note that ifΦ_ω was implemented with homogeneous non linearitiessuch as RELU, if we swap (v,ω) with (c v, c'ω) for any constants c,c'>0, the distance d_ℱ_v,ω remains unchanged, hence the scaling invariance. Constrained Form. Since the Rayleigh Quotient is not amenable to optimization, we will consider Fisher IPM as a constrained optimization problem. By virtue of the scaling invariance and the constrained form of the Fisher IPM given inEquation (<ref>), d_ℱ_v,ω can be written equivalently as:d_ℱ_v,ω(ℙ,ℚ)=max_ω,v,v^⊤(1/2Σ_ω(ℙ)+1/2Σ_ω(ℚ)+γ I_m )v=1vμ_ω(ℙ)-μ_ω(ℚ)Define the pooled covariance: Σ_ω(ℙ;ℚ)=1/2Σ_ω(ℙ)+1/2Σ_ω(ℚ)+γ I_m. Doing a simple change of variable u=(Σ_ω(ℙ;ℚ) )^1/2 v we see that:d_ℱ_u,ω(ℙ,ℚ)= max_ωmax_u,u=1 u(Σ_ω(ℙ;ℚ))^-1/2(μ_ω(ℙ)- μ_ω(ℚ))= max_ω(Σ_ω(ℙ;ℚ) )^-1/2(μ_ω(ℙ)- μ_ω(ℚ)),hence we see that fisher IPM corresponds to the worst case distance between whitened means. Since the means are white, we don't need to impose further constraints on ω as in <cit.>. Another interpretation of the Fisher IPM stems from the fact that:d_ℱ_v,ω(ℙ,ℚ)=max_ω√((μ_ω(ℙ)-μ_ω(ℚ))^⊤Σ_ω^-1(ℙ;ℚ)(μ_ω(ℙ)-μ_ω(ℚ))),from which we see that Fisher IPM is aMahalanobis distancebetween the mean feature embeddings of the distributions. The Mahalanobis distance is defined by the positive definite matrix Σ_w(ℙ;ℚ). We show in Appendix <ref> that the gradient penalty in Improved Wasserstein <cit.> gives rise to a similarMahalanobis mean matching interpretation.Learning GAN with Fisher IPM. Hence we see that learning GAN with Fisher IPM:min_g_θmax_ωmax_v, v^⊤(1/2Σ_ω(ℙ_r)+1/2Σ_ω(ℙ_θ)+γ I_m )v=1vμ_w(ℙ_r)-μ_ω(ℙ_θ)corresponds to a min-max game between a feature space and a generator. The feature space tries to maximize the Mahalanobis distance between the feature means embeddings of real and fake distributions. The generator tries to minimize the mean embedding distance.-0.1in § THEORY-0.1inWe will start first by studying the Fisher IPMdefined in Equation (<ref>) when the function space has full capacity i.e when the critic belongs to ℒ_2(X,1/2(ℙ+ℚ)) meaning that ∫_X f^2(x)(ℙ(x)+ℚ(x))/2dx <∞. Theorem <ref> shows that under this condition, the Fisher IPM corresponds to the Chi-squared distance between distributions, and gives a closed form expression of the optimal critic function f_χ(See Appendix <ref> for its relation with the Pearson Divergence). Proofs are given in Appendix <ref>.Consider the Fisher IPM for ℱ being the space of all measurable functions endowed by 1/2(ℙ+ℚ), i.e. ℱ:=ℒ_2(X,ℙ+ℚ/2). Define the Chi-squared distance between two distributions:χ_2(ℙ,ℚ)=√(∫_X(ℙ(x)-ℚ(x))^2/ℙ(x)+ℚ(x)/2 dx)-0.1in The following holds true for any ℙ,ℚ, ℙ≠ℚ: 1) The Fisher IPM for ℱ=ℒ_2(X,ℙ+ℚ/2) is equal to the Chi-squared distance defined above: d_ℱ(ℙ,ℚ)= χ_2(ℙ,ℚ).2) The optimal critic of the Fisher IPM on ℒ_2(X,ℙ+ℚ/2) is :f_χ(x)=1/χ_2(ℙ,ℚ)ℙ(x)-ℚ(x)/ℙ(x)+ℚ(x)/2. -0.14 in We note here that LSGAN <cit.> at full capacity corresponds to a Chi-Squared divergence, with the main difference that LSGAN has different objectives for the generator and the discriminator (bilevel optimizaton), and hence does not optimize a single objective that is adistance between distributions. The Chi-squared divergence can also be achieved in the f-gan framework from <cit.>. We discuss the advantages of the Fisher formulation in Appendix <ref>.Optimizing over ℒ_2(X,ℙ+ℚ/2) is not tractable, hence we have to restrict our function class, to a hypothesis class ℋ, that enables tractable computations. Here are some typical choices of the space ℋ: Linear functions in the input features, RKHS, a non linear multilayer neural network with a linear last layer (ℱ_v,ω). In this Section we don't make any assumptions about the function space and show in Theorem <ref> how the Chi-squared distance is approximated in ℋ, and how this depends on the approximation error of the optimal critic f_χin ℋ. -0.1 inLet ℋ be an arbitrary symmetric function space.We define the inner product ff_χ_ℒ_2(X,ℙ+ℚ/2)=∫_Xf(x)f_χ(x)ℙ(x)+ℚ(x)/2 dx, which induces the Lebesgue norm. Let 𝕊_ℒ_2(X,ℙ+ℚ/2) be the unit sphere in ℒ_2(X,ℙ+ℚ/2): 𝕊_ℒ_2(X,ℙ+ℚ/2)={f:X→ℝ,f_ℒ_2(X,ℙ+ℚ/2)=1 }.The fisher IPM defined on an arbitrary function space ℋ d_ℋ(ℙ,ℚ), approximates the Chi-squared distance. The approximation quality depends on the cosine of the approximation of the optimal critic f_χ in ℋ. Since ℋ is symmetric this cosine is always positive (otherwise the same equality holds with an absolute value)d_ℋ(ℙ,ℚ)=χ_2(ℙ,ℚ)sup_f ∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)ff_χ_ℒ_2(X,ℙ+ℚ/2), -0.15 inEquivalently we have following relative approximation error: χ_2(ℙ,ℚ)- d_ℋ(ℙ,ℚ)/χ_2(ℙ,ℚ) =1/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ/2). From Theorem <ref>, we know that we have always d_ℋ(ℙ,ℚ)≤χ_2(ℙ,ℚ). Moreover if the space ℋ was rich enough to provide a good approximation of the optimal critic f_χ, then d_ℋ is a good approximation ofthe Chi-squared distance χ_2. Generalization bounds for the sample quality of the estimatedFisher IPM from samples from ℙ and ℚ can be done akin to <cit.>, with the main difficulty that for Fisher IPM we have to boundthe excess risk of a cost function with data dependent constraintson the function class. We give generalization bounds for learning the Fisher IPM in the supplementary material (Theorem 3, Appendix <ref>).In a nutshell the generalization error of the critic learned in a hypothesisclass ℋ from samples of ℙ and ℚ,decomposes to the approximation error from Theorem <ref> and a statistical error that is bounded using data dependent local Rademacher complexities<cit.> and scales like O(√(1/n)),n=MN/M+N. We illustrate in Figure <ref> our main theoretical claims on a toy problem.§ FISHER GAN ALGORITHM USING ALMFor any choice of the parametric function class ℱ_p (for example ℱ_v,ω), note the constraint in Equation (<ref>) by Ω̂(f_p,g_θ) = 1/2 N∑_i=1^N f^2_p(x_i) + 1/2 N∑_j=1^N f^2_p(g_θ(z_j)). Define the Augmented Lagrangian <cit.> corresponding to Fisher GANobjective and constraint given in Equation (<ref>): L_F(p,θ,λ)= ℰ̂(f_p,g_θ)+ λ(1-Ω̂(f_p,g_θ))-ρ/2(Ω̂(f_p,g_θ)-1)^2-0.15 in where λ is the Lagrange multiplier and ρ>0 is the quadratic penalty weight. We alternate between optimizing the critic and the generator.Similarly to <cit.> we impose the constraint when training the critic only. Given θ, for training the critic we solve max_pmin_λL_F(p,θ,λ).Then giventhe critic parameters p we optimize the generator weightsθ to minimize the objective min_θℰ̂(f_p,g_θ) . We give in Algorithm <ref>, an algorithm for Fisher GAN, note that we use ADAM <cit.> for optimizing the parameters of the critic and the generator. We use SGD for the Lagrange multiplier with learning rate ρ following practices in Augmented Lagrangian <cit.>.-0.12in§ EXPERIMENTSWe experimentally validate the proposed Fisher GAN. We claim three main results: (1) stable training with a meaningful and stable loss going down as training progresses and correlating with sample quality, similar to <cit.>. (2) very fast convergence to good sample quality as measured by inception score. (3) competitive semi-supervised learning performance, on par with literature baselines, without requiring normalization of the critic.We report results on three benchmark datasets: CIFAR-10 <cit.>, LSUN<cit.> and CelebA <cit.>. We parametrize the generator g_θ and critic f with convolutional neural networks following the model design from DCGAN <cit.>. For 64 × 64 images (LSUN, CelebA) we use the model architecture in Appendix <ref>,for CIFAR-10 we train at a 32 × 32 resolution using architecture in <ref>for experiments regarding sample quality (inception score), while for semi-supervised learning we use a better regularized discriminator similar to the Openai <cit.> and ALI <cit.> architectures, as given in <ref>. We used Adam <cit.> as optimizer for all our experiments, hyper-parameters given in Appendix <ref>.Qualitative: Loss stability and sample quality. Figure <ref> shows samples and plots during training. For LSUN we use a higher number of D updates (n_c=5) ,since we see similarly to WGAN that the loss shows large fluctuations with lower n_c values. For CIFAR-10 and CelebA we use reduced n_c=2 with no negative impact on loss stability. CIFAR-10 here was trained without any label information. We show both train and validation loss on LSUN and CIFAR-10 showing, as can be expected, no overfitting on the large LSUN dataset and some overfitting on the small CIFAR-10 dataset. To back up our claim that Fisher GAN provides stable training, we trained both a Fisher Gan and WGAN where the batch normalization in the critic f was removed (Figure <ref>). Quantitative analysis: Inception Score and Speed. It is agreed upon that evaluating generative models is hard <cit.>. We follow the literature in using “inception score” <cit.> as a metric for the quality of CIFAR-10 samples. Figure <ref> shows the inception score as a function of number of g_θ updates and wallclock time. All timings are obtained by running on a single K40 GPU on the same cluster. We see from Figure <ref>, that Fisher GAN both produces better inception scores, and has a clear speed advantage over WGAN-GP.Quantitative analysis: SSL. One of the main premises of unsupervised learning, is to learn features on a large corpus of unlabeled data in an unsupervised fashion, which are then transferable to other tasks. This provides a proper framework to measure the performance of our algorithm. This leads us to quantify the performance of Fisher GAN by semi-supervised learning (SSL) experiments on CIFAR-10. We do joint supervised and unsupervised training on CIFAR-10, by adding a cross-entropy term to the IPM objective, in conditional and unconditional generation. Unconditional Generation with CE Regularization. We parametrize the critic f as in ℱ_v,ω. While training the critic using the Fisher GAN objective L_F given in Equation (<ref>), we train a linear classifier on the feature space Φ_ω of the critic, whenever labels are available (K labels). The linear classifier is trained with Cross-Entropy (CE) minimization. Then the critic loss becomes L_D = L_F - λ_D ∑_(x,y) ∈lab CE(x,y;S,Φ_ω), whereCE(x,y; S, Φ_ω) = -log[Softmax(SΦ_ω(x))_y ], where S ∈ℝ^K× m is the linear classifier and SΦ_ω∈ℝ^K with slight abuse of notation. λ_D is the regularization hyper-parameter. We now sample three minibatches for each critic update: one labeled batch from the small labeled dataset for the CE term, and an unlabeled batch + generated batch for the IPM.Conditional Generation with CE Regularization. We also trained conditional generator models, conditioning the generator on y by concatenating the input noise with a 1-of-K embedding of the label: we now have g_θ(z,y). We parametrize the critic in ℱ_v,ω and modify the critic objective as above.We also add a cross-entropy term for the generator to minimize during its training step: L_G = ℰ̂ + λ_G ∑_z ∼ p(z), y ∼ p(y) CE(g_θ(z,y),y;S,Φ_ω). For generator updates we still need to sample only a single minibatch since we use the minibatch of samples from g_θ(z,y) to compute both the IPM loss Ê and CE. The labels are sampled according to the prior y ∼ p(y), which defaults to the discrete uniform prior when there is no class imbalance. We found λ_D=λ_G=0.1 to be optimal.New Parametrization of the Critic: “K+1 SSL”. One specific successful formulation of SSL in the standard GAN framework was provided in <cit.>, where the discriminator classifies samples into K+1 categories: the K correct clases, and K+1 for fake samples.Intuitively this puts the real classes in competition with the fake class.In order to implement this idea in the Fisher framework, we define a new function class of the critic that puts in competition the K class directions of the classifier S_y, and another “K+1” direction v that indicates fake samples. Hence we propose the following parametrization for the critic: f(x) = ∑_y=1^K p(y|x)S_yΦ_ω(x) - vΦ_ω(x), where p(y|x)=Softmax(SΦ_ω(x))_y which is also optimized with Cross-Entropy. Note that this critic does not fall under the interpretation with whitened means from Section <ref>, but does fall under the general Fisher IPM framework from Section <ref>. We can use this critic with both conditional and unconditional generation in the same way as described above. In this setting we found λ_D=1.5, λ_G=0.1 to be optimal. Layerwise normalization on the critic. For most GAN formulations following DCGAN design principles, batch normalization (BN) <cit.> in the critic is an essential ingredient. From our semi-supervised learning experiments however, it appears that batch normalization gives substantially worse performance than layer normalization (LN) <cit.> or even no layerwise normalization. We attribute this to the implicit whitening Fisher GAN provides.Table <ref> shows the SSL results on CIFAR-10. We show that Fisher GAN has competitive results, on par with state of the art literature baselines. When comparing to WGAN with weight clipping, it becomes clear that we recover the lost SSL performance. Results with the K+1 critic are better across the board, proving consistently the advantage of our proposed K+1 formulation. Conditional generation does not provide gains in the setting with layer normalization or without normalization.§ CONCLUSIONWe have defined Fisher GAN, which provide a stable and fast way of training GANs. The Fisher GAN is based on a scale invariant IPM, by constraining the second order moments of the critic. We provide an interpretation as whitened (Mahalanobis) mean feature matching and χ_2 distance. We show graceful theoretical and empirical advantages of our proposed Fisher GAN. Acknowledgments. The authors thank Steven J. Rennie for many helpful discussions and Martin Arjovsky for helpful clarifications and pointers. unsrt Supplementary Materialfor Fisher GANYoussef Mroueh^*, Tom Sercu^*IBM Research AI § WGAN-GP VERSUS FISHER GAN Considerℱ_v,ω={f(x)=vΦ_w(x) , v∈ℝ^m, Φ_ω: X⊂ℝ^d→ℝ^m}LetJ_Φ_ω(x) ∈ℝ^m× d, [J_Φ_ω(x)]_i,j= ∂e_iΦ_ω(x) /∂ x_jbe the Jacobian matrix of the Φ_ω(.). It is easy to see that∇_xf(x)=J^⊤_Φ_ω(x)v ∈ℝ^d,and therefore∇_xf(x)^2=vJ_Φ_ω(x)J^⊤_Φ_ω(x) v,Note that ,J_Φ_ω(x)J^⊤_Φ_ω(x)is the so called metric tensor in information geometry (See for instance <cit.> and references there in).The gradient penalty for WGAN of <cit.> can be derived from a Rayleigh quotient principle as well, written in the constraint form:d_ℱ_v,ω(ℙ,ℚ)= sup_f ∈ℱ_v,ω, 𝔼_u∼ U[0,1]𝔼_x∼ uℙ+(1-u)ℚ∇_xf(x)^2=1𝔼_x∼ℙf(x)-𝔼_x∼ℚf(x)Using the special parametrization we can write: 𝔼_u∼ U[0,1]𝔼_x∼ uℙ+(1-u)ℚ∇_xf(x)^2= v^⊤(𝔼_u∼ U[0,1]𝔼_x∼ uℙ+(1-u)ℚJ_Φ_ω(x)J^⊤_Φ_ω(x)) vLetℳ_ω(ℙ;ℚ)= 𝔼_u∼ U[0,1]𝔼_x∼ uℙ+(1-u)ℚJ_Φ_ω(x)J^⊤_Φ_ω(x)∈ℝ^m× mis the expected Riemannianmetric tensor <cit.>. Hence we obtain:d_ℱ_v,ω(ℙ,ℚ)=max_wmax_v, v^⊤ℳ_ω(ℙ;ℚ)v=1vμ_ω(ℙ)-μ_ω(ℚ)=max_ωℳ^-1/2_ω(ℙ;ℚ)(μ_ω(ℙ))-μ_ω(ℚ)Hence Gradient penalty can be seen as well as mean matching in the metric defined by the expected metric tensor ℳ_ω.Improved WGAN <cit.> IPM can be written as follows :max_ω√((μ_ω(ℙ)-μ_ω(ℚ))^⊤ℳ_ω^-1(ℙ;ℚ)(μ_ω(ℙ)-μ_ω(ℚ)))to be contrasted with Fisher IPM:max_ω√((μ_ω(ℙ)-μ_ω(ℚ))^⊤Σ_ω^-1(ℙ;ℚ)(μ_ω(ℙ)-μ_ω(ℚ))) Both Improved WGAN are doing mean matching using different Mahalanobis distances! While improved WGAN uses anexpected metric tensor ℳ_ω to compute this distance, Fisher IPM uses a simple pooled covariance Σ_ω to compute this metric. It is clear that Fisher GAN has a computational advantage!§ CHI-SQUARED DISTANCE AND PEARSON DIVERGENCEThe definition of χ_2 distance:χ_2^2(ℙ,ℚ)=2∫_X(ℙ(x)-ℚ(x))^2/ℙ(x)+ℚ(x) dx.The χ_2Pearson divergence:χ^P_2(ℙ,ℚ)=∫_X(ℙ(x)-ℚ(x))^2/ℚ(x) dx.We have the following relation:χ^2_2(ℙ,ℚ)=1/4χ^P_2(ℙ,ℙ+ℚ/2). § FISHER GAN AND Φ-DIVERGENCE BASED GANS Since f-gan <cit.> also introduces a GAN formulation which recovers the Chi-squared divergence, we compare our approaches.Let us recall here the definition ofφ-divergence:d_φ(ℙ,ℚ)=∫_Xφ(ℙ(x)/ℚ(x))ℚ(x) dx,where φ :ℝ^+→ℝ is a convex, lower-semicontinuous function satisfying φ(1)=0. Let φ^* the Fenchel conjugate of φ:φ^*(t)=sup_u ∈ Dom_φut-φ(u)As shown in <cit.> and in <cit.>, for any function space ℱ we get the lower bound:d_φ(ℙ,ℚ)≥sup_f ∈ℱ𝔼_x∼ℙf(x)-𝔼_x∼ℚφ^*(f(x)) ,For the particular case φ(t)=(t-1)^2 and φ^*(t)=1/4t^2+t we have the Pearson χ_2 divergence:d_φ(ℙ,ℚ)=∫_X(ℙ(x)-ℚ(x))^2/ℚ(x)dx=χ^P_2(ℙ,ℚ)Hence to optimize the same cost function of Fisher GAN in the φ-GAN framework we have to consider:1/2√(χ^P_2(ℙ,ℙ+ℚ/2)),Fisher GAN gives an inequality for the symmetric Chi-squared and the φ-GAN gives a lower variational bound. i.e compare for φ-GAN:sup_f ∈ℱ𝔼_x∼ℙf(x)-𝔼_x∼ℙ+ℚ/2φ^*(f(x)) =sup_f ∈ℱ𝔼_x ∼ℙf(x)-𝔼_x∼ℙ+ℚ/2(1/4f^2(x)+f(x))= sup_f∈ℱ1/2(𝔼_x∼ℙ f(x)- 𝔼_x∼ℚf(x))-1/4𝔼_x∼ℙ+ℚ/2f^2(x)and for Fisher GAN:sup_f∈ℱ, 𝔼_x∼ℙ+ℚ/2f^2(x)=1𝔼_x∼ℙf(x)-𝔼_x∼ℚ(f(x))while equivalent at the optimum those two formulations for the symmetric Chi-squared given in Equations (<ref>), and (<ref>) have different theoretical and practical properties. On the theory side: * While the formulation in(<ref>) is aφ divergence, the formulation given by the Fisher criterium in (<ref>) is an IPM with a data dependent constraint. This is a surprising result because φ-divergences and IPM exhibit different properties and the only known non trivialφ divergence that is also an IPM with data independent function class is the total variation distance <cit.>. When we allow the function class to be dependent on the distributions, the symmetric Chi-squareddivergence (in fact general Chi-squared also) can be cast as an IPM! Hence in the context of GAN training we inherit the known stability of IPM based training for GANs. * Theorem<ref>for the Fisher criterium gives us an approximation error when we change the function from the space of measurable functions to a hypothesis class. It is not clear how tight the lower bound in the φ-divergence will be as we relax the function class.On the practical side: * Once we parametrize the critic f as a neural network with linear output activation, i.e. f(x)=vΦ_ω(x),we see that the optimization is unconstrained for the φ-divergence formulation (<ref>) and the weights updates can explode and have anunstable behavior. On the other hand in the Fisher formulation (<ref>) the data dependent constraint that is imposed slowly through the lagrange multiplier,enforces a variance control that prevents the critic from blowing up and causing instabilities in the training. Note that in the Fisher case we have three players: the critic,the generator and the lagrange multiplier. The lagrange multiplier grows slowly to enforce the constraint and to approach the Chi-squared distance as training converges.Note that the φ-divergence formulation (<ref>) can be seen as a Fisher GAN with fixed lagrange multiplier λ=1/2 that is indeed unstable in theory and in our experiments. Note that if the Neyman divergence is of interest, it can also be obtained as the following Fisher criterium:sup_f∈ℱ, 𝔼_x∼ℙf^2(x)=1𝔼_x∼ℙf(x)-𝔼_x∼ℚ(f(x)) ,this is equivalent at the optimum to:χ^N_2(ℙ,ℚ)=∫_X(ℙ(x)-ℚ(x))^2/ℙ(x) dx.Using a neural network f(x)=vΦ_ω(x), the Neyman divergence can be achieved with linear output activation and a data dependent constraint:sup_v,ω, 𝔼_x∼ℙ (vΦ_ω(x))^2=1v𝔼_x∼ℙΦ_ω(x)-𝔼_x∼ℚΦ_ω(x) To obtain the same divergence as a φ-divergence we need φ(u)=(1-u)^2/u, and φ^*(u)=2-2√(1-u) , (u<1) . Moreover exponential activation functions are used in <cit.>, which most likely renders this formulation also unstable for GAN training.§ PROOFSConsider the space of measurable functions, ℱ={f: X→ℝ, f measurable such that ∫_X f^2(x)(ℙ(x)+ℚ(x))/2dx <∞}meaning that f∈ℒ_2(X,ℙ+ℚ/2).d_ℱ(ℙ,ℚ) = sup_f ∈ℒ_2(X,ℙ+ℚ/2), f≠ 0 x∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)]/√(1/2𝔼_x∼ℙf^2(x)+1/2𝔼_x ∼ℚ f^2(x))= sup_f ∈ℒ_2(X,ℙ+ℚ/2), f_ℒ_2(X,ℙ+ℚ/2)=1 x∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)]= sup_f ∈ℒ_2(X,ℙ+ℚ/2), f_ℒ_2(X,ℙ+ℚ/2)≤1 x∼ℙ𝔼[ f(x)] - x∼ℚ𝔼[f(x)](By convexity of the cost functional inf)= sup_f ∈ℒ_2(X,ℙ+ℚ/2) inf _λ≥ 0L(f,λ),where in the last equation we wrotethe lagrangianof the Fisher IPM for this particular function class ℱ:=ℒ_2(X,ℙ+ℚ/2):L(f,λ)= ∫_X f(x) (ℙ(x)-ℚ(x)) dx + λ/2(1- 1/2 ∫_X f^2(x)(ℙ(x)+ℚ(x))dx), By convexity of the functional cost and constraints, and since f∈ℒ_2(X,ℙ+ℚ/2), we can minimize the inner loss to optimize this functional for each x∈X <cit.>. The first order conditions of optimality (KKT conditions) gives us for the optimum f_χ,λ_*:(ℙ(x)-ℚ(x)) -λ_*/2f_χ(x) (ℙ(x)+ℚ(x))) =0 , f_χ(x)= 2/λ_*ℙ(x)-ℚ(x)/ℙ(x)+ℚ(x).Using the feasibility constraint:∫_X f^2_χ(x)(ℙ(x)+ℚ(x)/2) =1, we get :∫_X4/λ^2_*(ℙ(x)-ℚ(x))^2/(ℙ(x)+ℚ(x))^2(ℙ(x)+ℚ(x)/2) =1,which gives us the expression of λ_*:λ_*=√(∫_X(ℙ(x)-ℚ(x))^2/ℙ(x)+ℚ(x)/2 dx).Hence for ℱ:=ℒ_2(X,ℙ+ℚ/2) we have:d_ℱ(ℙ,ℚ)=∫_X f_χ(x) (ℙ(x)-ℚ(x)) dx=√(∫_X(ℙ(x)-ℚ(x))^2/ℙ(x)+ℚ(x)/2 dx)=λ_*Define the following distance between two distributions:χ_2(ℙ,ℚ)= dℙ/dℙ+dℚ/2-dℚ/dℙ+dℚ/2_ℒ_2(X, ℙ+ℚ/2),We refer to this distance asthe χ_2 distance between two distributions. It is easy to see that :d_ℱ(ℙ,ℚ)= χ_2(ℙ,ℚ)and the optimal critic f_χ has the following expression:f_χ(x)=1/χ_2(ℙ,ℚ)ℙ(x)-ℚ(x)/ℙ(x)+ℚ(x)/2.Define the means difference functional ℰ:ℰ(f; ℙ,ℚ)= 𝔼_x∼ℙf(x) -𝔼_x ∼ℚf(x)Let𝕊_ℒ_2(X,ℙ+ℚ/2)={f:X→ℝ,f_ℒ_2(X,ℙ+ℚ/2)=1 }For a symmetric function class ℋ, the Fisher IPM has the following expression:d_ℋ(ℙ,ℚ) = sup_f∈ℋ, f_ℒ_2(X,ℙ+ℚ/2)=1ℰ(f;ℙ,ℚ)= sup_f ∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)ℰ(f;ℙ,ℚ).Recall that for ℋ=ℒ_2(X, ℙ+ℚ/2), the optimum χ_2(ℙ,ℚ) is achieved for :f_χ(x)=1/χ_2(ℙ,ℚ)ℙ(x)-ℚ(x)/ℙ(x)+ℚ(x)/2, ∀ x ∈X a.s.Let f∈ℋ such thatf_ℒ_2(X,ℙ+ℚ/2)=1 we have the following:ff_χ_ℒ_2(X,ℙ+ℚ/2) = ∫_Xf(x)f_χ(x)(ℙ(x)+ℚ(x))/2dx= 1/χ_2(ℙ,ℚ)∫_X f(x)(ℙ(x)-ℚ(x)) dx= ℰ(f;ℙ,ℚ)/χ_2(ℙ,ℚ). It follows that for any f ∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2) we have:ℰ(f;ℙ,ℚ)= χ_2(ℙ,ℚ) ff_χ_ℒ_2(X,ℙ+ℚ/2)In particular taking the sup over ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2) we have:d_ℋ(ℙ,ℚ)=χ_2(ℙ,ℚ)sup_f ∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)ff_χ_ℒ_2(X,ℙ+ℚ/2). note that since ℋ is symmetric all quantities are positive after taking the sup (if ℋ was not symmetric one can take the absolute values, and similar results hold with absolute values.)If ℋ is rich enough so that we find, for ε∈ (0,1), a1-ε approximation of f_χ in ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2), i.e:sup_f ∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)ff_χ_ℒ_2(X,ℙ+ℚ/2)=1-εwe have therefore that d_ℋ is a 1-ε approximation of χ_2(ℙ,ℚ):d_ℋ(ℙ,ℚ)= (1-ε) χ_2(ℙ,ℚ).Since f and f_χ are unit norminℒ_2(X,ℙ+ℚ/2) we have the following relative error:χ_2(ℙ,ℚ)- d_ℋ(ℙ,ℚ)/χ_2(ℙ,ℚ) =1/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ)/2. § THEOREM 3: GENERALIZATION BOUNDS Let ℋ be a function space of real valued functions on X. We assume that ℋ is bounded, there exists ν>0, such that f_∞≤ν. Since the second moments are bounded we can relax this assumption using Chebyshev'sinequality, we have:ℙ{x ∈X,f(x)≤ν}≤x∼ℙ+ℚ/2𝔼f^2(x)/ν^2=1/ν^2,hence we have boundedness with high probability. Define the expected mean discrepancy ℰ(.) and the second order norm Ω(.):ℰ(f)= 𝔼_x∼ℙf(x) -𝔼_x ∼ℚf(x) , Ω(f)=1/2(𝔼_x∼ℙf^2(x)+ 𝔼_x∼ℚf^2(x) )and their empirical counterparts, given N samples {x_i}^N_i=1∼ℙ,{y_i}^M_i=1∼ℚ :Ê(f)= 1/N∑_i=1^N f(x_i) -1/M∑_i=1^M f(y_i), Ω̂(f)= 1/2N∑_i=1^N f^2(x_i) +1/2M∑_i=1^M f^2(y_i), Let n=MN/M+N. Let ℙ,ℚ∈𝒫(X), ℙ≠ℚ, and let χ_2(ℙ,ℚ) be their Chi-squared distance.Let f^*∈max_f∈ℋ,Ω(f)=1ℰ(f), and f̂∈max_f∈ℋ,Ω̂(f)=1ℰ̂(f). Define the expected mean discrepancy of the optimal empirical critic f̂:d̂_ℋ(ℙ,ℚ)=ℰ(f̂)For τ>0. The following generalization bound on the estimation of the Chi-squared distance, with probability 1-12e^-τ:χ_2(ℙ,ℚ)-d̂_ℋ(ℙ,ℚ)/χ_2(ℙ,ℚ)≤1/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ/2)_ approximation error +ε_n /χ_2(ℙ,ℚ)_Statistical Error.where ε_n = c_3ℛ_M,N(f;{f∈ℋ, Ω̂(f) ≤ 1+ν^2+2η_n},S)+ c_4(1+2νλ̂) ℛ_M,N(f;{f ∈ℋ,Ω̂(f)≤ 1+ν^2/2+η_n},S)+O(1/√(n))and η_n ≥ c_1 νℛ_N,M(f; f∈ℋ,S) +c_2ν^2τ/n,λ̂ is the Lagrange multiplier, c_1,c_2,c_3,c_4 are numerical constants,and ℛ_M,Nisthe rademacher complexity:ℛ_M,N(f;ℱ,S)= E_σsup_f ∈ℱ[ ∑_i=1^N+Mσ_i Ỹ_if(X_i)|S],Ỹ=(1/N,…1/N_N,-1/M…-1/M_M),S={x_1… x_N, y_1… y_M},σ_i=± 1 with probability 1/2, that are iids.For example:ℋ={f(x)=vΦ(x), v∈ℝ^m}Note that for simplicity here we assume that the feature map is fixed Φ: X→ℝ^m, and we parametrize the class function only with v. ℛ_M,N(f;{ℋ, Ω̂(f) ≤ R},S)) ≤√(2Rd(γ)/n),where d(γ)=∑_j=1^m σ^2_j/σ^2_j+γis the effective dimension (d(γ)<<m). Hence we see that typically ε_n= O(1/√(n)).Let {x_i}_i=1^N ∼ℙ, {y_i}_i=1^M ∼ℚ. Define the following functionals:ℰ(f)= 𝔼_x∼ℙf(x) -𝔼_x ∼ℚf(x) , Ω(f)=1/2(𝔼_x∼ℙf^2(x)+ 𝔼_x∼ℚf^2(x) )and their empirical estimates:Ê(f)= 1/N∑_i=1^N f(x_i) -1/M∑_i=1^M f(y_i), Ω̂(f)= 1/2N∑_i=1^N f^2(x_i) +1/2M∑_i=1^M f^2(y_i)Define the following Lagrangians:L(f,λ)=ℰ(f)+λ/2(1-Ω(f)), L̂(f,λ)=Ê(f)+λ/2(1-Ω̂(f))Recall some definitions of the Fisher IPM:d_ℋ(ℙ,ℚ)=sup_f∈ℋinf_λ≥ 0L(f,λ) achieved at(f_*,λ_*)We assume that a saddle point for this problem exists and it is feasible. We assume also that λ̂ is positive and bounded.d_ℋ(ℙ,ℚ)=ℰ(f_*)and Ω(f_*)=1 L(f,λ_*)≤L(f_*,λ_*)≤L(f_*,λ)The fisher IPMempirical estimate is given by:d_ℋ(ℙ_N,ℚ_N)=sup_f∈ℋinf_λ≥ 0L̂(f,λ), achieved at(f̂,λ̂)hence we have: d_ℋ(ℙ_N,ℚ_N)=Ê(f̂)and Ω̂(f̂)=1.The Generalization error of the empirical critic f̂ is the expected mean discrepancy ℰ(f̂). We note d̂_ℋ(ℙ,ℚ)=ℰ(f̂), the estimated distance using the critic f̂, on out of samples: χ_2(ℙ,ℚ)-d̂_ℋ(ℙ,ℚ) =ℰ(f_χ)-ℰ(f̂)=ℰ(f_χ)-ℰ(f^*)_Approximation Error+ℰ(f^*)-ℰ(f̂)_Statistical Error Bounding the Approximation Error. By Theorem <ref> we know that:ℰ(f_χ)-ℰ(f^*)= χ_2(ℙ,ℚ)- d_ℋ(ℙ,ℚ) =χ_2(ℙ,ℚ)/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ/2).Hence we have for ℙ≠ℚ:χ_2(ℙ,ℚ)-d̂_ℋ(ℙ,ℚ)/χ_2(ℙ,ℚ)=1/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ/2)+ℰ(f^*)-ℰ(f̂)/χ_2(ℙ,ℚ)_Statistical ErrorNote that this equation tells us that the relative error depends on the approximation error of the the optimal critic f_χ, and the statistical error coming from using finite samples in approximating the distance. We note that the statistical error is divided by the Chi-squared distance, meaning that we need a bigger sample size when ℙand ℚ are close in the Chi-squared sense, in order to reduce the overall relative error. Hence we are left with bounding the statistical error using empirical processes theory. Assume ℋ is a space of bounded functions i.e f_∞≤ν.Bounding the Statistical Error. Note that we have: (i) L̂(f^*,λ̂)≤L̂(f̂,λ̂) and (ii) Ω(f^*)=1.ℰ(f^*)-ℰ(f̂) = (ℰ(f^*)- Ê(f^*) )+(Ê(f^*)+λ̂/2(1-Ω̂(f^*))_L̂(f^*,λ̂)- Ê(f̂)_L̂(f̂,λ̂) ) +( Ê(f̂)-ℰ(f̂))+λ̂/2(Ω̂(f^*)-1)≤sup_f ∈ℋ, Ω(f)≤ 1Ê(f)-ℰ(f) + sup_f ∈ℋ, Ω̂(f)≤ 1Ê(f)-ℰ(f) + λ̂/2(Ω̂(f^*)-Ω(f^*)) Using (i) and (ii)≤sup_f ∈ℋ, Ω(f)≤ 1Ê(f)-ℰ(f) + sup_f ∈ℋ, Ω̂(f)≤ 1Ê(f)-ℰ(f) + λ̂/2sup_f∈ℋ,Ω(f)≤ 1Ω̂(f)-Ω(f). Let S={x_1… x_N, y_1… y_M}. Define the following quantities: Z_1(S) = sup_f ∈ℋ,Ω(f)≤ 1Ê(f)-ℰ(f) ,Concentration of the cost on data distribution dependent constraint Z_2(S)= sup_f ∈ℋ,Ω̂(f)≤ 1Ê(f)-ℰ(f) , Concentration of the cost on an empirical data dependent constraint Z_3(S)=sup_f∈ℋ,Ω(f)≤ 1Ω̂(f)-Ω(f), λ̂ Z_3(S)is the sensitivity of the cost as the constraint set changesWe have:ℰ(f^*)-ℰ(f̂)≤Z_1(S)+Z_2(S)+λ̂ Z_3(S), Note that the sup in Z_1(S) andZ_3(S) is taken with respect to class function {f, Ω(f)=f^2_ℒ_2(X,ℙ+ℚ/2)≤ 1} hence we will bound Z_1(S), and Z_3(S) using local Rademacher complexity. InZ_2(S) the sup is taken on a data dependent function class and can be bounded with local rademacher complexity as well but needs more careful work.Bounding Z_1(S), and Z_3(S) Let Z(S)= sup_f∈ℱℰ(f)-ℰ̂(f), Assume that f_∞≤ν, for all f∈ℱ. * For any α,τ>0. Define variances var_ℙ(f), and similarly var_ℚ(f).Assume max(var_ℙ(f),var_ℚ(f))≤ r for any f∈ℱ. We have with probability 1-e^-τ : Z(S)≤(1+α) E_SZ(S)+ √(2rτ(M+N)/MN)+ 2 τν(M+N)/MN(2/3+1/α)The same result holds for :Z(S)= sup_f∈ℱℰ̂(f)-ℰ(f).* By symmetrization we have:𝔼_S Z(S)≤ 2 E_Sℛ_M,N(f;ℱ,S) where ℛ_M,Nisthe rademacher complexity:ℛ_M,N(f;ℱ,S)= E_σsup_f ∈ℱ[ ∑_i=1^N+Mσ_i Ỹ_if(X_i)|S],Ỹ=(1/N,…1/N_N,-1/M…-1/M_M), σ_i=± 1 with probability 1/2, that are iids.* We have with probability 1-e^-τ for all δ∈ (0,1):E_Sℛ_M,N(f;ℱ,S) ≤ℛ_M,N(f;ℱ,S)/1-δ+τν (M+N)/MNδ(1-δ).Let ϕ be a contraction, that is ϕ(x)-ϕ(y)≤ L x-y. Then, for every class ℱ,ℛ_M,N(f;ϕ∘ℱ,S)≤ L ℛ_M,N(f;ℱ,S),ϕ∘ℱ={ϕ∘ f, f∈ℱ}. Let n=MN/M+N. Applying Lemma <ref> for ℱ={f∈ℋ,Ω(f)≤ 1}. Since Ω(f)≤ 1, var_ℙ(f)≤Ω(f)≤1, and similarly for var_ℚ(f). Hence max(var_ℙ(f),var_ℚ(f))≤ 1. Putting all together we obtain with probability 1-2e^-τ:Z_1(S) ≤2(1+α)/1-δℛ_M,N(f;{f ∈ℋ, Ω(f) ≤ 1},S) + √(2τ/n) + 2τν/n(2/3+1/α+1+α/δ(1-δ))Now tuning to Z_3(S) applying Lemma <ref> for {f^2, f∈ℋ,Ω(f)≤ 1}. Note that Var(f^2)≤𝔼f^4 ≤Ω(f)ν^2 ≤ν^2. We have that for α>0, δ∈ (0,1) and with probability at least 1-2e^-τ:Z_3(S) ≤2(1+α)/1-δℛ_N,M(f^2;{ f∈ℋ,Ω(f)≤ 1},S)+ √(2τν^2/n)+ 2τν^2/n(2/3+1/α+ 1+α/δ(1-δ))Note that applying the contraction Lemma for ϕ(x)=x^2 (with lipchitz constant 2ν on [-ν,ν]) we have: ℛ_N,M(f^2; { f∈ℋ,Ω(f)≤ 1},S)≤ 2νℛ_N,M(f;{f∈ℋ,Ω(f)≤ 1},S),Hence we have finally:Z_3(S) ≤4(1+α)ν/1-δℛ_N,M(f; {f∈ℋ,Ω(f)≤ 1},S)+ √(2τν^2/n) + 2τν^2/n(2/3+1/α+ 1+α/δ(1-δ))Note that theof complexity of ℋ, depends also upon the distributions ℙ and ℚ, since it is defined on the intersection of ℋ and the unityball in ℒ_2(X,ℙ+ℚ/2). From Distributions to Data dependent Bounds. We study how the Ω̂(f) concentrates uniformly on ℋ. Note that in this case to apply Lemma <ref>, we use r≤𝔼(f^4)≤ν^4.We have with probability 1-2e^-τ:Ω̂(f)≤Ω(f)+ 4(1+α)ν/1-δℛ_N,M(f; f∈ℋ,S) + √(2τν^4r/n) + 2τν^2/n(2/3+1/α+ 1+α/δ(1-δ))Now using that for any α>0:2√(uv)≤α u+v/α we have for α=1/2: √(2τν^4/n)≤ν^2/2 + 4τν^2/n. For some universal constants, c_1,c_2, let:η_n ≥ c_1 νℛ_N,M(f; f∈ℋ,S) +c_2ν^2τ/n,we have therefore with probability 1-2e^-τ:Ω̂(f)≤Ω(f)+ν^2/2+η_n,note that Typically η_n=O(1/√(n)).The same inequality holds with the same probability:Ω(f)≤Ω̂(f)+ν^2/2+η_n,Note that we have now the following inclusion using Equation (<ref>): {f,f∈ℋ, Ω(f) ≤ 1 }⊂{f, f∈ℋ, Ω̂(f)≤ 1+ν^2/2+η_n }Hence:ℛ_M,N(f;{f∈ℋ, Ω(f) ≤ 1},S) ≤ℛ_M,N(f;{f ∈ℋ,Ω̂(f)≤ 1+ν^2/2+η_n},S)Hence we obtain a data dependent bound in Equations (<ref>),(<ref>) with a union bound with probability 1-6e^-τ.Bounding Z_2(S). Note that concentration inequalities don't apply to Z_2(S) since the cost function and the function class are data dependent. We need to turn the constraint to a data independent constraint i.e does not depend on the training set.For f,Ω̂(f)≤ 1, by Equation (<ref>) we have with probability 1-2e^-τ:Ω(f)≤ 1+ν^2/2+η_n,we have therefore the following inclusion with probability 1-2e^-τ:{f∈ℋ,Ω̂(f)≤ 1}⊂{f ∈ℋ,Ω(f)≤ 1+ν^2/2+η_n}Recall that:Z_2(S) =sup_f∈ℋ, Ω̂(f)≤ 1Ê(f)-ℰ(f)Hence with probability 1-2e^-τ: Z_2(S)≤Z̃_2(S)= sup_f,f∈ℋ,Ω(f)≤ 1+ν^2/2+η_nÊ(f)-ℰ(f) Applying again Lemma <ref>on Z̃_2(S)we have with probability 1-4e^-τ:Z_2(S)≤Z̃_2(S) ≤2(1+α)/1-δℛ_M,N(f;{ f∈ℋ, Ω(f) ≤ 1+ν^2/2+η_n},S) + √(2τ(1+ν^2/2+η_n)/n)+ 2τν/n(2/3+1/α+1+α/δ(1-δ)).Now reapplying the inclusion using Equation (<ref>), we get the following bound on the local rademacher complexitywith probability 1-2e^-τ:ℛ_M,N(f;{f ∈ℋ, Ω(f) ≤ 1+ν^2/2+η_n},S) ≤ℛ_M,N(f;{f ∈ℋ, Ω̂(f) ≤ 1+ν^2+2η_n},S)Hence with probability 1-6e^-τ we have:Z_2(S) ≤2(1+α)/1-δ ℛ_M,N(f;{f ∈ℋ, Ω̂(f) ≤ 1+ν^2+2η_n},S) + √(2τ(1+ν^2/2+η_n)/n)+ 2τν/n(2/3+1/α+1+α/δ(1-δ)).Putting all together. We have with probability at least 1-12e^-τ, for universal constants c_1,c_2,c_3,c_4η_n ≥ c_1 νℛ_N,M(f; f∈ℋ,S) +c_2ν^2τ/n,ℰ(f^*)-ℰ(f̂) ≤Z_1(S)+Z_2(S)+λ̂ Z_3(S)≤ε_n= c_3ℛ_M,N(f;{f∈ℋ, Ω̂(f) ≤ 1+ν^2+2η_n},S)+ c_4(1+2νλ̂) ℛ_M,N(f;{f ∈ℋ,Ω̂(f)≤ 1+ν^2/2+η_n},S) +O(1/√(n)).Note that typically ε_n= O(1/√(n)). Hence it follows that:χ_2(ℙ,ℚ)-d̂_ℋ(ℙ,ℚ)/χ_2(ℙ,ℚ)≤1/2inf_f∈ℋ∩ 𝕊_ℒ_2(X,ℙ+ℚ/2)f-f_χ^2_ℒ_2(X,ℙ+ℚ/2)_ approximation error +ε_n /χ_2(ℙ,ℚ)_Statistical Error.If ℙ and ℚ are close we need more samples to estimate the χ_2 distance and reduce the relative error.Example: Bounding local complexity for a simple linear function class.ℋ={f(x)=vΦ(x), v∈ℝ^m}Note that for simplicity here we assume that the feature map is fixed Φ: X→ℝ^m, and we parametrize the class function only with v. Note that sup_v, v^⊤(Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)v≤ 2R v∑_i=1^N σ_iỸ_i Φ(X_i)= sup_v,v≤ 1v (Σ(ℙ_N)+Σ(ℚ_M)+γ I_m/2R)^-1/2∑_i=1^N σ_iỸ_i Φ(X_i) = (Σ(ℙ_N)+Σ(ℚ_M)+γ I_m/2R)^-1/2∑_i=1^N+Mσ_iỸ_i Φ(X_i) = √(2R)√(∑_i,j=1^N+Mσ_i σ_j Ỹ_i Ỹ_jΦ(X_i)^⊤(Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1Φ(X_j))It follows by Jensen inequality that 𝔼_σsup_v, v^⊤(Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)v ≤√(2R)v∑_i=1^N σ_iỸ_i Φ(X_i) ≤√(2R)√(𝔼_σ∑_i,j=1^N+Mσ_i σ_j Ỹ_iỸ_j Φ(X_i)^⊤(Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1Φ(X_j))= √(2R)√(∑_i=1^N+MỸ^2_i Φ(X_i)^⊤(Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1Φ(X_i))= √(2R)√(Tr((Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1(1/NΣ(ℙ_N)+1/MΣ(ℚ_M) )))≤√(2R M+N/MN)√(Tr((Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1( Σ(ℙ_N)+ Σ(ℚ_M) ))) Letd(γ)= Tr((Σ(ℙ_N)+Σ(ℚ_M)+γ I_m)^-1( Σ(ℙ_N)+ Σ(ℚ_M) )), d(γ) is the so called effective dimension in regression problems.Let Σ be the singular values of Σ(ℙ_N)+Σ(ℚ_M),d(γ)=∑_j=1^m σ^2_j/σ^2_j+γ Hence we obtain the following bound on the local rademacher complexity: ℛ_M,N(f;{ℋ, Ω̂(f) ≤ R},S)) ≤√(2R (M+N) d(γ)/MN)= √(2Rd(γ)/n) Note that without the local constraint the effective dimension d(γ) (typically d(γ)<< m) is replaced by the ambient dimension m.§ HYPER-PARAMETERS AND ARCHITECTURES OF DISCRIMINATOR AND GENERATORS For CIFAR-10 we use adam learning rate η=2e-4, β_1=0.5 and β_2=0.999, and penalty weight ρ=3e-7, for LSUN and CelebA we use η=5e-4, β_1=0.5 and β_2=0.999, and ρ=1e-6. We found the optimization to be stable with very similar performance in the range η∈[1e-4 , 1e-3] andρ∈[1e-7 , 1e-5] across our experiments. We found weight initialization from a normal distribution with stdev=0.02 to perform better than Glorot <cit.> or He <cit.> initialization for both Fisher GAN and WGAN-GP. This initialization is the default in pytorch, while in the WGAN-GP codebase He init <cit.> is used. Specifically the initialization of the generator is more important.We used some L2 weight decay: 1e-6 on ω (i.e. all layers except last) and 1e-3 weight decay on the last layer v.§.§ Inception score WGAN-GP baselines: comparison of architecture and weight initialization As noted in Figure <ref> and in above paragraph, we used intialization from a normal distribution with stdev=0.02 for the inception score experiments for both Fisher GAN and WGAN-GP. For transparency, and to show that our architecture and initialization benefits both Fisher GAN and WGAN-GP, we provide plots of different combinations below (Figure <ref>).Architecture-wise, F64 refers to the architecture described in Appendix <ref> with 64 feature maps after the first convolutional layer. F128 is the architecture from the WGAN-GP codebase <cit.>, which has double the number of feature maps (128 fmaps) and does not have the two extra layers in G and D (D layers 2-7, G layers 9-14). The result reported in the WGAN-GP paper <cit.> corresponds to . For WGAN (Figure <ref>) the 64-fmap architecture gives some initial instability but catches up to the same level as the 128-fmap architecture. §.§ LSUN and CelebA.### LSUN and CelebA: 64x64 dcgan with G_extra_layers=2 and D_extra_layers=0 G ( (main): Sequential ( (0): ConvTranspose2d(100, 512, kernel_size=(4, 4), stride=(1, 1), bias=False) (1): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True) (2): ReLU (inplace) (3): ConvTranspose2d(512, 256, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (4): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True) (5): ReLU (inplace) (6): ConvTranspose2d(256, 128, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (7): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True) (8): ReLU (inplace) (9): ConvTranspose2d(128, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (10): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (11): ReLU (inplace) (12): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (13): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (14): ReLU (inplace) (15): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (16): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (17): ReLU (inplace) (18): ConvTranspose2d(64, 3, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (19): Tanh () ) ) D ( (main): Sequential ( (0): Conv2d(3, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (1): LeakyReLU (0.2, inplace) (2): Conv2d(64, 128, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (3): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True) (4): LeakyReLU (0.2, inplace) (5): Conv2d(128, 256, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (6): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True) (7): LeakyReLU (0.2, inplace) (8): Conv2d(256, 512, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (9): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True) (10): LeakyReLU (0.2, inplace) ) (V): Linear (8192 -> 1) )§.§ CIFAR-10: Sample Quality and Inceptions Scores Experiments ### CIFAR-10: 32x32 dcgan with G_extra_layers=2 and D_extra_layers=2. For samples and inception. G ( (main): Sequential ( (0): ConvTranspose2d(100, 256, kernel_size=(4, 4), stride=(1, 1), bias=False) (1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True) (2): ReLU (inplace) (3): ConvTranspose2d(256, 128, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (4): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True) (5): ReLU (inplace) (6): ConvTranspose2d(128, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (7): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (8): ReLU (inplace) (9): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (10): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (11): ReLU (inplace) (12): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (13): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (14): ReLU (inplace) (15): ConvTranspose2d(64, 3, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (16): Tanh () ) ) D ( (main): Sequential ( (0): Conv2d(3, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (1): LeakyReLU (0.2, inplace) (2): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (3): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (4): LeakyReLU (0.2, inplace) (5): Conv2d(64, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (6): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (7): LeakyReLU (0.2, inplace) (8): Conv2d(64, 128, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (9): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True) (10): LeakyReLU (0.2, inplace) (11): Conv2d(128, 256, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (12): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True) (13): LeakyReLU (0.2, inplace) ) (V): Linear (4096 -> 1) (S): Linear (6144 -> 10) )§.§ CIFAR-10: SSL Experiments ### CIFAR-10: 32x32 D is in the flavor OpenAI Improved GAN, ALI.G same as above.D ( (main): Sequential ( (0): Dropout (p = 0.2) (1): Conv2d(3, 96, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1)) (2): LeakyReLU (0.2, inplace) (3): Conv2d(96, 96, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (4): BatchNorm2d(96, eps=1e-05, momentum=0.1, affine=True) (5): LeakyReLU (0.2, inplace) (6): Conv2d(96, 96, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False) (7): BatchNorm2d(96, eps=1e-05, momentum=0.1, affine=True) (8): LeakyReLU (0.2, inplace) (9): Dropout (p = 0.5) (10): Conv2d(96, 192, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (11): BatchNorm2d(192, eps=1e-05, momentum=0.1, affine=True) (12): LeakyReLU (0.2, inplace) (13): Conv2d(192, 192, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), bias=False) (14): BatchNorm2d(192, eps=1e-05, momentum=0.1, affine=True) (15): LeakyReLU (0.2, inplace) (16): Conv2d(192, 192, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False) (17): BatchNorm2d(192, eps=1e-05, momentum=0.1, affine=True) (18): LeakyReLU (0.2, inplace) (19): Dropout (p = 0.5) (20): Conv2d(192, 384, kernel_size=(3, 3), stride=(1, 1), bias=False) (21): BatchNorm2d(384, eps=1e-05, momentum=0.1, affine=True) (22): LeakyReLU (0.2, inplace) (23): Dropout (p = 0.5) (24): Conv2d(384, 384, kernel_size=(3, 3), stride=(1, 1), bias=False) (25): BatchNorm2d(384, eps=1e-05, momentum=0.1, affine=True) (26): LeakyReLU (0.2, inplace) (27): Dropout (p = 0.5) (28): Conv2d(384, 384, kernel_size=(1, 1), stride=(1, 1), bias=False) (29): BatchNorm2d(384, eps=1e-05, momentum=0.1, affine=True) (30): LeakyReLU (0.2, inplace) (31): Dropout (p = 0.5) ) (V): Linear (6144 -> 1) (S): Linear (6144 -> 10) ) § SAMPLE IMPLEMENTATION IN PYTORCHThis minimalistic sample code is based on <https://github.com/martinarjovsky/WassersteinGAN> at commit d92c503.Some elements that could be added are: Validation loop Monitoring of weights and activations Separate weight decay for last layer v (we trained with 1e-3 weight decay on v). Adding Cross-Entropy objective and class-conditioned generator.§.§ Main loopFirst note the essential change in the critic's forward pass definition: [basicstyle=,language=Python,firstline=183,lastline=185]code/diff.txtThen the main training loop becomes:[basicstyle=,language=Python,firstline=155,lastline=226]code/main.py §.§ Full diff from referenceNote that from the arXiv source, the filecould be used in combination with .[basicstyle=,language=Python]code/diff.txt | http://arxiv.org/abs/1705.09675v3 | {
"authors": [
"Youssef Mroueh",
"Tom Sercu"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170526182224",
"title": "Fisher GAN"
} |
[email protected]@ist.ac.at IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, AustriaThe formation of vortices is usually considered to be the main mechanism of angular momentum disposal in superfluids. Recently, it was predicted that a superfluid can acquire angular momentum via an alternative, microscopic route – namely, through interaction with rotating impurities, forming so-called `angulon quasiparticles' [Phys. Rev. Lett. 114, 203001 (2015)]. The angulon instabilities correspond to transfer of a small number of angular momentum quanta from the impurity to the superfluid, as opposed to vortex instabilities, where angular momentum is quantized in units of ħ per atom. Furthermore, since conventional impurities (such as molecules) represent three-dimensional (3D) rotors, theangular momentum transferred is intrinsically 3D as well, as opposed to a merely planar rotation which is inherent to vortices.Herein we show that the angulon theory can explain the anomalous broadening of the spectroscopic lines observed for CH_3 and NH_3 molecules in superfluid helium nanodroplets, thereby providing a fingerprint of the emerging angulon instabilities in experiment.Fingerprints of angulon instabilities in the spectra of matrix-isolated molecules Mikhail Lemeshko December 30, 2023 =================================================================================One of the distinct features of the superfluid phase is the formation of vortices – topological defects carrying quantized angular momentum, which arise if the bulkof the superfluid rotates faster than some critical angular velocity <cit.>. Vortex nucleation has been considered to be the main mechanism angular momentum disposal in superfluids <cit.>.Recently, it was predicted that a superfluid can acquire angular momentum via a different, microscopic route, which takes effect in the presence of rotating impurities, such as molecules <cit.>. In particular, it was demonstrated that a rotating impurity immersed in a superfluid forms the `angulon' quasiparticle, which can be thought of as a rigid rotor dressed by a cloud of superfluid excitations carrying angular momentum <cit.>.The angulon theory was able to describe,in good agreement with experiment, renormalization of rotational constants <cit.> and laser-induced dynamics <cit.> of molecules in superfluid helium nanodroplets. One of the key predictions of the angulon theory are the so-called`angulon instabilities' <cit.> that occur at some critical value of the molecule-superfluid coupling where the angulon quasiparticle becomes unstable and one or a few quanta of angular momentum are resonantly transferred from the impurity to the superfluid. These instabilities are fundamentally different from the vortex instabilities, associated with the transfer of angular momentum quantised in units of ħ per atom of the superfluid. Furthermore, vortices can be thought of as planar rotors, i.e., the eigenstates of the L̂_z operator. Angulons, on the other hand, are the eigenstates of the total angular momentum operator, 𝐋̂^2, andtherefore the transferred angular momentum is three-dimensional.While vortex instabilities have been subject to several experimental studies in the context of superfluid helium <cit.>, ultracold quantum gases <cit.>, and superconductors <cit.>, the transfer of angular momentum to a superfluid via the angulon instabilities has not yet been observed in experiment.In this Letter we provideevidence for the emergence of the angulon instabilities in experiments on CH_3 <cit.> and NH_3 <cit.> molecules trapped in superfluid helium nanodroplets. Spectroscopy of molecules matrix-isolated in ^4He has been an active area of research during the last two decades <cit.>. In general, it is believedthat the superfluid helium environment alters the molecular rovibrational spectra only weakly, the main effect being the renormalization of the molecular moment of inertia <cit.>, which is somewhat analogous to the renormalization of effective mass of electrons propagating in crystals <cit.>. It has been shown, however, that superfluid ^4He leads to homogeneous broadening of some spectroscopic lines. While the inhomogeneous line broadening is known to arise due to the size distribution of the droplets <cit.>, the mechanisms of the homogeneous broadening have been under active discussion <cit.>andtheir convincing microscopic interpretation has been wanting.Our aim here is to explain the anomalously large broadening of the ^R R_1 (1) transition, recently observed in in ν_3 rovibrational spectra of CH_3 in helium droplets <cit.>. The experimental spectrum is shown in Fig. <ref>(a) by the black solid line; Fig. <ref>(c) provides a schematic illustration of the molecular levels in the gas phase.One can see that all the spectroscopic lines are left intact by the helium environment, except the ^R R_1 (1) line, which is broadened by∼ 50 GHz <cit.> compared to the gas-phase simulation (blue dots). In Ref. <cit.> this feature was qualitatively explained by the coupling between the 2_2 and 1_1 molecular levels induced by the V_33(r) anisotropic term of the CH_3–He potential energy surface (PES)<cit.>, based on the theory of Ref. <cit.>. A similar effect was also present in earlier experiments on NH_3 <cit.>, see Fig. <ref>(b).Our goalis to provide a microscopic description of thespectra shown using the angulon theory and thereby demonstrate that the broadening is due to an angulon instability, accompanied by a resonant transfer of3ħ of angular momentum from the molecule to the superfluid. It is important to note that the angulon quasiparticle theory described below is substantially simpler – and therefore more transparent – thannumerical calculations based onMonte-Carlo algorithms <cit.>.We start by generalizing the angulon Hamiltonian, derived in Refs. <cit.> for linear-rotor molecules, to the case of symmetric tops such as CH_3 and NH_3 <cit.>: Ĥ=BĴ^2+(C-B)Ĵ^'2_z+∑_q λμω (q) b̂^†_q λμb̂_q λμ+ + α∑_q λμξ v_λξ(q) (b̂^†_q λμ [D^λ_μξ (Ω̂)+(-1)^ξD^λ_μ -ξ (Ω̂)]+h.c.)where we introduced the notation ∑_q ≡∫ dq and set ħ≡ 1. The first two terms of Eq. (<ref>) correspond to the kinetic energy of a symmetric-top impurity, with Ĵ and Ĵ'̂ the angular momentum operators acting in the laboratory and impurity frames, respectively <cit.>. B and C are rotational constants determined by the corresponding moments of inertia as B=1/2I_x'=1/2I_y' and C=1/2I_z'. The energies of the free impurity states are given by E_JK=BJ(J+1)+(C-B)K^2, and correspond to (2J+1)–fold degenerate states, | JMK ⟩. Here J is the angular momentum of the molecule, M gives its projection on the z-axis of the laboratory frame, and K gives its projection on the z'-axis of the molecular frame. The third term of the Hamiltonian represents the kinetic energy of the bosons in the superfluid, as given by the dispersion relation, ω (q). Here the boson creation and annihilation operators, b̂^†_q λμ and b̂_q λμ, are expressed in the angular momentum basis, where q=|q| labels the boson's linear momentum, λ is the angular momentum, and μ is the angular momentum projection onto the z-axis, see Ref. <cit.> for details. The last term of Eq. (<ref>) defines the interactions between the molecular impurity and the superfluid, where we have introduced an auxiliary parameter α, which for comparison with experiment will be set to α≡ 1. D^λ_μξ (Ω̂) are Wigner D-matrices, and Ω̂≡ (θ̂, ϕ̂, γ̂) are the angle operators defining the orientation of the molecular axis in the laboratory frame. It is important to note that Eq. (<ref>) becomes quantitatively accurate for a symmetric-top molecule immersed in a weakly-interacting Bose-Einstein Condensate <cit.>. It has been demonstrated, however, that one can develop a phenomenological theory based on the angulon Hamiltonian that describes rotations of molecules in superfluid ^4He in good agreement with experiment <cit.>. Here we pursue a similar route, i.e., we fix the interaction parameters, v_λξ(q), based on ab inito PES's <cit.> in such a way that the depth of the trapping potential (mean-field energy shift) for the molecule is reproduced <cit.>. Let us proceed with calculating the spectrum of a symmetric-top impurity in ^4He in the weakly-interacting regime, applicable to both CH_3 and NH_3 <cit.>. We start from constructing a variational ansatz based on single-boson excitations, analogous to that used in Ref. <cit.> for linear molecules:|ψ_LMk_0⟩= Z^1/2_LMk_0 | 0 ⟩| LMk_0 ⟩ ++∑_kqλμjmk β_λ j k (q) C^LM_jm, λμb̂^†_q λμ| 0 ⟩| jmk ⟩Here | 0 ⟩ is the vacuum of bosonic excitations, and Z^1/2_LMk_0 and β_λ j k (q) are the variational parameters obeying the normalization condition, Z_LMk_0=1-∑_qλ jk|β_λ jk(q) | ^2. The coefficient Z_LMk_0 is the so-called quasiparticle weight <cit.>, i.e., the overlap between the dressed angulon state, |ψ_LMk_0⟩, and the free molecular state, | LMk_0 ⟩| 0 ⟩.The angulon state (<ref>) is an eigenstate of the total angular momentum operators, 𝐋̂^2 and L̂_z, which correspond to good quantum numbers L and M. In the absence of external fields, the quantum number M is irrelevant and will be omitted hereafter.In addition, we introduce approximate quantum numbers j and k, describing the angular momentum of the molecule and its projection on the molecular axis z' (k = k_0 for j=L), and λ giving angular momentum of the excited boson. The idea of approximate quantum numbers in the present context is analogous to (and inspired by) Hund's cases of molecular spectroscopy <cit.>. As a result, we can label the angulon statesas L_k_0 (j_k, N_λ^λ), where N_λ gives the number of phonons in a state with angular momentum λ. The ansatz of Eq. (<ref>) restricts the possible values ofN_λ to 0 or 1.After the minimization of the energy, E = ⟨ψ_LMk_0|Ĥ|ψ_LMk_0⟩/ ⟨ψ_LMk_0|ψ_LMk_0⟩, with respect to Z^1/2 ∗_LMk_0 and β_λ j k^∗ (q), we arrive to the Dyson-like equation <cit.>:E=BL(L+1)+(C-B)k_0^2-Σ_Lk_0 (E)Here Σ_Lk_0 (E) is the angulon self-energy containing all the information about the molecule-helium interaction:Σ_Lk_0 (E)=∑_q λ j k ξξ'v_λξ (q) v_λξ' (q) /Bj(j+1) +(C-B)k^2 -E+ω (q)× × (C ^jk_LK, λξ+(-1)^ξ C^jk_LK, λ -ξ)(C^jk_LK, λξ'+(-1)^ξ' C^jk_LK, λ -ξ')In the limit of k=0, K=0, ξ=0, and ξ'=0,Eqs. (<ref>) and (<ref>) reduce to the equations derived in Ref. <cit.> for a linear molecule.Within the electric dipole approximation, the angulon excitation spectrumis given by the following expression:S^v' v_LMk_0(E)=|⟨ v' |⟨ ψ_L'M'k_0'|μ̂|ψ_LMk_0⟩| v ⟩| ^2 × × ImΣ_L'k_0'(E)/(γ_L'k_0'(E)-E)^2+[ImΣ_L'k_0'(E)]^2where γ_L'k_0'(E)=BL'(L'+1)+(C-B)k_0'^2-ReΣ_L'k_0', μ̂ is the dipole moment operator, and| v ⟩ and | v' ⟩ label the initial and final vibrational states. We assume that only one initial state, |ψ_LMk_0⟩| v ⟩, is populated and the optical transition occurs to all excited states, |ψ_L'M'k_0'⟩| v' ⟩, in accordance with the selection rules determined by the electric dipole matrix elements. As can be seen, the imaginary part of the self-energy, ImΣ_L'k_0'(E), gives the width of the spectral lines.It is important to note that the angulon Hamiltonian (<ref>) describes solely the rotational motion and does not explicitly take into account any effects related to molecular vibrations. While the vibrational corrections due to helium are relatively small <cit.>, we have included them into our model for a more accurate comparison with experiment <cit.>. Let us comparethe prediction of the angulon theorywith the experimental data of Fig. <ref>(a),(b). In order to obtain quantitative results, we need to fix the model parameters. For the ν_3vibrational band of CH_3the rotational constants are B=9.47111(2) cm^-1 and C=4.70174(3) cm^-1 <cit.>;for NH_3, B=9.76647(17) cm^-1 andC=6.23370(21) cm^-1 <cit.>.For ω (q) we substitute the empirical dispersion relation <cit.>. The couplingconstants, v_λξ(q), can be derived from the Fourier transforms of the spherical components of the PES <cit.>,v_λξ(q) = √(nq^4/π m ω(q))(1+δ_ξ 0)^-1∫ dr r^2 f_λξ(r) j_λ (r q), where m is the mass of a helium atom, j_λ (rq) are the spherical Bessel functions, and f_λξ(r) determines the components of the spherical harmonics expansion of themolecule-helium potentials <cit.>. In order to derive the simplest possible model, we take into account only the isotropic term, λ = ξ = 0, as well as the leading anisotropic term, λ = ξ = 3.It has been previously shown <cit.> that the effects of helium can be parametrized by a few characteristic properties of the molecule-helium potential, such as the PES anisotropy and the depth of its minima, which renders the fine details of the PES irrelevant. Therefore, in order to further simplify the model, we choose effective potentials characterized by the Gaussian form-factors, f_λξ(r)=u_λξ (2π)^-3/2 e^-r^2/2r_λξ^2, such that their magnitude, u_λξ, and range, r_λξ, reproduce known properties of the molecule-helium interaction. In particular, we set r_00=r_33=3.45 Å (r_00=r_33=3.22 Å), to the position of the global minimum of the CH_3-He <cit.> (NH_3-He <cit.>) PES. The magnitude of the isotropic potential, u_00= 23.2 B for CH_3 andu_00= 26.0 B for NH_3, was chosen so as to reproduce the mean-field shift (`trapping depth' or `impurity chemical potential') of 40 cm^-1, typical for small molecules dissolved in helium nanodroplets <cit.>. Finally, the anisotropy ratio, u_33/u_00=0.22 for CH_3 and u_33/u_00=0.25 for NH_3, was chosen to reproduce the ratio of the areas under the corresponding ab initio PES components <cit.>.Red lines in Fig. <ref>(a), (b) show the results of the angulon theory from Eqs. (<ref>)–(<ref>), with α≡ 1. One can see that the angulon theory is in a good agreement with experiment for allthe spectroscopic lines considered. In particular, for the broadened ^RR_1(1) line, we obtainthe linewidth of 50 GHz for CH_3 and 47 GHz for NH_3, which is close to the experimental values of 57 GHz and 50 GHz, respectively. In order to gain insight into the origin of the line broadening, let us study how the angulon spectral function <cit.>changes with the molecule-helium interaction strength. The spectral function can be obtained from Eq. (<ref>) by setting ⟨ v' |⟨ψ_L'M'k_0'|μ̂|ψ_LMk_0⟩| v ⟩≡ 1, which corresponds to neglecting all the spectroscopic selection rules. Fig. <ref>(a) shows the spectral function for the parameters of the CH_3 molecule listed above, as a function ofenergy, E/B, and the molecule-helium interaction parameter, α.The corresponding spectral function for NH_3 looks qualitatively similar.The limit of α→ 0 corresponds to the states of a free molecule, shown in Fig. <ref>(c). For finite α, however, the angulon levels develop an additional fine structure, which was discussed in detail inRefs. <cit.>. Of a particular interest is the region in the vicinity of α=1, where the so-called `angulon instability' occurs. In this region,the state with total angular momentum L=2 (which is a good quantum number) changes its composition: for α≲ 0.8, the angulon state corresponds to 2_2 (2_2, 0 ), i.e., it is dominated by the molecular state 2_2.In the region of 0.8 ≲α≲ 1.3, the L=2 angulon state crosses the phonon continuum attached to the j=1 molecular state, which results in the phonon excitation. The collective state in the instability region is 2_2 (1_1, 1^3). That is, while the total angular momentum L=2 is conserved,it is shared between the molecule and the superfluid due to the molecule-helium interactions. Effectively, the molecule finds itself in the 1_1 state, which is accompanied by a creation of one phonon with angular momentum λ=3. Fig. <ref> (b) shows thephonon density, |β_3 1 1 (q)|^2, in the vicinity of the instability, which is dominated by short-wavelength excitations with q ∼ 2.5 Å located in the `beyond the roton' region <cit.>. It is important to note that the qualitative discussion of Ref. <cit.> attributed the broadening to excitation of at least two phonons,since the splitting between the 2_2 and 1_1 states (∼ 21 cm^-1) exceeds the maximum energy of an elementary excitation in superfluid ^4He (∼ 14 cm^-1 <cit.>). The results presented above demonstrate that the broadening can be explained as a one-phonon transitionbetween two many-particle states, 2_2 (2_2, 0 ) and 2_2 (1_1, 1^3). In other words, in the presence of the superfluid, the level structure of the dressed molecule changes, and the energy conservation arguments have to be modified accordingly.Thus, we have generalized the angulon theory to the case of light symmetric-top molecules and demonstrated that angulon instabilities predicted in Refs. <cit.> have in fact been observed in the spectra of CH_3 and NH_3 immersed in superfluid helium nanodroplets. This paves the way to studying the decay of angulon quasiparticles and other microscopic mechanisms of the angular momentum transfer in experiments on quantum liquids, with possible applications to phonon quantum electrodynamics <cit.>.Furthermore, the angulon instabilities have been predicted to lead to anomalous screening of quantum impurities <cit.> as well as to the emergence of non-abelian magnetic monopoles <cit.>, which opens the door for the study of exotic physical phenomena in helium droplet experiments. Future measurements on isotopologues, such as CD_3 or ND_3, would allow the variation of themolecular rotational constants without altering the molecule-helium interactions,thereby providing an additional test of the model.It would be of great interest to perform quench experiments involving short laser pulse excitations <cit.> of CH_3 and NH_3 in helium nanodroplets aiming to observe the dynamical emergence of the angulon instability.We thank Richard Schmidt for comments on the manuscript and Gary Douberly for insightful discussions and providing the experimental data from Ref. <cit.>. This work was supported by the Austrian Science Fund (FWF), project Nr. P29902-N27 and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 665385.§ SUPPLEMENTAL MATERIAL§.§ Derivation of the angulon hamiltonianThe Hamiltonian for a rotating molecule interacting with a bath of bosons has the following structure: Ĥ= Ĥ_mol+Ĥ_bos+Ĥ_mol-bos. The present derivation for a symmetric-top impurity is analogous to the linear impurity case described in detail in Refs. <cit.>.For a symmetric-top impurity, the anisotropic molecule-helium potential is expanded in spherical harmonics as follows <cit.>:V_mol-He(r, θ, ϕ)= ∑_λξV_λξ (r)(1+δ_ξ 0)^-1 [ Y_λξ (θ, ϕ)+ Y_λξ^* (θ, ϕ)]Here (r, θ, ϕ) describe the position of the helium atom with respect to the center of mass of the molecule in the molecular (body-fixed) coordinate system. Spherical components, V_λξ(r), of the PES for the CH_3-He and NH_3-He complexes <cit.> are shown in Fig. <ref>. The pairwise interaction potential determines the explicit form for the last term of the Hamiltonian (<ref>), which contains the coupling constants v_λξ(q), defined as:v_λξ(q) = √(2nq^2 ϵ (q)/πω (q))(1+δ_ξ 0)^-1∫ dr r^2 f_λξ(r) j_λ (rq)Here ϵ (q)=q^2/2m is the kinetic energy of a boson of mass m, and j_λ (rq) are spherical Bessel functions. We choose model interaction potentials characterized by the Gaussian form-factors, f_λξ(r)=u_λξ(2π)^-3/2e^-r^2/2r_λξ^2, with parameters r_λξ corresponding to the global minimum of the CH_3–He (NH_3–He) PES. The isotropic component, u_00, was chosen such that itreproduces the mean-field shift (`trapping potential') of 40 cm^-1, typical for small molecules in helium nanodroplets <cit.>. The mean-field shift can be expressed in terms of f_00 (r) as follows <cit.>:E_mf= √(4 π) n ∫ r^2 f_00(r)drThe ratio of u_33/u_00 was fixed to satisfy the following condition:∫^∞_0 dr r^2 f_33(r) j_3 (r q)/∫^∞_0 dr r^2 f_00(r) j_0 (r q) = ∫_r_c^∞ dr r^2V_33(r) j_3 (r q)/∫_r_c^∞ dr r^2V_00(r) j_0 (r q),where V_λξ(r) are the spherical components of the ab initioPES. The cut-off distance, r_c, was set to the classical turning point for a collision at the temperature inside a helium droplet, i.e. such that V_00(r_c)=k_B × 0.4 K, with k_B the Boltzmann constant <cit.>.§.§ The Dyson equation Minimization of energy, E = ⟨ψ_LMk_0|Ĥ|ψ_LMk_0⟩/ ⟨ψ_LMk_0|ψ_LMk_0⟩, with respect to Z^1/2 ∗_LMk_0 and β_λ j k^∗ (q) leads to the Dyson-like equation <cit.>:[G^ang_Lk_0 (E)]^-1=[G^0_Lk_0 (E)]^-1-Σ_Lk_0(E)=0,whereG^0_Lk_0 (E)=1/BL(L+1)+(C-B)k_0^2-Eis the Green's function of the unperturbed molecule and G^ang_Lk_0 (E) is the angulon Green's function. The energy can be found self-consistently, as a set of solutions toEq. (<ref>) for a given total angular momentum L, which is the conserved quantity of the problem. Alternatively, one can reveal stable and meta-stable states of the system bycalculating the spectral function <cit.>:𝒜_Lk_0=Im[G^ang_Lk_0 (E+i0^+)]The spectral function (<ref>) corresponds to Eq. (<ref>) with ⟨ v' |⟨ψ_L'M'k_0'|μ̂|ψ_LMk_0⟩| v ⟩≡ 1.§.§ Matrix elements for spectroscopic transitionsWithin the electric dipole approximation, the probability of a perpendicular optical transition between twoangulon states, |ψ_LMk_0⟩| v ⟩ and |ψ_L'M'k'_0⟩| v' ⟩, is given by:I_LMk_0,v^L'M'k'_0,v'∼g'/g|⟨ v' |⟨ψ_L'M'k'_0|μ̂|ψ_LMk_0⟩| v ⟩|^2where g and g' give the degeneracies of the |ψ_LMk_0⟩| v ⟩ and |ψ_L'M'k'_0⟩| v' ⟩ states, respectively, μ̂ is the dipole moment operator. Substituting the angulon wavefunctions from Eq. (<ref>), for a perpendicular optical transition, we obtain <cit.>:I_LMk_0,v^L'M'k'_0,v'∼g'/g|⟨ v'|μ̂| v ⟩| ^2 [ Z^1/2*_LMk_0 Z^1/2_L'M'k'_0M_LMk_0^L'M'k'_0+∑_ q qλμ jmk j'm'k'β^*_λ jk (q) β_λ j'k' (q) C^LM_jm,λμ C^L'M'_j'm',λμM^j'm'k'_jmk]^2where M^jmk_j'm'k' is a rotational matrix element for the transition between the molecular states | jmk ⟩ and | j'm'k' ⟩ <cit.>.M^jmk_j'm'k'=√(2j+1/8π^2)√(2j'+1/8π^2)[ 1jj'0m-m'-1jj'0k-k' + 1jj'0m-m'1jj'0k-k' ]§.§ Corrections to the angulon energyWe corrected the energies of angulons states for the vibrational shift in He droplets, inversion splitting of NH_3 and Coriolis coupling. The shift of the vibrational frequency and ground-state inversion splitting of NH_3 were set to their empirical values: δν=ν_He-ν_gas=0.08 cm^-1 for CH_3 <cit.>,δν=-0.5 cm^-1, Δ^inv_0=0.8 cm^-1for NH_3 <cit.>. Therotation-vibration Coriolis coupling is given by the following matrix element <cit.>:⟨ v |⟨ψ_Lk_0| - 2ζ C π̂'_z Ĵ'̂_̂ẑ|ψ_Lk_0⟩| v⟩ = -2ζ C l(Z^∗_Lk_0k_0+∑_qjk|β_λ jk(q) |^2 k)where |ψ_Lk_0⟩ is the angulon state of Eq. (<ref>), ζ is the constant parametrizing the Coriolis coupling, π̂'_z is the vibrational angular momentum operator with respect to the symmetric top axis with eigenvalues l. For a given vibrational state | v ⟩, the vibrational angular momentum | l| = v,v-2,…,1 or 0. For the ground vibrational state l=0, and | l |=1 for the excited state under consideration, |ν_3=1 ⟩. We used the Coriolis constants ζ C= 0.35 cm^-1 for CH_3 <cit.>, and ζ C = 0.29 cm^-1 for NH_3 <cit.>.§.§ Comparison to experiment Table <ref> lists the spectral characteristics of the experimental lines shown in Fig. <ref>(a), (b), as well as the results of the angulon theory. One can see that the angulon theory is able to reproduce the width of the ^RR_1(1) line, which is approximately one order of magnitude broader compared to other transitions.The model tends to underestimate the line broadening by a few GHz, which we attribute to the fact that only single-phonon excitations are included into the ansatz (<ref>). Using more involved, diagrammatic approaches <cit.> to the Hamiltonian (<ref>) is expected to further improve the agreement. | http://arxiv.org/abs/1705.09220v2 | {
"authors": [
"Igor N. Cherepanov",
"Mikhail Lemeshko"
],
"categories": [
"physics.chem-ph",
"cond-mat.quant-gas",
"physics.atm-clus"
],
"primary_category": "physics.chem-ph",
"published": "20170525151756",
"title": "Fingerprints of angulon instabilities in the spectra of matrix-isolated molecules"
} |
plaintop C>c<ATLAS CMS LHC SMPhysik Department T31, Technische Universität München, James-Franck-Straß e 1, D-85748 Garching, GermanyInstitute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, Sommerfeldstraß e 16, D-52056 Aachen, GermanyInstitute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, Sommerfeldstraß e 16, D-52056 Aachen, GermanyMax-Planck-Institut für Kernphysik,Saupfercheckweg 1, D-69117 Heidelberg, GermanyTUM-HEP 1085/17TTK-17-18 Chemical equilibrium is a commonly made assumption in the freeze-out calculation of coannihilating dark matter. We explore the possible failure of this assumption and find a new conversion-driven freeze-out mechanism. Considering a representative simplified model inspiredby supersymmetry with a neutralino- and sbottom-like particle we find regions in parameter space with very small couplings accommodating the measured relic density. In this region freeze-out takes place out of chemical equilibrium and dark matter self-annihilation is thoroughly inefficient. The relic density is governed primarily by the size of the conversion terms in the Boltzmann equations. Due to the small dark matter coupling the parameter region is immune to direct detection but predicts an interesting signature of disappearing tracks or displaced vertices at the LHC. Unlike freeze-in or superWIMP scenarios, conversion-driven freeze-out is not sensitive to the initial conditions at the end of reheating. Coannihilation without chemical equilibrium Stefan Vogl=========================================== § INTRODUCTIONThe origin and the nature of dark matter (DM) in the Universe is one of the most pressing questions inparticle- and astrophysics. Despite impressive efforts to uncover its interactions with the Standard Model (SM) of particle physics in (in)direct detection and accelerator-based experiments,DM remains elusive and, so far, our understanding is essentially limited to its gravitational interactions (see e.g. <cit.>).It is therefore of utmost interest toinvestigate mechanisms for the generation of DM in the early Universe that go beyond the widely studied paradigm of thermal freeze-out, and that can point towards non-standard signatures. In this spirit we subject the well-known coannihilation scenario <cit.> to further scrutiny and investigate the importance of the commonly made assumption of chemical equilibrium (CE) between DM and the coannihilation partner. This requires solving the full set of coupled Boltzmann equations which has been performed in the context of specific supersymmetric scenarios <cit.>. Here we consider a simplified DM model and explore the break-down of CE in detail finding a new, conversion-driven solution for DM freeze-out which points towards a small interaction strength of the DM particle with the SM bath. While the smallness of the coupling renders most of the conventional signatures of DM unobservable, new opportunities for collider searches arise. In particular we find that searches for long-lived particles at the LHC are very powerful tools for testing conversion-driven freeze-out.The structure of the paper is as follows: We begin by introducing a simplified model for coannihilations in Sec. <ref>. In Sec. <ref> we present the Boltzmann equations which govern the DM freeze-out and investigate conversion-driven solutions before we confront the regions of parameter space which allow for a successful generation of DM with LHC searches in Sec. <ref>. Finally, we summarize our results and conclude. In the appendix we provide further details and results. In particular, we discuss the appearance of divergencies in the conversion rates in Appendix <ref>, describe our treatment of Sommerfeld enhancement in Appendix <ref> and justify the assumption of kinetic equilibrium by providing solutions of the full momentum-dependentBoltzmann equation in Appendix <ref>. § SIMPLIFIED MODEL FOR COANNIHILATION While the precise impact of the breakdown of CE between DM and its coannihilation partner will in general dependon the details of the considered model, the key aspects of the phenomenology can be expected to be universal.As a representative case we choose a simplified model for DM interacting with quarks.We extend the matter content of the SM minimally by a Majorana fermion χ, being a singlet under the SM gauge group, and a scalar quark-partnerq, mediating the interactions with the SM and acting as the coannihilation partner. The interactions of the new particles among themselves and with the SM are given by <cit.> ℒ_int = |D_μq|^2 - λ_χqq̅1-γ_5/2χ +h.c.,where q is a SM quark field, D_μ denotes the covariant derivative, which contains the interactions of q with the gauge bosons as determined by its quantum numbers, and λ_χ is a Yukawa coupling. Here we choose q=b and Y=-1/3. For the coupling λ_χ =1/3√(2)e/cosθ_W≈ 0.17 our simplified model makes contact with the Minimal Supersymmetric SM (MSSM) wherecan be identified with a right-handed sbottom and χ with a bino-like neutralino. However, here we will not consider the MSSM but vary λ_χ freely.[For a realization of small λ_χ in extensions of the MSSM see <cit.>.] Note that choosing a top-partner mediator instead yields similar results althoughquantitative differences arise due to the large top mass.On top of the gauge and Yukawa interactions described above a Higgs-portal interaction given by ℒ_h=λ_h h^† h ^†is also allowed. This interaction does not involve χ directly and has no impact on the conversion rates χ↔, that are responsible for establishing CE. Nevertheless, it canmodify the annihilation rate of . Since the additional contributions involving the scalar coupling compete with QCD processes, i.e.^†→ g g, they are sub-leading unless λ_h is very large. Even in this case we do not expect qualitative differences, and therefore neglect this contribution in the following.§ FREEZE-OUT WITHOUT CHEMICAL EQUILIBRIUM For coannihilation to be effective the coannihilating particles – here χ and –have to be in thermal contact through efficientconversion rates χ↔. For couplings λ_χ of the order of the electroweak coupling strength, conversion ratesare much larger than the Hubble rate H during freeze-out, guaranteeing CE, i.e. n_χ/n_χ^=n_/n_^ ,where n (n^) is the (equilibrium) number density. While CE holds the results are not sensitive to the size of the conversion rates and the Boltzmann equations can be reduced to a single equation that does not contain conversion terms <cit.>. This approach is solved in standard tools <cit.>.For smaller couplings, however, CE can break down and the full coupled set of Boltzmann equations has to be solved <cit.>. In our case:Y_χ/ x = 1/ 3 H s/ x [ σ_χχv(Y_χ^2-Y_χ^ 2)+σ_χ v(Y_χY_-Y_χ^Y_^) + Γ_χ→/s(Y_χ-Y_Y_χ^/Y_^) -Γ_/s(Y_-Y_χY_^/Y_χ^)+σ_χχ→^†v(Y_χ^2-Y_^2Y_χ^ 2/^ 2) ] Y_/ x = 1/ 3 H s/ x [ 1/2σ_^†v(Y_^2-Y_^ 2)+σ_χv(Y_χY_-Y_χ^Y_^)- Γ_χ→/s(Y_χ-Y_Y_χ^/Y_^) +Γ_/s(Y_-Y_χY_^/Y_χ^) -σ_χχ→^†v(Y_χ^2-Y_^2Y_χ^ 2/Y_^ 2) ] , where Y= n/s is the comoving number density, s the entropy density and x=m_χ/T. We take the internal degrees of freedom g_χ=2 and g_=3. Y_ represents the summed contribution of the mediator and its anti-particle. Since the cross sections are averaged over initial state degrees of freedom, this leads to the factor 1/2 in the respective Boltzmann equation,Eq. (<ref>). Equally,Γ_χ→ is understood to contain the conversion into both.Apart from the familiar annihilation and coannihilation terms displayed in the first lines of Eqs. (<ref>) and (<ref>) three additional rates for the conversion processes enter in the second lines. The first term includes all the scattering processes which convert DM in its coannihilation partner. Thescattering rate is given by Γ_χ→=2 ∑_k,l⟨σ_χ k→ lv ⟩ n_k^ ,where k,l denote SM particles. The factor of two arises from annihilation into the mediator and its anti-particle. Neglecting quantum statistical factors and assuming Boltzmann distributions the thermal average for scattering reads <cit.> σ_ij vn_i^n_j^=T g_i g_j/256π^5∫p_ijp_ab/√(s)|M|^2K_1(√(s)/T)scosθ,where K_i denotes amodified Bessel function of the second kind. Here g_i are the internal degrees of freedom of species i, p_ij and p_ab denote the absolute value of the three momentum of the initial and final state particles in the center-of-mass frame, respectively. In certain scattering processes soft- or t-channel divergences can appear in the thermal averages. We discuss these issues and how we resolve them in detail in Appendix <ref>. The next term in the second lines of Eqs. (<ref>) and (<ref>) captures the conversion induced by the decay and inverse decay of . This rate is controlled by the thermally averaged decay width Γ_. The thermally averaged decay rate is given byΓ_≡Γ1/γ = ΓK_1(/T) / K_2 (/T ),where γ is the Lorentz factor. Finally, the last term takes the scattering processes in the odd-sector into account. We include all diagrams that are allowed at tree-level and use FeynRules <cit.> and CalcHEP <cit.> togenerate the squared matrix elements |M|^2, see Tables <ref> and <ref> for a summary of the included processes. We take into account the Sommerfeld enhancement of the ^† annihilation rates as detailed in Appendix <ref>.A first naive estimate that allows us to determine the ballpark of the parameters where conversion processes become relevant follows from demanding Γ_Y_^/Y_χ^∼ H ,for temperatures relevant to the freeze-out dynamics (m_χ/T∼ 30). Usingas a representative benchmark m_χ=500GeV and a value of the mediator mass that allows for coannihilations (m_= 510 GeV) this relation indicates λ_χ∼𝒪(10^-7). The order of magnitude is largely insensitive to the precise choice of masses, as long as coannihilations can occur.For such a small couplinga clear hierarchy emerges between the different rates, see left panel of Fig. <ref>. The annihilation χχ→ SM SM and χχ→^† that are proportional to λ_χ^4 and thermally suppressed by n_χ^ are exceedingly small and cannot compete with the Hubble expansion.Even though the coannihilation rate χ→ SM SM, which scales asσ_χ v ∝λ_χ^2 g^2 (where g is a SM gauge coupling) is enhanced relative to this by many orders of magnitudeit is also negligible compared to H. In contrast, the leading contribution to ^†→ SM SM is set by the gauge interactions ofand, therefore, the rate remains comfortably larger than H untilT≈ m_χ/30.The most important annihilations, especially for very small λ_χ are theannihilations into gluons.Since the interaction rates are suppressed exponentially by the masses of external particles, it is clear that the conversion processes containing external gluons dominate over the rates containing weak scale particles.The conversion ratesare close to the Hubble rate and, for this choice of couplings, just about sufficient to make conversion processes relevant for the freeze-out. Taking these rates and solving the Boltzmann equations we find the results presented in the right-hand side of Fig. <ref>. We solve the system of coupled equations from x=1 up to x=1000.[Due to efficient annihilations, theabundance is very close to equilibrium at early times. For numerical convenience, it is sufficient to track its deviationfrom equilibrium starting from x∼ 15.] The χ abundance leaves its equilibrium value already at rather high temperatures, well before the freeze-out of a typical thermal relic or thefreeze-out. The slow decline of the χ abundance after this point isdue to the close-to-inefficient conversion terms which remove over-abundant χs. In Fig. <ref> we show the dependence of the final freeze-out density on the coupling λ_χ (red solid line). For large enough coupling, the solution coincides with the result that would be obtained when assuming CE (blue dotted line). The relic density is in this case largely set by the strength ofself-annihilation into gluons. When lowering the value of λ_χ, conversions χ↔ become less efficient and one obtains a relic density that lies above the value expected for CE.For the benchmark scenario shown in Fig. <ref>, the freeze-out density matches the value determined by Planck <cit.> for a coupling of λ_χ≈ 2.6 × 10^-7.Above we assumed that both χ andhave thermal abundances for T≫ m_χ. While this assumption is certainly well justified for , one may question the dependence on the initial condition for χ due to its small coupling to the thermal bath. We check the dependence on this assumption by varying the initial abundance at T=m_χ within the range (0-100)× Y_χ^. The evolution of the abundances for our benchmark point are shown in Fig. <ref>, for early times (x<20). We find that all trajectories converge before x≲ 5, thereby effectively removing any dependence of the final density on the initial condition at x=1. The dependence of the final freeze-out density on the initial condition is also indicatedin Fig. <ref> by the area shaded in red, and is remarkably small. Therefore, conversion-driven freeze-out is largely insensitive to details of the thermal historyprior to freeze-out and in particular to a potential production during the reheating process. Note that this feature distinguishes conversion-driven freeze-out from scenarios for which DM has an even weaker coupling such that it was never in thermal contact (e.g. freeze-in production <cit.>). Thus, while requiring a rather weak coupling, the robustness of the conventional freeze-out paradigm is preserved in the scenario considered here.As discussed before, conversions χ↔ are driven by two types of processes, decay and scattering. It turns out that quantitatively both are important for determining the freeze-out density. To illustrate the importance of scattering processes, we show the freeze-out density that would be obtained when only taking decays into account by the gray dashed line in Fig. <ref>. Furthermore, the gray shaded area indicates the dependence on initial conditions that would result neglecting scatterings. We find that scattering processes, which are active at small x, are responsible for wiping out the dependence on the initial abundance in the full solution of the coupled Boltzmann equations. While Eqs. (<ref>) and (<ref>) do not require chemical equilibrium to hold, they rely on the implicit assumption of kinetic equilibrium, i.e. the assumption that the momentum dependence of the distribution functions is proportional to a thermal distribution. For the freeze-out of weakly interacting massive particles, this assumption is typically well justified because scattering processes are enhanced compared to annihilations by a Boltzmann factor of order e^m_χ/T. However, in the present scenario, the small coupling λ_χ renders elastic scatterings of the form χ b↔χ b inefficient. Instead, the leading processes to establish kinetic equilibrium are the inelastic conversion processes discussed above. Since their rate is, by definition, comparable to the Hubble rate in the interesting regime of parameters, one may wonder whether the treatment based on integrated Boltzmann equations is justified for χ. In order to check this point, we solved the full, momentum-dependent Boltzmann equation for χ, taking the leading decay and scattering processes into account. We find that, while the distribution function can indeed deviate from the thermal distribution at intermediate times, the final relic abundance differs only mildly from the integrated treatment (below the 10% level). The main reason is that the collision operator does not depend strongly on the momentum mode, such that all modes behave in a similar way.For a detailed discussion we refer to appendix <ref>. Let us briefly comment on possible refinements. Apart from quantum statistics, also thermal effects could play a role at small x. In particular, the thermal mass for the b-quark can lead to a thermal blocking of the decay at high temperatures and for very small mass splitting. Since a consistent inclusion of this effect would require to take also further thermal processes into account, and since (hard) scatterings dominate for small x, we do not expect these corrections to significantly affect our conclusion. Additionally, bound state effects could play a role for theannihilation <cit.>.§ VIABLE PARAMETER SPACEWe will now explore the parameter space consistent with a relic density that matches the DM density measured by Planck, Ω h^2 = 0.1198± 0.0015 <cit.>. In the considered scenario, for small couplings, ^† annihilation is the only efficient annihilation channel. Hence the minimal relic density that can be obtained for a certain point in the m_χ-m_ plane is the one for a coupling λ_χ that just provides CE (but is still small enough so that χχ- and χ-annihilation is negligible). The curve for which this choice provides the right relic density defines the boundary of the valid parameter space and is shown as a black, solid curve in Fig. <ref>. Below this curve a choice of λ_χ sufficiently large to support CEwould undershoot the relic density. In this region a solution with small λ_χ exists that renders the involved conversion rates just large enough to allow for the rightportion of thermal contact betweenand χ to provide the right relic density. The value of λ_χ ranges from 10^-7 to 10^-6 (from small to large m_χ). These values lie far beyond the sensitivity of direct or indirect detection experiments.For the solutions providing the right relic density, during typical freeze-out (i.e. when T∼ m_χ/30) the conversion rates have to be on the edge of being efficient, cf. Eq. (<ref>). From this simple relation (and assuming that the decay width, Γ_, is similar in size as the other conversion rates) we can already infer that the decay length ofis of the order of 1–100 cm for a DM particle with a mass of a few hundred GeV predicting the signature of disappearing tracks or displaced vertices at the LHC.The decay length in our model is shown as the gray dotted lines in Fig. <ref>. It ranges from 25 cm to below 2.5 cm for increasing mass difference (the dependence on the absolute mass scale is more moderate).In proton collisions at the LHC pairs of s could be copiously produced. They willhadronize to form R-hadrons <cit.> which will, for the relevant decay length, either decay inside or traverse the sensitive parts of the detector.Accordingly, the signatures of displaced vertices and (disappearing) highly ionizing tracks provide promising discovery channels.Similar searches have, e.g., been performed for a gluino R-hadron (decaying into energetic jets) <cit.> or a purely electrically chargedheavy stable particle <cit.> but have not been performed for the model under consideration (see also <cit.> for a recent account on simplified DM models providing displaced vertices). However, some constraints on the model can already be derived from existing searches.Searches for detector-stable R-hadrons <cit.> can be reinterpreted for finite decay lengths by convoluting the signal efficiency with the fraction of R-hadronsthat decay after traversing the relevant parts of the detector. This reinterpretation provides limits down to a decay length of cτ≃0.1m for an R-hadron mass around 100 GeV and can be used to estimate excluded parameter regions in our model.To this end we compute the weighted fraction of R-hadronsthat decay after traversing the relevant parts of the detector in a Monte Carlo simulation as follows.For a given R-hadron in an event i this fraction isℱ^i_pass = e^-ℓ/(cτβγ) ,where ℓ=ℓ(η) is the travel distance to pass the respective part of the detector which depends on the pseudo-rapidity η while γ is the Lorentz factor according to the velocity β. We use a simple cylindrical approximation for the CMStracker[We considered the tracker-onlyand tracker+muon-system analysis of <cit.> finding the higher sensitivity for the former one.] with a radius and length of 1.1 m and 5.6 m, respectively. For the weighting we compute[ For simplicity we display the formula for one R-hadron candidate per event, for events with two candidateswe follow the prescription in <cit.> (with the replacement 𝒫^i_off→ℱ^i_pass𝒫^i_off in the respectivesum in the numerator of Eq. (<ref>)).]ℱ_pass= ∑_iℱ^i_pass𝒫^i_on𝒫^i_off/∑_i𝒫^i_on𝒫^i_off ,where 𝒫^i_on and 𝒫^i_off are the probabilities of the respective event to be triggered and pass the selection cuts, respectively, and the sum runs over all generated events. We use the tabulated probabilities 𝒫^i_on, 𝒫^i_off for lepton-like heavy stable charged particles following the prescription in <cit.> (see also <cit.> for details of the implementation of isolation criteria and validation).We expect this to be a good approximation as the selection criteria for lepton-like heavy stable charged particles and R-hadronsare identical and differences in the overall detector efficiency cancel out in Eq. (<ref>). We simulate events with MadGraph5_aMC@NLO <cit.>, performing showering and hadronization with Pythia 6 <cit.>.We use the cross section predictions from NLLFast <cit.> and rescale the signal by ℱ_pass. The 95% CL exclusionlimits are then obtained from a comparison to the respective cross section limitsfrom searches for (top-squark) R-hadrons presented in <cit.>.The results are shown in Fig. <ref>. We show limits for two models regarding the hadronization and interaction of the R-hadron with the detector material, the generic model <cit.> and Regge (charge-suppressed) model <cit.> as the red solid and blue dashed line, respectively. In addition to the results for the 8 TeV LHC we show results from an analogous reinterpretation of the preliminary results from 12.9 fb^-1 of data from the 13 TeV LHCrun <cit.>. Since the tabulated probabilities in <cit.> are only provided for 8 TeV we use these also for the analysis of the 13 TeV simulation assuming a similar detector efficiencyfor R-hadrons in both runs. The fraction of R-hadrons passing the tracker is exponentially suppressed for smalllife-times significantly weakening the respective sensitivity. However, there are twocompeting factors that nevertheless result in meaningful limits for cτ smaller than the detector size. On the one hand, for small masses the production cross section rises quickly. On the other hand, for smaller masses a larger fraction of R-hadrons is significantly boosted enhancing the travel distance in the detector. However, this (latter) effect does not significantly enhance the sensitivity as the signal efficiency for largely boosted R-hadronsdecreases rapidly (as tracks become indistinguishable from minimal ionizing tracks for β→1). Note that the above CMS analysis has been interpreted for R-hadrons containing top-squarks.As discussed in <cit.> the expected energy loss for an R-hadron containing sbottoms is smaller. This results in an efficiency around 30–40% smaller relative to the case of the stop and therefore inslightly weaker limits on the sbottom mass, see e.g. <cit.>.However, we use the above limit taking the result for the Regge model (that provides the weaker limits) as a realistic estimate of the LHC limits for the (sbottom-like) mediator in our model considering the fact that the uncertainties in the hadronizationare of similar size as the difference between the sbottom and stop case. The resulting limits from the 8 TeV <cit.> and 13 TeV <cit.> LHC data are superimposed in Fig. <ref>. For mass splittings below m_b (below gray dashed curve) the 2-body decay is not allowed and the R-hadrons can be considered detector-stable. Towards large mass splittings (smaller life-times) the limits fall off significantly providing noconstraint above ≃ 13 GeV.In addition, a large number of experimental results for a sbottom-neutralino simplified model assuming a prompt sbottom decay exist, see e.g. <cit.>. While most of these searches are not applicable to non-prompt decays, monojet searches, targeting small mass splittings, have been performedthat do not rely on the prompt decay of the mediator <cit.>. We superimpose the (stronger) limit from <cit.> that uses 3.2 fb^-1 of 13 TeV data. § CONCLUSION In this work we have considered the possibility that the common assumption of chemical equilibrium during DM freeze-out does not hold. For definiteness, we have focused on a simplified model with particle content inspired by supersymmetry, comprising a neutral Majorana fermion as the DM candidate and a colored scalar particle that mediates a coupling to bottom quarks. For small mass splitting between the mediator and the DM particle, the freeze-out is dominated by self-annihilation of the mediator. This process can be efficient enough to deplete the DM density below the observed value, thus giving rise to a portion of parameter space in which thermal freeze-out cannot account for all of the DM abundance. In this work we have demonstrated that this conclusion hinges on the assumption of chemical equilibrium, and that the freeze-out process can account for the DM density determined by Planck when relaxing this assumption. This occurs when the DM particle interacts very weakly with both the SM and the mediator, such that conversion processes have to be taken into account explicitly.We find that this opens up new regions in parameter space which lead to characteristic signatures of long-lived particles at collider experiments. R-hadron searches performed at the 8 and 13 TeV LHC runs already constrain part of the parameter space providing conversion-driven freeze-out. A dedicated search for disappearing R-hadron tracks and displaced vertices targeting decay lengths in the range 1–100 cm is expected to probe an even larger portion of the allowed parameter space. The mechanism discussed here is distinct from the freeze-in scenario <cit.>, for which the DM particle was never in thermal equilibrium, and which would require a much smaller coupling strength than considered here. On the other hand, it shares some similarities with the superWIMP scenario (see e.g. <cit.>), but also differs in various respects. In particular, the relic density is set by the interplay of conversion and annihilation processes during freeze-out, unlike for superWIMPs, where DM is produced from the late decay of a heavier state that undergoes a standard thermal freeze-out. In addition, for the mechanism considered here, the life-time of the coannihilation partner is short enough such that constraints from Big Bang nucleosynthesis are generally avoided, provided that the decay rate gives a sizable contribution to conversion. Unlike both freeze-in and superWIMP scenarios in the considered mechanism the final relic density is insensitive to the initial condition of the abundance at the end of the reheating process.We expect that conversion-driven freeze-out can be realized generically in DM models featuring strong coannihilations. If the coannihilation partner is not colored but only electrically charged, one may expect signatures related to lepton-like highly ionizing tracks. Finally, it is possible that the efficient self-annihilation of the coannihilation partner is itself driven by a new interaction beyond the SM <cit.>. In this case the mechanism described here can be relevant even if the coannihilating state is a SM singlet with macroscopic decay length, potentially leading to displaced vertex signatures.Note added: Reference <cit.>, which appeared recently, discusses a related mechanism.§ ACKNOWLEDGEMENTS We thank Michael Krämer, Björn Sarrazin, Pasquale Serpico and Wolfgang Waltenberger for helpful discussions.We acknowledge support by the German Research Foundation DFG through theresearch unit “New physics at the LHC”. § DIVERGENCES IN CONVERSION RATES Due to the inclusion of scattering processes, two issues arise in the thermal averages. Since we do not consider loop corrections to the two-body decay or 1→ 3 processes →χ b g, we cannot use them to cancel soft contributions from g ↔ b χ scatterings (of course, the γ scattering also has this problem, as noted by ^* in Table <ref>). Instead, we regularize these processes by imposing a cut on the minimal process energy of s_min=(+m_cut)^2, with fiducial valuem_cut=0.5 GeV. We checked that our numerical results are stable when varying m_cut over a wide range, see Fig. <ref>, indicating that the bulk of the scattering processes occurs at energies above the b mass. On top of this, we find that in processes of the type b̅↔χ H the b-quark in the t-channel is allowed to go on-shell for some center-of-mass energies (the affected processes are marked with ^** in Tab. <ref>). This corresponds to a double counting of the on-shell two-body decay. We choose to suppress the on-shell part by introducing a large Breit-Wigner width for the b-quark, taking Γ_b= m_b in our numerical calculations.Since this issue occurs only in processes involving weak scale particles (H,W,Z) and the scattering rate is dominated by gluons theprecise value for the width does not have an appreciableimpact on the results. § SOMMERFELD ENHANCEMENTIn the presence of light degrees of freedomnon-perturbative corrections to the annihilation rates are known to become relevant in the non-relativistic limit <cit.>. Between pairs of color charged particles the exchange of gluons generates a potential which modifies the wave function of the initial state particles and leads to a non-negligible correction of the tree-level cross section <cit.>.To leading order the effect of the QCD potential can be described by a Coulomb-like potential <cit.> V(r)≈α_s/2 r[ C_Q- C_R - C_R'] where C_R and C_R' denote the Casimir coefficients of the incoming particles while C_Q is the Casimir coefficient of the final state.For a general Coulomb-potential with V(r)= α/r the s-wave Sommerfeld correction factor S_0 is given by <cit.> S_0 =-πα/β/1-e^πα /β ,where β =v /2 and the total annihilation cross section of particles moving in this potential is given byσ_Somm = S_0 ·σ_tree.[In principle the Sommerfeld factors have to be determined separately for each partial wave. For the model considered here the total Sommerfeld effect can be approximated to good accuracy by applying the s-wave correction to the full cross section.]For final states which are exclusively in a singlet, i.e. ZZ,W^+ W^-,γγ, or an octet representation, i.e. γ g, Z g, the enhancement is given by Eq. (<ref>) with α=-4/3α_s or α=1/6 α_s, respectively. The gg final state is slightly more complicated since it can be in a singlet or octet representation.After summing over the different contributions the total Sommerfeld correction factor for this case reads <cit.> S_0→2/7S_0|_α=-4/3α_s+5/7S_0|_α=1/6 α_s . Since this channel dominates the annihilation rates by orders of magnitude, we only takethe correction for annihilation to gluons into account.§ KINETIC EQUILIBRIUMIn this section we compare solutions of the differential, momentum-dependent Boltzmann equation for χ with the integrated Boltzmann equation used in the main text. The latter relies on the assumption of kinetic equilibrium. This assumption may be questionable for the range of parameters we are interested in, because the leading interactions of χ are the conversion processes that become inefficient around the time of freeze-out. If a particle X is in kinetic equilibrium with the thermal bath of SM particles at temperature T, its distribution function is given byf_X(p,t)=f_X^(p,T)Y_X(t)/Y_X^(T) ,where f_X(p,t) is the phase-space density. We assume that due to efficient coupling to the SM, the mediatoris in kinetic equilibrium for all relevant times. When dropping the assumption of kinetic equilibrium for χ, the unintegrated Boltzmann equation for f_χ,( ∂_t -Hp∂_p)f_χ(p,t)=1/E_χC[f_χ] ,has to be solved, see e.g. <cit.>. The collision operator C determines the differential interaction rate ofeach momentum mode of χ. Introducing the notation x(t,p)=/T , q(t,p)=p/T ,the Liouville operator on the left-hand side of Eq. (<ref>)can be brought into the form( ∂_t -Hp ∂_p) =H/1-x/3/ x(x∂_x+ x/3/ xq∂_q)≈ H x ∂_x .In the last line we assumeto be constant during freeze-out which is approximatelysatisfied above temperatures of 𝒪(GeV).Here we adapt the notation of <cit.> for the effective degrees of freedomandassociated with the energy and entropy densities, respectively. §.§ Collisional operator for conversion For the purpose of comparing the integrated with the differential Boltzmann equation,we focus on the dominant interaction processes that drive the distribution function towards kinetic equilibrium.For the parameter range we are interested in, these are conversion processes,in particular (inverse) decays χ b ↔, and, for earlier times, also inelastic scatteringsχ A ↔ B. We neglect the small contributions from χχ annihilation, χ coannihilation andelastic scattering χ b ↔χ b.Note that if we would take these additional processes into account, one expects them to drive the distribution function closer toits equilibrium shape. Therefore the following analysis may be regarded as a conservative estimate of the deviations from kinetic equilibrium. With these assumptions Eq. (<ref>) becomes a linear differential equation and the remaining contributions to C[f_χ] are the (inverse) decay χ b ↔,C_12[f_χ]= 1/2∫Π_bΠ_ (2π)^4δ^4(∑ p_i) |M|^2[ f_ - f_χ f_b ] ,and the inelastic scatteringsχ A ↔ B,C_22[f_χ]= 1/2∫Π_A Π_ Π_B (2π)^4δ^4(∑ p_i) |M|^2 ×[ f_ f_B -f_χ f_A ] ,where Π_X=g_X ^3p_X/((2π)^3E_X). Apart from the different integrations, the terms in brackets differ from those in the case of the integrated Boltzmann equation.For the decay term we findf_ - f_χ f_b = f_^Y_/Y_^ - f_χ f_b^=f_b^( f_χ^Y_/Y_^-f_χ).using the relation of detailed balance for the equilibrium distributions. A similar simplification can be performed for inelastic scattering such that C factorizes intoC[f_χ]= C(q,x) (f_χ^Y_/Y_^-f_χ) E_χ.As before, we neglect quantum statistical factors in the calculation. The unintegrated Boltzmann equation (<ref>) can hence be written in the formH x∂_x f_χ(q,x)= C(q,x)(f_χ^Y_/Y_^-f_χ).This ordinary differential equation together with the boundary condition f_χ(q,x_0)=f^_χ(q,x_0)=exp( -√(q^2+x_0^2))can be solved with separation of variables and variation of constants. The result reads (cf. <cit.>)f_χ(q,x)= f^_χ(q,x)Y_/Y_^-∫_x_0^x(f^_χ(q,y)Y_(y)/Y_^(y)) / y×exp( -∫_y^xC(q,z)/z H(z) z )y.§.§ Simplifications for the numerical solution In order to trace the evolution of the number density n_χ(x)=4πg_χ m_χ^3/x^3∫q^2/(2π)^3f_χ(q,x) q ,the phase space density f_χ(q,x) has to be computed for a largenumber of momentum modes q and temperature parameters x providing a sufficiently accurate numerical approximation.This is a computationally expensive task. In the following we therefore introduce analytic simplifications of the collision operators.For the two-body decay we can find the analytic result for the collision operator, neglecting the b-quark mass,C_12=2 g_g_b |M|^2 T/16π p_χ E_χ(e^-p_min/T-e^-p_max/T) ,where|M|^2 = λ_χ^2/g_χg_b(m_^2-m_χ^2) , p_min/max = m_^2-m_χ^2/2m_χ^2(E_χ∓ p_χ)and g_χ=2, g_b=6, g_=3.The factor of 2 in the numerator of Eq. (<ref>) accounts for the two different processes when considering the mediator and its anti-particle, which is not included in g_.For the inelastic scattering we consider the process χ b ↔ g.Neglecting, again, the bottom mass we can express the collision operator asC_22 =2 g_qT/16π^2 p_χ E_χ∫_m_^2^∞(s-m_χ^2) σ(s)×(e^-p_min(s)/T-e^-p_max(s)/T)swhere σ(s) = 1/4| p_χ· p_q | ∫Π_Π_g (2π)^4δ(p_χ+p_q-p_-q_g)| M|^2and p_min/max(s) = s-m_χ^2/2m_χ^2(E_χ∓ p_χ) .Note that while C_12 is an analytic function, C_22 has to be numerically evaluated. We choose to precompute C_22 on a two-dimensional grid in x and q=p_χ/T and use an interpolation forthe numerical evaluation of Eq. (<ref>). We provide most of the discussion for the case when taking intoaccount the decay term only, which is expected to capture the main effects. We comment on the effect of scatterings at the end of this section.§.§ Iterative solution of the coupled system The solution Eq. (<ref>) of the Boltzmann equation for f_χ(q,x) requiresas an input the evolution of the mediator abundance, Y_(x). The latter can be obtained bysolving the corresponding integrated Boltzmann equation, which in turninvolves Y_χ(x), that is determined by integrating f_χ(q,x) over all momentum modes. Therefore the equations for f_χ(q,x) and Y_(x) form a coupled set of equations.Here we solve this coupled set of differential equations in an iterative process.We start with an initial “guess” for Y_(x), which we take to be the solution when assuming kinetic equilibrium (see below for a discussion of different choices).We then solve for f_χ(q, x) on a momentum-grid, and numerically compute Y_χ(x)using Eq. (<ref>) as described in the last subsection. With this solution forY_χ(x) we recalculate Y_(x) using the integrated Boltzmann equation. We subsequentlyiterate between solving for f_χ(q, x) and Y_(x), until we encounter sufficient convergence.In order to solve the differential Boltzmann equation in an acceptable time, we neglect the bottom massand chooseandto be evaluated at x=50 and constant for all times. We do not expecta strong dependence on these simplifications.The resulting evolution of the abundance Y_χ(x) for the benchmark point m_χ=500GeV, =510GeV is shown in Fig. <ref> (upper panel) as a red solid curve. We compare the result to the solution of the coupled integrated Boltzmann equation (red dotted curve) obtained under the same approximations.We adjust the coupling λ_χ=4.03×10^-7 so as to obtain the measured DM relic densityfor the solution of the coupled integrated Boltzmann equation. The lower panel of Fig. <ref> shows the ratio of the differential and integrated solutions for Y_χ(x). While the dark matter abundance differs by up to a factor of two at intermediate times, the final relic abundance agrees well with the corresponding result when assuming kinetic equilibrium, with deviations below the 10% level.The main reason is that, for the process and the kinematical situation that is relevant here, the collision term does not depend strongly on the momentum mode, see Fig. <ref> (dot-dashed lines). In the same figure, we also show the result for f_χ(q, x) at various times x (blue lines), which indeed differs from an equilibrium distribution (indicated by the red dashed lines) at intermediate times (upper and middle panel in Fig. <ref>). Nevertheless,around the time when the dark matter abundance freezes out, the remaining decays of thermalizedtend to restore an equilibrium distribution (lowest panel). It is interesting to observe that the total abundance obtained from the unintegratedBoltzmann equation is slightly below the result when assuming kinetic equilibrium. This can also be understood from Fig. <ref>.For high temperatures, the momentum modes obtained from the differentialsolution essentially change only due to redshift. In contrast, the kinetic equilibrium distribution populates somewhat higher modes.By the time the conversion gets efficient, the collision term is larger for smallermomentum modes. Therefore, the conversion into s is stronger for thedifferential solution, rendering a slightly smaller abundance.Let us now discuss the impact of the initial “guess” for Y_(x) used for the iterative solution. We check that the converged result is independent of the starting point of the iteration by using tworather different initial abundances. First, we use the equilibrium abundance Y_^(x) as a starting point.The results are shown in the left panel of Fig. <ref>.The evolution of Y_χ(x) obtained in the first iteration step is shown by the thick orange line,and the successive iterations are indicated by the thin orange lines. The final, converged result(thick red line) agrees well with the result obtained in Fig. <ref>. The same is true forY_(x) (solid blue line). On the other hand, we would like to point out that the first iteration and the converged result are rather far apart. This means that it is crucial to solve for the coupled set of equations, allowing for deviations Y_(x)≠ Y_^(x). For curiosity, we note that if one would compare the differential with the integrated result for Y_χ(x) while fixing Y_(x)= Y_^(x), one would find an O(10) difference in the final abundance (see orange dotted versus solid line inthe left panel of Fig. <ref>), while the corresponding difference for the converged results is below ∼ 10%. Hence, the partial freeze-out of the mediatorand its subsequent decay into χ are crucial for the conclusion that the impact of deviations from kinetic equilibrium on the relic density is small.Second, we consider an extreme possibility and initially set Y_χ(x) to be constant and equal to the relativistic equilibrium density. In this case we start the iteration with the computation of Y_(x). The resulting iterative solutions for Y_χ(x) are shown in right panel of Fig. <ref> (orange lines). Again, the converged result for Y_χ(x) (red solid line) and Y_(x) (solid blue line) agree well with those shown in Fig. <ref>.The convergence of the final relic density for the three different choices of initial abundances is shown in Fig. <ref>.Indeed, the converged results agree, indicating that the iterative scheme is stable and leads to a unique result. Next we want to check if the situation changes drastically when including also 2→ 2 scattering processes.Due to the increase in numerical complexity described above, we consider the leading processχ b ↔ g expected to capture the main effects. In order to estimate the physical contributions from hard scatterings, we perform regularizations on the level of the scattering cross section byintroducing a cut-off s_min=(+1 GeV)^2 and additionally a regulator at the matrixelement level of 1/t^2→ 1/(t^2+(1 GeV)^4).In addition, we restrict the integration over the angle θ_t between b and g in the center-of-mass frame to cosθ_t ∈ [-0.9,0.9]. Again, we solve the coupled system in an iterative approach as described above, but taking scatterings into account. As before, we then compare the converged result for the final relic density with the corresponding result obtained when assuming kinetic equilibrium. We find that the relative deviations in the resultingrelic densitystay below 10%. Furthermore, the deviation for Y_χ(x) for intermediate times becomes smaller. This is expected, because scatterings increase the conversion rates at smaller x.Altogether, we find that the impact of deviations from kinetic equilibrium on the final relic abundance is rather mild, below the 10% level. This justifies using integrated Boltzmann equations for Y_χ(x)and Y_(x). | http://arxiv.org/abs/1705.09292v2 | {
"authors": [
"Mathias Garny",
"Jan Heisig",
"Benedikt Lülf",
"Stefan Vogl"
],
"categories": [
"hep-ph",
"astro-ph.CO"
],
"primary_category": "hep-ph",
"published": "20170525180001",
"title": "Coannihilation without chemical equilibrium"
} |
Optimal Transport Theory for Cell Association in UAV-Enabled Cellular Networks Mohammad Mozaffari^1, Walid Saad^1, Mehdi Bennis^2, and Mérouane Debbah^3 ^1 Wireless@VT, Electrical and Computer Engineering Department, Virginia Tech, VA, USA,Emails:<mmozaff , [email protected]>. ^2 CWC - Centre for Wireless Communications, Oulu, Finland, Email: <[email protected]>. ^3 Mathematical and Algorithmic Sciences Lab, Huawei France R & D, Paris, France, and CentraleSup´elec,Universit´e Paris-Saclay, Gif-sur-Yvette, France, Email: <[email protected]>. ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ In this paper, a novel framework for delay-optimal cell association in unmanned aerial vehicle (UAV)-enabled cellular networks is proposed. In particular, to minimize the average network delay under any arbitrary spatial distribution of the ground users, the optimal cell partitions of UAVs and terrestrial base stations (BSs) are determined. To this end, using the powerful mathematical tools of optimal transport theory, the existence of the solution to the optimal cell association problem is proved and the solution space is completely characterized. The analytical and simulation results show that the proposed approach yields substantial improvements of the average network delay. § INTRODUCTION The use of unmanned aerial vehicles (UAVs) such as drones and balloons is an effective technique for improving the quality-of-service (QoS) of wireless cellular networks due to their inherent ability to create line-of-sight (LoS) communication links<cit.>. Nevertheless, there are many technical challenges associated with the UAV-based communication systems, which include deployment, path planning, flight time constraints, and cell association.In <cit.> and <cit.>, the authors studied the efficient deployment of aerial base stations to maximize the coverage performance.The path planning challenge and optimal trajectory of UAVs were addressed in <cit.> and <cit.>. Moreover, UAV communications under flight time considerations was studied in <cit.>. Another important challenge in UAV-based communications is cell (or user) association. In <cit.>, the authors analyzed the user-UAV assignment for capacity enhancement of heterogeneous networks. However, this work is limited to the case in which users are uniformly distributed within a geographical area. In <cit.>, the authors proposed a power-efficient cell association scheme while satisfying the rate requirement of users in cellular networks. However, in <cit.>, the authors do not consider the presence of UAVs and their objective function does not account for network delay. In <cit.>, the optimal deployment and cell association of UAVs are determined with the goal of minimizing the UAVs' transmit power while satisfying the users' rate requirements. However, the work in <cit.> mainly focused on the optimal deployment of the UAVs and does not analyze the existence and characterization of the cell association problem. Therefore, our work is different from <cit.> in terms of the system model, the objective function, the problem formulation as well as analytical results.In fact, none of the previous studies in <cit.>, addressed the delay-optimal cell association problem considering both UAVs and terrestrial base stations, for any arbitrary distribution of users.The main contribution of this paper is to introduce a novel framework for delay-optimal cell association in a cellular network in which both UAVs and terrestrial BSs co-exist. In particular, given the locations of the UAVs and terrestrial BSs as well as any general spatial distribution of users, we find the optimal cell association by exploiting the framework of optimal transport theory <cit.>. Within the framework of optimal transport theory, one can address cell association problems for any general spatial distribution of users. In fact, the main advantage of optimal transport theory is to provide tractable solutions for a variety of cell association problems in wireless networks. In our problem, we first prove theexistence of the optimal solution to the cell association problem, and, then, we characterize the solution space. The results show that, our approach results in a significantly lower delay compared to a conventional signal strength-based association. § SYSTEM MODEL AND PROBLEM FORMULATION Consider a geographical area 𝒟⊂R^2 in whichK terrestrial BSs in set 𝒦 are deployed to provide service for ground users that are spatially distributed according to a distribution f(x,y) over the two-dimensional plane. In addition to the terrestrial BSs, M UAVs in set ℳ are deployed as aerial base stations to enhance the capacity of the network. We consider a downlink scenario in which the BSs and the UAVs use a frequency division multiple access (FDMA) technique to service the ground users. The locations of BS i∈𝒦 and UAV j∈ℳ are, respectively, given by (x_i,y_i,h_i) and (x^uav _j,y^uav _j,h^uav_j), with h_i and h^uav_j being the heights of BS i and UAV j.The maximum transmit powers of BS i and UAV j areP_i and P_j^uav. Let W_i and W_j be the total bandwidth available for each BS i and UAV j. Our performance metric is the transmission delay, which is referred to as the time needed for transmitting a given number of bits. In this case, the delay is inversely proportional to the transmission rate. We use A_i and B_j to denote, respectively, the area (cell) partitions in which the ground users are assigned to BS i and UAV j. Hence, the geographical area is divided into M+K disjoint partitions each of which is served by one of the BSs or the UAVs. Given this model, our goal is to minimize the average network delay by optimal partitioning of the area.Based on the spatial distribution of the users,we determine the optimal cell associations to minimize the average network delay. Note that, the network delay significantly depends on the cell partitions due to the following reasons. First, the cell partitions determine the service area of each UAV and BS thus impacting the channel gain that each user experiences. Second, the number of users in each partition depends on the cell partitioning. In this case, since the total bandwidth is limited, the amount of bandwidth per user decreases as thenumber of users in a cell partition increases. Thus, users in the crowded cell partitions achieve a lower throughput which results in a higher delay. Next, we present the channel models.§.§ UAV-User and BS-User path loss modelsIn UAV-to-ground communications, the probability of having LoS links to users depends on the locations, heights, and the number of obstacles, as well as the elevation angle between a given UAV and it’s served ground user. In our model, we consider a commonly used probabilistic path loss model provided by International Telecommunication Union (ITU-R), and the work in <cit.>. The path loss between UAV j and a user located at (x,y) is <cit.>: Λ_j = {[ K_o^2(d_j/d_o)^2μ _LoS, LoS link,; K_o^2(d_j/d_o)^2μ _NLoS, NLoS link, ]. where K_o=(4π f_cd_o/c)^2, f_c is the carrier frequency, c is the speed of light, and d_o is the free-space reference distance. Also, μ_LoS and μ_ NLoS are different attenuation factors considered for LoS and NLoS links.d_j=√((x-x^uav _j)^2+(y-y^uav _j)^2+h^uav_j^2) is the distance between UAV j and an arbitrary ground user located at (x,y). For the UAV-user link, the LoS probability is <cit.>:P_LoS,j = α( 180/πθ_j- 15)^γ,θ_j>π/12, where θ_j=sin ^- 1( h_j/d_j) is the elevation angle (in radians) between the UAV and the ground user. Also, α and γ are constant values reflecting the environment impact. Note that, the NLoS probability isP_NLoS,j=1-P_LoS,j.Considering d_o=1 m, the average path loss is K_od_j^ 2[ P_LoS,jμ _LoS + P_NLoS,jμ _NLoS]. Therefore, the received signal power from UAV j considering an equal power allocation among its associated users will be: P̅_r,j^uav = P_j^uav/( N_j^uav K_od_j^2[ P_LoS,jμ _LoS + P_NLoS,jμ _NLoS]),where P_j^uav is the UAV's total transmit power, and N_j^uav =N ∬_B_jf(x,y)dxdy is the average number of users associated with UAV j, with N being the total number of users. For the BS-user link, we use the traditional path loss model. In this case, the received signal power from BS i at user's location (x,y) will be: P_r,i= P_i K_o ^-1 d_i^-n/N_i,where d_i=√((x-x_i)^2+(y-y_i)^2+h_i^2) is the distance between BS i and a given user, N_i =N ∬_A_if(x,y)dxdy is the average number of users associated with BS i, and n is the path loss exponent. §.§ Maximum Coverage Range Here, we determine the maximum achievable coverage range of each UAV and terrestrial BS. The coverage range of a UAV j or a BS i is the maximum radius within which the averagereceived signal-to-noise-ratio (SNR) remains above a specified threshold, γ_th. Clearly, to find the maximum coverage range, we must set theSNR to γ_th. For a UAV j, let R_j = √(d_j^2 - h_j^2) be the coverage range. Then, given (<ref>) and (<ref>), the maximum coverage range R_j^uav is the solution to the following equation: [ α( 180/πtan^ - 1( h_j/R_j) - 15)^γ( μ _LoS - μ _NLoS) + μ _NLoS] ×( h_j^2 + R_j^2) - γ _thW_j^uavN_oK_o/P_j^uav = 0. Given h_j, the left side of equation (<ref>) is a monotonically increasing function of R_j. Hence, (<ref>) admits a unique solution. Clearly, the coverage range depends on the altitude of the UAV. Therefore, unlike terrestrial BSs, one can change the coverage performance of the UAV by varying its altitude.In particular, the maximum coverage range of each UAV that can be achieved by optimally adjusting the altitude, is given by: The maximum coverage range of any UAV placed at its optimal altitude is: R_j^uav =(P_j^uavcos^2θ_j^* /K_oγ _thW_j^uavN_o( α( 180/πθ_j^*- 15)^γ( μ _LoS - μ _NLoS) + μ _NLoS))^0.5,where θ_j ^* = _θ_j{∂ R_j(θ_j)/∂θ_j= 0}.Given (<ref>), and consideringh_j = R_jtanθ_j, we can write the coverage radius as: R_j(θ_j)= (P_j^uavcos^2θ_j /K_oγ _thW_j^uavN_o( α( 180/πθ_j- 15)^γ( μ _LoS - μ _NLoS) + μ _NLoS))^0.5,it can be shown that, given the specified parameters α and γ for the air-to-ground communication, and ∂ ^2R_j(θ_j )/∂θ_j ^2≤ 0 and, hence, R_j(θ_j) is a concave function of θ_j. Therefore, the optimal elevation angle is given by:θ_j ^* = _θ_j{∂ R_j(θ_j)/∂θ_j= 0}.Subsequently, the maximum coverage range will be R_j^uav=R_j(θ_j ^*), with the corresponding optimal altitude of R_j(θ_j ^*)tan( θ_j ^*). For a terrestrial BS i, the maximum coverage range is: R_i^BS = ( P_iK_o/γ _thN_oW_i)^1/n. Note that, (<ref>) and (<ref>) provide the necessary conditions for assigning each user to the UAVs and BSs. Next, based on the maximum coverage range of UAVs and BSs given in (<ref>) and (<ref>), we formulate our throughput-optimal association problem.§.§ Problem formulation Given the average received signal power in the UAV-user communication, the average throughput of a user located at (x,y) connecting to a UAV j can be approximated by:C_j^uav= W_j/N_j^uavlog_2( 1 + P̅_r,j^uav/σ^2),where σ^2 is the noise power for each user which is linearly proportional to the bandwidth allocated to the user.The throughput of the user if it connects to a BS i is: C_i = W_i/N_ilog _2( 1 + P_r,i/σ^2). Now, let ℒ=𝒦∪ℳ be the set of all BSs and UAVs. Also, here, the location of each BS or UAV is denoted by s_k, k∈ℒ. We also consider D_k = {[ A_k,ifk ∈𝒦,; B_k, if k ∈ℳ, ]. denoting all the cell partitions, and Q( v,s_k,D_k) = {[ b/ C_k,ifk ∈𝒦,; b/ C_k^uav,ifk ∈ℳ, ]. where v=(x,y) is the 2D locations of the ground users, and b is the number of bits that must be transmitted to location v. Then, our optimization problem that seeks to minimize the average network delay over the entire area will be: min_D_k∑_k ∈ℒ∫_D_kQ( v,s_k,D_k)f(x,y)dxdy, s.t. ⋃_k ∈ℒD_k= 𝒟, D_l∩D_m = ∅ , ∀ lm ∈ℒ.where both constraints in (<ref>)guarantee that the cell partitions are disjoint and their union covers the entire area, 𝒟. § OPTIMAL TRANSPORT THEORY FOR CELL ASSOCIATION Given the locations of the BSs and the UAVs as well as the distribution of the ground users, we find the optimal cell association for which the average delay of the network is minimized.Let g_k(z) = Nz/W_k, with W_k being the bandwidth for each BS or UAV k and z is a generic argument.Also, we consider:F(v,s_k) = {[ b/log _2(1+ P_r,k(v,s_k)/σ^2 ),if k∈𝒦,; b/log _2(1+ P̅_r,j^uav(v,s_k)/σ^2 ),ifk∈ℳ. ].Now, the optimization problem in (<ref>) can be rewritten as:min_D_k∑_k ∈ℒ∫_D_k[ g_k( ∫_D_kf(x,y)dxdy)F(v,s_k)]f(x,y)dxdy,s.t. ⋃_k ∈ℒD_k= 𝒟, D_l∩D_m = ∅ , ∀ lm ∈ℒ, where D_k is the cell partition of each BS or UAV k.Solving the optimization problem in (<ref>) is challenging and intractable due to various reasons. First, the optimization variables D_k, ∀ k ∈ℒ, are sets of continuous partitions which are mutually dependent. Second, f(x,y) can be any generic function of x and y that leads to the complexity of the given two-fold integrations. To overcome these challenges, next, we model this problem by exploiting optimal transport theory <cit.> in order to characterize the solution.Optimal transport theory <cit.> allows analyzing complex problems in which, for two probability measures f_1 and f_2 on Ω⊂R^n, one must find the optimal transport map T from f_1 to f_2that minimizes the following function:min_T ∫_Ωc( x,T(x)) f_1(x)dx;T:Ω→Ω, where c(x,T(x)) denotes the cost of transporting a unit mass from a location x to a location T(x).Our cell association problem can be modeled as a semi-discrete optimal transport problem. In this case, the users follow a continuous distribution, and the base stations can be considered as discrete points. Then, we need to map the users to the BSs and UAVs such that the total cost function is minimized. In this case,the optimal cell partitions are directly determined by the optimal transport map <cit.>. Next, we prove the existence of the optimal solution to the problem in (<ref>). The optimization problem in (<ref>) admits an optimal solution given N0, and σ 0.Let a_k = ∫_D_kf(x,y)dxdy, and for ∀ k ∈ℒ,E ={a = ( a_1,a_2,...,a_K + M) ∈R^K + M;a_k≥ 0,∑_k = 1^K + Ma_k = 1}.Now, considering f(x,y)=f(v) and c( v,s_k) = g_k(a_k)F( v,s_k), for any given vector a, problem (<ref>) can be considered as a classical semi-discrete optimal transport problem.First, we prove that c( v,s) is a semi-continuous function. Considering the fact that s_k is discrete, we have: lim_(v,s) → (v^*,s_k) F( v,s)=lim_v→v^* F( v,s_k).Note that, given any s_k, k belongs to only of 𝒦 and ℳ sets. Given s_k, F(v,s_k) is a continuous function of v. Then, considering the fact that given a_k, g_k(a_k) is constant, we have lim_(v,s) → (v^*,s_k) g_k(a_k) F( v,s) = g_k(a_k)F( v^*,s_k). Therefore,c(v,s) is a continuous function and, hence, is also a lower semi-continuous function.Now, we use the following lemma from optimal transport theory:Consider two probability measures f and λ on 𝒟⊂R^n. Let f be continuous and λ = ∑_k ∈Na_kδ _s_k be a discrete probability measure. Then, for any lower semi-continuous cost function, there exists an optimal transport map from f to λ for which ∫_𝒟c( x,T(x)) f(x)dx is minimized <cit.>.Considering Lemma 1, for any a∈ E, the problem in (<ref>) admits an optimal solution. Since E is a unit simplex in R^M+K which is a non-empty and compact set, the problem admits an optimal solution over the entire E.Next, we characterize the solution space of (<ref>). To acheive the delay-optimal cell partitions in (<ref>), each user located at (x,y) must be assigned to the following BS (or UAV): k=min_l ∈ℒ{a_l/W_lF(v_o,s_l)}, Given (<ref>), the optimal cell partition D_k includes all the points which are assigned to BS (or UAV) k. As proved in Theorem 1, there exist optimal cell partitions D_k, k∈ℒ which are the solutions to (<ref>). Now, consider two partitions D_l and D_m, and a point v_o=(x_o,y_o)∈ D_l. Also, let B_ϵ(v_o) be a ball with a center v_o and radius ϵ >0. Now, we generate the following new cell partitions 3pt⌢ D_k (which are variants of the optimal partitions): {[ 3pt⌢ D_l = D_l\B_ε(v_o),;3pt⌢ D_m = D_m ∪B_ε(v_o),; 3pt⌢ D_k = D_k,kl,m. ]. Let a_ε = ∫_B_ε(v_o)f(x,y)dxdy, and 3pt⌢ a_k = ∫_3pt⌢ D_kf(x,y)dxdy. Considering the optimality of D_k, k∈ℒ, we have: ∑_k ∈𝒦∫_D_kg_k( a_k)F(v,s_k) f(x,y)dxdy≤^(a)∑_k ∈𝒦∫_3pt⌢ D_kg_k( 3pt⌢ a_k)F(v,s_k) f(x,y)dxdy.Now, canceling out the common terms in (<ref>) leads to:∫_D_lg_l( a_l)F(v,s_l)f(x,y)dxdy+ ∫_D_mg_m( a_m)F(v,s_m) f(x,y)dxdy≤∫_D_m∪B_ε(v_o)g_m( a_m + a_ε)F(v,s_m) f(x,y)dxdy+∫_D_l\B_ε(v_o)g_l( a_l - a_ε)F(v,s_l) f(x,y)dxdy,∫_D_l( g_l( a_l) - g_l( a_l - a_ε))F(v,s_l) f(x,y)dxdy +∫_B_ε(v_o)g_l( a_l - a_ε)F(v,s_l) f(x,y)dxdy≤∫_D_m( g_m( a_m+ a_ε) - g_m( a_m))F(v,s_m) f(x,y)dxdy + ∫_B_ε(v_o)g_m( a_m + a_ε)F(v,s_m) f(x,y)dxdy,where (a) comes from the fact that D_k, ∀ k ∈ℒ are optimal and, hence, any variation of such optimal partitions, shown by 3pt⌢ D_k, cannot lead to a better solution. Now, we multiply both sides of the inequality in (<ref>) by 1/a_ϵ, take the limit when ϵ→ 0, and use the following equalities:lim_ε→ 0a_ε = 0,lim_a_ε→ 0g_l(a_l) - g_l(a_l - a_ε)/a_ε = g'_l(a_l),lim_a_ε→ 0g_m(a_m + a_ε) - g_m(a_m)/a_ε = g'_m(a_m), then we have: g'_l( a_l)∫_D_lF(v_o,s_l)f(x,y)dxdy+ g_l( a_l)F(v_o,s_l)≤g'_m( a_m) ∫_D_mF(v_o,s_m)f(x,y)dxdy+ g_m( a_m)F(v_o,s_m). Now, given g_k(z) = Nz/W_k, we can compute g'_l(a_l) = . dg_l(z)/dz|_z = a_l=N/W_k, then, using a_k=∫_D_kf(x,y)dxdy leads to: N/W_la_lF(v_o,s_l) + Na_l/W_lF(v_o,s_l)≤N/W_ma_mF(v_o,s_m) + Na_m/W_mF(v_o,s_m), as a result: a_l/W_lF(v_o,s_l) ≤a_m/W_mF(v_o,s_m). Finally, (<ref>) leads to (<ref>) that completes the proof. Theorem 2 provides a precise cell association rule for ground users that are distributed following any general distribution f(x,y). In fact, the inequality given in (<ref>) captures the condition under which the user is assigned to a BS or UAV l. Under the special case of a uniform distribution of the users, the result in Theorem 2 leads to the classical SNR-based association in which users are assigned to base stations that provide strongest signal. From Theorem 2, we can see that there is a mutual dependence between a_l and D_l (i.e. cell association), ∀ l∈ℒ. To solve the equation given in Theorem 2, we adopt an iterative approach which is shown to converge to the global optimal solution <cit.>. In this case, we start with initial cell partitions (e.g. Voronoi diagram), and iteratively update the cell partitions based on Theorem 2. § SIMULATION RESULTS AND ANALYSIS For our simulations, we consider an area of size 4 km× 4km in which 4 UAVs and 2 macrocell base stations are deployed based on a traditional grid-based deployment.The ground users are distributed according to a truncated Gaussian distribution with a standard deviation σ_o. This type of distribution which is suitable to model a hotspot area. The simulation parameters are given as follows. f_c=2 GHz, transmit power of each BS is 40 W, and transmit power of each UAV is 1 W. Also, N=300, W_j=W_i=1 MHz, and the noise power spectral density is -170 dBm/Hz. We consider a dense urban environment with n=3, μ_LoS=3 dB, μ_NLoS=23 dB, α=0.36, and γ=0.21 <cit.>. The heights of each UAV and BS are, respectively, 200 m and 20 m <cit.>. All statistical results are averaged over a large number of independent runs. In Fig. <ref>, we compare the delay of the proposed cell association with the traditional SNR-based association. We consider a truncated Gaussian distribution with a center (1300 m,1300 m), and σ_o varying from 200 m to 1200 m. Lower values of σ_o correspond to scenarios in which users are more concentrated around the hotspot center. Fig. <ref> showsthat the proposed cell association significantly outperforms the SNR-based association in terms of the average delay. For low σ_o values, the average delay decreases by 72% compared to the SNR-based association. This is due to the fact that, in the proposed approach, the impact of network congestion is taken into account.Hence, the proposed approach avoids creating highly loaded cells. In contrast, an SNR-based association can yield highly loaded cells. As a result, in the congested cells, each user will receive a low amount of bandwidth that leads a low transmission rate or equivalently high delay. In fact, compared to the SNR-based association case, our approach is more robust against network congestion and its performance is significantly less affected by changing σ_o. As an illustrative example, Fig. <ref> shows the locations of the BSs and UAVs as well as the cell partitions obtained using SNR-based association and the proposed delay-optimal association. In this case, users are distributed based on a 2D truncated Gaussian distribution with mean values of (1300 m,1300 m), and σ_o=1000 m. As shown in Fig. <ref>, the size and shape of cells are different in these two association approaches. For instance, the red cell partition in the proposed approach is smaller than the SNR-based case. In fact, the red partition in the SNR-based approach is highly congested and, consequently, its size is reduced in the proposed case so as to decrease the congestion as well as the delay.§ CONCLUSION In this paper, we have proposed a novel framework for delay-optimal cell association in UAV-enabled cellular networks. In particular, to minimize the average network delay based on the users' distribution, we have exploited optimal transport theory to derive the optimal cell associations for UAVs and terrestrial BSs. Our results have shown that, the proposed cell association approach results in a significantly lower network delay compared to an SNR-based association.IEEEtran | http://arxiv.org/abs/1705.09748v1 | {
"authors": [
"Mohammad Mozaffari",
"Walid Saad",
"Mehdi Bennis",
"Merouane Debbah"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170527005928",
"title": "Optimal Transport Theory for Cell Association in UAV-Enabled Cellular Networks"
} |
[Corresponding author ]e-mail: [email protected]. of Mathematics, Imperial College London, South Kensington Campus, London, U.K.Dept. of Mathematical Sciences, University of Liverpool, Liverpool, U.K.MN2S, Femto-st Besancon, 25030 Besancon CEDEX France Aix-Marseille Université, Centrale Marseille, Institut Fresnel-CNRS (UMR 7249), 13397 Marseille cedex 20, FranceISTerre, CNRS, Univ. Grenoble Alpes, France, BP 53 38041 Grenoble CEDEX 9Hap2U, CIME Nanotech, 3 parvis Louis Néel 38000 Grenoble, France It is well known in metamaterials that local resonance and hybridization phenomena dramatically influence the shape of dispersion curves; the metasurface created by a cluster of resonators, subwavelength rods, atop an elastic surface being an exemplar with these features. On this metasurface, band-gaps, slow or fast waves, negative refraction and dynamic anisotropy can all be observed by exploring frequencies and wavenumbers from the Floquet-Bloch problem and by using the Brillouin zone. These extreme characteristics, when appropriately engineered, can be used to design and control the propagation of elastic waves along the metasurface. For the exemplar we consider, two parameters are easily tuned: rod height and cluster periodicity. The height is directly related to the band-gap frequency, and hence to the slow and fast waves,while the periodicity is related to the appearance of dynamic anisotropy. Playing with these two parameters generates a gallery of metasurface designs to control the propagation of both flexural waves in plates and surface Rayleigh waves for half-spaces. Scalability with respect to the frequency and wavelength of the governing physical laws allows the application of these concepts in very different fields and over a wide range of lengthscales. Elastic wave control beyond band-gaps: shaping the flow of waves in plates and half-spaces. Matthieu Rupin December 30, 2023 ===========================================================================================§ INTRODUCTION Recent years have witnessed the increasing popularity of metamaterial concepts, based on so-called local resonance phenomenon, to control the propagation of electromagnetic <cit.>, acoustic and elastic <cit.> waves in artificially engineered media. Initiallyattention focused on the existence of subwavelength band-gaps generated by the resonators <cit.>, and resulting frequency dependent effective material parameters for negative refraction and focusing effects <cit.>, and now consideration is transitioning to methods for achieving more complete forms of wave control by encompassing tailored graded designs to obtain spatially varying refraction index <cit.>, wide band-gaps and mode conversion. In the fields of photonics and acousticsthis transition has already taken place and new graded designs allow for thetailored control of the propagation of light<cit.>, micro-waves <cit.>, water waves <cit.> and sound <cit.>.Elastodynamic media have, in contrast to acoustic and electromagnetic systems, additional complexity such as supporting both compressional and shear wave speeds that differ and which mode convert at interfaces <cit.>. On one hand this makes elastic metamaterials complex to model and require the use of computational elastodynamic techniques <cit.>, on the other hand it offers new control possibilities not achievable in the electromagnetic or acoustic cases. Wave control has implications in several disciplines and the discoveries of metasurface science are currently being translated into several applications.If we limit our discussion to elastic metamaterials, potential applications can be implemented at any lengthscale. On the large-scale, seismic metamaterials have become very popular <cit.>. At smaller scale, in mechanical engineering, applications based on wave re-routing and protection are currently being explored <cit.> to reduce vibrations in high precision manufacturing and in laboratories for high precision measurements (e.g. interferometry) or in the field of ultrasonic sensing to amplify signal to noise ratio. In the field of acoustic imaging, the tailored control of hypersound (elastic waves at GHz frequencies) used for cell or other nano-compound imaging <cit.>, is emerging as one of the most promising applications of energy trapping and signal enhancement through metamaterials. Furthermore, at this small scale, novel nanofabrication techniques deliver the tailoring possibilities required for graded devices <cit.>. Among the possible resonant metasurface designs for elastic waves proposed in recent years <cit.>, the one made of a cluster of rods (the resonators) on an elastic substrate has revealed superior characteristics and versatility of use in particular towards the fabrication of graded design.The physics of this metasurface is well described through a Fano-like resonance <cit.>. A single rod attached to an elastic surface couples with the motion of both the A_0 mode in a plate or the Rayleigh wave on a thick elastic substrate (half-space). This coupling is particularly strong at the longitudinal resonance frequencies of the rod. At this point, the eigenvalues of the equation describing the motion of the substrate and the rod are perturbed by the resonance and become complex leading to the formation of a band-gap <cit.>. When the resonators are arranged on a subwavelength cluster (i.e. with λ, the wavelength ≫ than the resonator spacing), as in the metasurface discussed here, the resonance of each rod acts constructively enlarging the band-gap until, approximately, the rod's anti-resonance <cit.>. Thus the resulting band-gap is broad and subwavelength.Because the resonance frequency of the rod drives the band-gap position, a spatially graded metasurface is simply obtained by varying the length of the rods, which directly underpins the resonance frequency.Thus the length of the rod appears to be the key parameter for the metasurface tunability, although theperiodicity and distribution of the rods cannot be ignored as they also influence the dispersion curves leading to zone characterised by dynamic anisotropy and negative refraction <cit.>. These effects are important as they may be used to generate highly collimated waves or for subwavelength imaging. Our purpose in this work is to complement the research on local resonance and slow and fast waves, with the study of the dynamic anisotropy effect <cit.> when the rods are periodically arranged on the elastic surface. In fact, it has been recently realized that many novel features of hyperbolic metamaterials such as superlensing and enhanced spontaneous emission <cit.> could be achieved thanks to dynamic anisotropy in photonic <cit.> and phononic crystals <cit.>. For instance, the high-frequency homogenization theory <cit.> establishes a correspondence between anomalous features of dispersion curves on band diagrams with effective tensors in governing wave equations: flat band and inflexion (or saddle) points lead to extremely anisotropic and indefinite effective tensors, respectively, that change the nature of the wave equations (elliptic partial differential equations can turn parabolic or hyperbolic depending upon effective tensors). This makes analysis of dynamic anisotropy a potentially impactful work.The first half of the article is dedicated to the review of the state of the art on the control of flexural and Rayleigh waves with rods on an elastic substrate. This part will collect the major achievements and milestones obtained by our research group in the past 3 years. In the second part we will present another characteristic of this metasurface analysing the 2D dispersion curves and the effect of dynamic anisotropy in the subwavelength regime.§ EARLY RESULTS: PLATE VS. INFINITE HALF-SPACE METAMATERIALWe start by recalling results obtained with a metamaterial, introduced in 2014, made from a thin elastic plate and a cluster of closely spaced resonators (see model in Fig. 1(a)) both made of aluminium. At that time, despite the limited knowledge of the metasurface dispersion properties, the cluster of resonators immediately showed surprising phenomena such as the presence, in the Fourier spectra, of large subwavelength band-gaps <cit.> affecting the propagation of the A_0 mode in the thin plate in the kHz range.Around the same time, <cit.> demonstrated that by exploiting the stop band,waves can be trapped in a very subwavelength cavity and that energy could be tunneled through the metasurface by inserting a defect, with an approach reminiscent of phononic crystal applications.These early attempts to compute the dispersion curves of the metasurface for a given rod size and spacing were based on array methods that projected the time series recorded from either experiment or numerical simulation on the frequency wavenumber plane (f-k plane). These preliminary results confirmed the resonant nature of the band-gap and uncovered another striking characteristic of the metamaterial: the nearly flat branches occurring at edges of the Brillouin zone before and after the band-gap. These flat branches represent, for high wavenumbers, very slow modes. Conversely for k approaching the origin, these modes travel very fast.The analytical calculation of the dispersion properties by <cit.> (e.g. the plot in Fig. 1(a)), mean we can now fully harness the power of this metasurface and use the concept of fast and slow modes to fully control the propagation of waves in a plate <cit.>. Before showing the effects of such tailored wave control we continue our digression into the important applications of elastic resonators on an elastic surface. It has been known since <cit.> that short pillars (or other type of resonators <cit.>) on an elastic half-space canalter the dispersion curves by introducing Bragg and resonant band-gaps for Rayleigh waves. However, the use of longitudinally elongated resonators, such as the rods shown in Fig 1b, allow for a much clearerseparation of the longitudinal mode (responsible for the band-gap) from other flexural resonances that will be discussed in the last section of this article. This has the twofold advantage of pushing the band-gap to the subwavelength scale, simultaneously increasing its breadth, and also simplifying the analytical description of the metamaterial.From an analytical point of view the thin elastic plate metasurface can still be treated as a scalar problem as one can use Kirchhoff's plate theory coupled with a longitudinal wave equation for the rod. In an elastic half-space this is no longer possible and the full elastic equation must be used to describe its physics. With this concept in mind <cit.> constructed an analytical formulation for the dispersion curve of a 1D array of resonators on the half-space considering only the longitudinal modes of the rods.From visual inspection of the plot in Fig. 1(b), besides the obvious lack of dispersion for body and Rayleigh waves in the half-space (in contrast the A_0 mode in Fig. 1(a) is strongly dispersive) and the different frequency and size of the model (metres and kHz for the plate and centimetres and MHz for the half-space), a similar hybridization mechanism <cit.> creates the band-gap in both systems. However, in the half-space, the maximum speed of the system is bounded by the shear S-wave line. These observations are consistent with the physical interpretation that the vertical component of the elliptically polarised Rayleigh waves, usually travellingslower then the shear wave, couples with the longitudinal motion of the resonator. The presence of these band-gaps have inspired the development of so-called seismic metamaterials for Rayleigh waves <cit.> where the close relationship between shear S and Rayleigh waves in the half-space lead to unexpected wave phenomena in the metamaterial. As chiefly demonstrated in <cit.> and <cit.>, the resonance creates an hybrid branch bridging the Rayleigh line with the S-wave line. Through a graded resonators design (e.g. decreasing or increasing rod's height) the conversion becomes ultra-broadband, a key requirement for practical engineering applications. § GALLERY OF CONTROL POSSIBILITIES ACHIEVED BY TUNING THE ROD LENGTHThe rich physics encoded within the hybrid dispersion curves that we have just described for the plate and half-space cases, can be translated into extraordinary wave propagation phenomena. Furthermore, scalability is one of the strong characteristics of metamaterials which makes them applicable in different wave realms and lengthscales. With the following examples wedemonstrate that applications for the two different settings and lengthscale introduced in Figs. 1(a) and (b), namely the elastic plate and the half-space. This choice is made to remain coherent with our previous laboratory and numerical studies on these structures <cit.>. The description starts from Fig. 2(a), snapshots extracted from a numerical simulation show the band-gap created by a small cluster of resonators located on top of a thin elastic plate. The field has been filtered inside the band-gap at a frequency between 2 and 3 kHz (6 mm thick plate and 60 cm long rods, both made of aluminium). The band-gap frequency f directly depends on the resonator length h and therefore can be easily tuned by selecting longer or shorter rods using the well known formula:f=1/4 h√(E/ρ), whereE its Young's modulus and ρ its density. This formula is valid when the substrate is sufficiently stiff, for seismic metamaterials, where the resonator might be supported by a soft sediment layer, the contribution of the substrate must be taken into account when calculating the resonance frequency <cit.>.In Fig. 2(b) and (c), we exploit the effective wave velocity (slow waves) that is locally achieved in the metamaterial. In these figures we show two types of so-called graded index lenses <cit.> well known for their capacity to focus and re-route waves without aberration and reflection. These lenses are characterised by a radially varying velocity profile decreasing from the outside to the inside (for 4 kHz flexural waves h varies approximately between 60 and 80 cm whilein Fig. 2(b) while between 60 and 90 cm for the case in Fig. 2(c)). In practice, such a material is very difficult to fabricate unless one uses layers of different material or a graded thickness profile for the plate case. For the half-space this is clearly not possible. By using the slow modes of the flat branch occurring before the band-gap (see dispersion curves in Figs. 1(a) and (b)) these velocity gradients can be achieved by tailoring the resonator height distribution to the velocity profile required by the lens. This step is better achieved using the analytical form of the dispersion curve as shown in <cit.> derived using the theory from <cit.>. Although only the results for the plate have been currently published, the same method can be applied to Rayleigh waves too with the theory developed by <cit.>. In the remaining three figures the description moves to the control of Rayleigh waves. Unlike the plate case where the physics can be captured in the plane, here it is important to describe the wholewavefield inside the half-space. For this reason 2D simulations in the P-SV plane (planestrain) are now shown.As already anticipated in Fig. 1(b), the first snapshot shows the band-gap (here the field is filtered between 0.35-0.4 MHz) produced by an array of resonators of constant height. In Fig. 2(e) we show the well-known phenomena of the rainbow trapping but now for elastic waves. As for the lens case, this effect is completely due to the slow branch occurring below the band-gap. The graded array of resonator (resonant metawedge), enhances this effect and makes this device completely broadband (inversely proportional to the height). Note that,compared to the band-gap described in Fig. 2(d), here the Rayleigh wave remains confined to the surface while a broadband band-gap is produced after the wedge; for clarity of presentation we have used a monochromatic source of Rayleigh waves at 0.5 MHz. When the wedge orientation is reversed, as in Fig. 2(f), the surprising phenomenon of modal conversion is obtained and the graded profile enhances the conversion on a large frequency band; in the previous section this was already anticipated from the analysis of the dispersion curves.The control possibilities emerging from this discussion suggest tremendous potential for applications of these metamaterials toward vibration reduction and enhanced sensing. In this section we have not specified yet whether these phenomena depend, or not, on the periodicity of the resonator distribution in the metamaterial.Because local resonance is at the origin of the effects presented so far the answer is no for all of them. However, in the next section we will explore the important implications of periodicity. § PERIODICITY, DYNAMIC ANISOTROPY AND HYPERBOLIC BEHAVIOURThe height of the rods is not the only parameter available in terms of design of the metasurface. Solid state physics informs us that the lattice periodicity and spacing also matter as that generates, in particular, Bragg-type scattering. Dynamic anisotropy, that is anisotropy observed in the wavefield, that changes as frequency varies, is a common feature in phononic crystals with the most extreme situation being that where the wave energy is confined to “rays” with the field taking a cross-like form. Despite this, it has only marginally been associated with subwavelength metamaterials <cit.> with most work carried out in the context of phononic crystals. This section explores how anisotropy is obtained with this metasurface design. We introduce in Figs. 3(a) and (b) the dispersion curves for a 2D array of resonators respectively on a plate and on an infinite elastic support (half-space). The analysis is carried out inside the well-known irreducible Brillouin zone defined on the wavevector plane 𝐤=(k_x,k_y) by the three points of coordinats: Γ=(0,0), M=(π/d,π/d) and X=(π/d,0) where d is the pitch of the array of resonators. Given the complexity of the 3D problem, the model is solved numerically and includes all the admissible modes of the unit cell, not only, as previously done, the elongation of the rods. The resulting dispersion curves are characterised by several resonances that makes it hard to distinguish the longitudinal one. To aid interpretation we plot, along with the curve, the ratio between the vertical and the longitudinal value of the eigenfunction measured at the top of the resonator (where for all modes, the displacement reach a maximum <cit.>). High values mean that the motion is vertically polarised, conversely low value means that motion is horizontal; this interpretation is further confirmed associating to each resonance its modal deformation. The size of the unit cell in Fig. 3(a) is chosen to be similar to the cluster configuration in our previous work <cit.> where we have used a 6 mm plate and 60 cm rods both made of aluminium. The eigenvalue analysis is done using COMSOL and we make use of the built-in Bloch-Floquet boundary conditions to mimic an infinite 2D array of rods that are3-cm-spaced.The bare plate dispersion curve is shown in red for the Γ-X direction that is equal to the configuration in Fig. 1(a) (although without flexural resonances). Thanks to the colorcode used, the longitudinal modes are clearly identified in the dispersion curves. Given the lattice size, the first longitudinal mode is very subwavelength ∼λ/8. While the zoomed detail around this resonance is shown in Fig. 4, we can already distinguish the change in curvature that is responsible for the dynamic anisotropy behaviour. The other flat branches are mainly flexural modes (except for some breathing mode of the resonator). These are all double modes because the resonator is free to move in both directions. In Fig. 3(b) we repeat the same analysis for the half-space. The dimensions of the unit cell are similar to those for the plate although the spacing is slightly larger to improve the visualization of the anisotropy in Fig. 4(b). A technical detail is that,to mimic the infinite character of the half-space, we have applied an absorbing boundary at the lower side of the computational cell (see COMSOL Structural Mechanics Module documentation).The physics of the wave propagation in the half-space differs from the plate case mainly because of the lack of dispersion (see the straight dispersion curves for the bare half-space) and the higher speed of the waves. In this configuration however, the longitudinal resonance is only slightly subwavelength ∼λ/3. Clearly, by using a longer resonator, the band-gap can be pushed to a much lower frequency but the curvature of the longitudinal resonance is then shrunk down to a fraction of the Hertz, making the visualizationof the anisotropy practically impossible as the effect is so sensitive that small numerical or manufacturing variations would spoil the expected result. We now focus on the anisotropic behaviour by zooming in to frequencies close to the longitudinal modes. A detailed view of the first mode of the plate is depicted in Fig. 4(a). We can clearly appreciate the slope change that occurs before the resonance. A spectral element simulation in the time domain shows a snapshot of the wavefield filtered around the inflexion point of the mode. An array of 20×20 resonators, spaced and sized according to Fig. 3(a), is placed at the center of the plate. Because the plate boundaries are reflecting, to improve the visualization despite the reverberations, we have smoothed the square array removing the corner. The cluster is in fact octagonal. The shape and size of the plate is identical to the one used in <cit.>, so this phenomenon could be easily verified experimentally. The source is located in the middle of the array and in our case it is broadband Gaussian pulse.The cross-shaped anisotropic profile, as well as the strong contrast between the wavelength inside and outside the plate is clearly visible, and reminiscent of wave patterns in negatively refracting and hyperbolic metamaterials. Using the same modeling technique, dynamic anisotropy also characterizes the half-space and it is indeed visible in the numerical results of Fig. 4(b). The half-space is simulated applying perfectly matched layers on the side and on the bottom surface. The snapshot show the vertical component of the displacement filtered at the inflexion point. As for the case of the plate, a similar cross is visible. However, here we observe a strong spatial attenuation of the field due to the fact that waves are free to propagate or scatter downward, while in the plate they were guided (e.g. Fig. 2(d)).At this stage, we note that there is a vast literature on electromagnetic hyperbolic metamaterials, which were theorized by David Smith and David Schurig almost fifteen years ago in the context of negatively refracting media described by electric permittivity and magnetic permeability tensors with eigenvalues of opposite signs <cit.>. These media originally thought of as an anisotropic extension of John Pendry's perfect lens <cit.> take their name from the topology of the isofrequency surface. In an isotropic homogeneous medium (e.g. vacuum in electromagnetics and air in acoustics), the linear dispersion and isotropic behaviour of transversely propagating (electromagnetic or sound) waves implies a circular isofrequency contour given by the dispersion equation k_x^2+k_y^2=ω^2/c^2 with k=(k_x,k_y) the wavevector, ω the angular wave frequency and c the wavespeed of light or sound waves. In a transversely anisotropiceffective medium, one has T_yy k_x^2+T_xx k_y^2=ω^2/c^2, where T_xx and T_yy are entries of the (inverse of) effective tensor of permittivity or mass density, shear or Young moduli etc. depending upon the wave equation. It is well known that thecircular isofrequency contour of vacuum distorts to an ellipse for the anisotropic case. However, when we have extreme anisotropy such that T_xxT_yy<0 the isofrequency contour opens into an open hyperbole. In electromagnetics, such a phenomenon requires the metamaterial to behave like a metal in one direction (along which waves are evanescent) and a dielectric in the other and similarly, in acoustics and platonics. A hallmark of hyperbolic media is an X-shape wave pattern for emission of a source located therein <cit.>, reminiscent of the hyperboles arising from the dispersion relations. Note of course, that if both entries of the effective tensor are negative, this means waves are evanescent in all directions, what corresponds to a metal in electromagnetics. In the case of structured plates, as aforementioned the high-frequency homogenization theory predicts that at the inflexion pointin Fig. 4(a), the effective tensor (that encompasses an anisotropic Young's modulus) has eigenvalues of opposite sign in the framework of the simplified model of Kirchhoff-Love <cit.>, and thus Lamb waves propagate like in a hyperbolic medium, and the similar features observed in Fig. 4(b) lead to an analogous conclusion for Rayleigh waves in structured halfspaces.§ FUTURE PERSPECTIVES Devices based upon exploiting band-gap phenomena, as seismic shields using ideas from Bragg-scattering <cit.> or zero-frequency stop-bands <cit.>, are gaining in popularity. Given the nuisance of ground vibration, and the importance of elastic wave control, for the urban environment this will be an area of growing importance; the additional degrees of freedom, control over sub-wavelength behaviour, and the broadband features that can be utilised using the resonant sub-wavelength structures discussed herein make them very attractive alternatives. At smaller scale one moves toward the manipulation of mechanical waves in vibrating structures, again it is the long-wave and low frequency waves that one often wants to control and, again, these are precisely the waves that are targeted by sub-wavelength resonator array devices. The ability to spatially segregate waves by frequency, the field enhancement and potential to mode convert surface to bulk waves, Fig. <ref>(d-f), are all phenomena with practical importance. Similarly, the ability to control surface waves to create concentrators, surface lenses and to redirect waves, using sub-wavelength arrays, Fig. <ref>(a-c), are powerful examples to draw upon for devices.The combined features of a flat band and a change of curvature near the inflexion point in Fig. 4 means that we are in a position to achieve effective parameters with eigenvalues of opposite sign exhibiting very different absolute values. So one can imagine controlling Rayleigh waves that would undergo simultaneously positive and negative refraction on the subwavelength scale, and this could lead to cloaking devices analogous to hyperbolic cloaks in electromagnetics <cit.>. At the geophysics scale applications of hyperbolic cloaks for Rayleigh waves are in seismic protection. It has been also suggested that one can achieve black hole effects <cit.> in hyperbolic metamaterials <cit.>, and this would have interesting applications in energy harvesting for Rayleigh waves propagating through arrays of rods at critical frequencies.Given the relative youth of metamaterials, as a field, and the very recent translation of metamaterial concepts to elastic plate, and elastic bulk, media there are undoubtedly many phenomena that will translate across from the more mature optical metamaterial field. Metasurfaces have become popular in optics as they can be created to combine the vision of sub-wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. 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Capasso, titleFlat optics with designer metasurfaces, journalNature Materials volume13, pages139–149 (year2014). § ACKNOWLEDGEMENTSAll of the computations presented in this paper were performed using the Froggy platform of the CIMENT infrastructure (https://ciment.ujf-grenoble.fr), supported by the Rhone-Alpes region (GRANT CPER07_13 CIRA), the OSUG2020 labex (reference ANR10 LABX56) and the EquipMeso project (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. A.C and R.C. thanks the EPSRC for their support through research grantEP/L024926/1. A.C. was supported by the Marie Curie Fellowship ”Metacloak". A.C., P.R., S.G and R.C. acknowledge thesupport of the French project Metaforet (reference ANR) that facilitates the collaboration between Imperial College, ISTerre and Institut Fresnel. | http://arxiv.org/abs/1705.09288v1 | {
"authors": [
"Andrea Colombi",
"Richard Craster",
"Daniel Colquitt",
"Younes Achaoui",
"Sebastien Guenneau",
"Philippe Roux",
"Matthieu Rupin"
],
"categories": [
"physics.class-ph",
"cond-mat.mtrl-sci",
"physics.app-ph",
"physics.comp-ph",
"physics.geo-ph"
],
"primary_category": "physics.class-ph",
"published": "20170525094204",
"title": "Elastic wave control beyond band-gaps: shaping the flow of waves in plates and half-spaces"
} |
Addicted To The Artist: Modeling Music Listening Behavior from Play LogsNational Institute of Advanced Industrial Science and Technology (AIST), Japan Taste or Addiction?: Using Play Logs to Infer Song Selection Motivation Kosetsu Tsukuda and Masataka Goto May 24, 2017 ======================================================================= Online music services are increasing in popularity. They enable us to analyze people's music listening behavior based on play logs. Although it is known that people listen to music based on topic (, rock or jazz), we assume that when a user is addicted to an artist, s/he chooses the artist's songs regardless of topic. Based on this assumption, in this paper, we propose a probabilistic model to analyze people's music listening behavior. Our main contributions are three-fold. First, to the best of our knowledge, this is the first study modeling music listening behavior by taking into account the influence of addiction to artists. Second, by using real-world datasets of play logs, we showed the effectiveness of our proposed model. Third, we carried out qualitative experiments and showed that taking addiction into account enables us to analyze music listening behavior from a new viewpoint in terms of how people listen to music according to the time of day, how an artist's songs are listened to by people, etc. We also discuss the possibility of applying the analysis results to applications such as artist similarity computation and song recommendation. § INTRODUCTION Among various leisure activities such as watching movies, reading books, and eating delicious food, listening to music is one of the most important for people <cit.>. In terms of the amount of accessible music, the advent of online music services (, Last.fm[<http://www.last.fm>], Pandora[<http://www.pandora.com>], and Spotify[<http://www.spotify.com>]) has made it possible for people to access millions of songs on the Internet, and it has become popular to play music using such services rather than physical media like CDs <cit.>. When users play music online, such services record personal musical play logs that show when users listen to music and what they listen to.Since personal music play logs have become available, it has become popular to use session information to analyze and model people's music listening behavior <cit.>. Here, a session is a sequence of logs within a given time frame. Zheleva <cit.> were the first to model listening behavior using a topic model based on session information. They revealed that a user tends to choose songs in a session according to the session's specific topic such as rock or jazz. However, it is not always correct to assume that a user chooses songs according to the session's topic. For example, after a user buys an artist's album or temporarily falls in love with an artist, s/he will be addicted to the artist and repeatedly listen to the artist's songs regardless of topic.In light of the above, this paper proposes a model that can deal with both a session topic and addiction to artists. Our proposed model uses the model proposed by Zheleva <cit.> as the starting point. We present each song-listening instance in terms of the corresponding song artist. In our model, each user has a distribution over topics that reflects the user's usual taste in music and a distribution over artists that reflects the user's addiction to artists. In addition, each user has a different ratio between usual taste and addiction, and probabilistically chooses a song in a session based on this ratio. That is, if a user has a high addiction ratio, s/he will probably choose a song of an artist from his/her artist distribution for addiction. Modeling people's music listening behavior by considering addiction is worth studying from various viewpoints: * Our model can show topic characteristics (, the rock topic has a high ratio of addiction) and artist characteristics (, most users choose an artist's songs when addicted to that artist). It is important to understand such characteristics from the social scientific viewpoint.* Our model can also show user characteristics (, a user chooses songs based on addiction in a session). There are many applications that could use this data such as advertisements and recommendation systems. For example, if a user chooses songs of an artist based on addiction in a session, it would be useful to recommend songs of that artist; if s/he chooses songs based on a topic, it would be better to recommend other artists' songs in the same topic. Our main contributions in this paper are as follows. * To the best of our knowledge, this is the first study modeling music listening behavior by considering both the usual taste in music and the addiction to artists.* We quantitatively evaluated our model by using real-world music play logs of two music online services. Our experimental results show that the model adopting both factors achieves the best results in terms of the perplexity computed by using test data.* We carried out qualitative experiments in terms of user characteristics, artist characteristics, and topic characteristics and show that our model can be used to analyze people's music listening behavior from a new viewpoint. The remainder of this paper is organized as follows. Section <ref> presents related work on analyzing music play logs and on modeling music listening behavior. Section <ref> describes the model that extends the model by Zheleva <cit.> by considering the addiction phenomenon. Section <ref> presents a procedure to infer the parameters. Section <ref> and <ref> report on our quantitative and qualitative experiments, respectively. Finally, Section <ref> concludes this paper.§ RELATED WORK§.§ Analysis of Music Listening BehaviorAnalyzing people's music listening behavior has attracted a lot of attention because (1) understanding how people listen to music is important from the social scientific viewpoint and (2) the analysis results can give useful insight into various applications such as music player interfaces and recommender systems.People's music listening behavior has been analyzed from various viewpoints. Rentfrow and Gosling <cit.> carried out a questionnaire-based survey and revealed the correlations between music preferences and personality, self-views (, wealthy and politically liberal), and cognitive ability (, verbal skills and analytical skills). Renyolds <cit.> made an online survey and reported that environmental metadata such as the user's activity, weather, and location affect the user's music selection. Analysis by Berkers <cit.> using Last.fm play logs showed the significant differences between male and female in terms of their music genre preferences. More recently, Lee <cit.> collected responses from users of commercial cloud music services and reported the criteria for generating playlists: personal preference, mood, genre/style, artists, etc. Among various factors, time information has received a lot of attention. Herrera <cit.> analyzed play counts from Last.fm and discovered that a non-negligible number of listeners listen to certain artists and genres at specific moments of the day and/or on certain days of the week. Park and Kahng <cit.> used log data of a commercial online music service in Korea and showed that there existed seasonal and time-of-day effects on users' music preference. Baur <cit.> also showed the importance of seasonal aspects, which influence music listening, using play logs from Last.fm.In spite of the variety of listening behavior analyses, to the best of our knowledge, no work has focused on users' addiction to, for example, songs and artists. In this work, we deal with this factor and analyze people's music listening behavior from a new perspective. §.§ Application Based on Music Listening LogsListening logs have been used for various applications, including the detection of similar artists. Schedl and Hauger <cit.> crawled Twitter[<http://twitter.com/>] for the hash tag #nowplaying and computed artist similarity using co-occurrence-based methods. Their experimental results showed that listening logs can be used to derive similarity measures for artists. Another application is playlist generation. Liu <cit.> proposed a playlist generation system informed by time stamps of a user's listening logs in addition to the user's music rating history and audio features such as wave forms. The most popular application is music recommendation. Since personal music play logs have become available, it has become popular to use session information to recommend songs. Park <cit.> proposed Session-based Collaborative Filtering (SSCF), which extends traditional collaborative filtering techniques by using preferred songs in the similar session. Dias and Fonseca <cit.> proposed temporal SSCF, where for each session, a feature vector is created consisting of five properties including time of day and song diversity. The work closest to ours is that of Zheleva <cit.>, who proposed a statistical model to describe patterns of song listening. They showed that a user tends to choose songs in a session according to the session's specific topic. We will describe the details of their model in Section <ref>.Although none of these applications used addiction information, we believe that this information could improve the usefulness of these applications. We discuss the possibility of using our analysis results to improve these applications in Section <ref>.§ MODEL As was mentioned earlier, our model builds on the one proposed by Zheleva <cit.>. After summarizing the notations used in our model in Section <ref>, we first describe the model by Zheleva <cit.> in Section <ref> and then propose our model in Section <ref>. §.§ Notations Given a music play log dataset, let U be a set of users in the dataset. Let l_un = (u, a, t_un) denote the nth play log of u ∈ U. More specifically, user u plays a song of artist a ∈ A at time t_un. Here, A is the set of artists in the dataset. Without loss of generality, we assume that play logs are sorted in ascending order of their timestamps: t_un < t_un' for n < n'.To capture user's listening preferences over time, we divide user's play logs into sessions. Following Zheleva <cit.> and Baur <cit.>, we use the time gap approach to generate sessions. If the gap between t_un and t_un+1 is less than 30 minutes, l_un and l_un+1 belong to the same session; otherwise, they belong to different sessions. Let S_ur be the rth session of u where S_ur consists of one or more of u's logs. Let R_u be the total number of u's sessions; then the set of u's sessions is given by D_u = { S_ur}_r=1^R_u. Hence, the set of sessions of all users is given by D = { D_u}_u ∈ U. §.§ Session Model The model proposed by Zheleva <cit.>, which is called the session model, is a probabilistic graphical model based on the Latent Dirichlet Allocation (LDA) <cit.>. The session model assumes that for each session, there is a latent topic (, rock or love song) that guides the choice of songs in the session. Figure <ref>(a) shows the graphical model of the session model, where shaded and unshaded circles represent observed and unobserved variables, respectively. In the figure, K is the number of topics, V_ur is the number of logs in the rth session of u, θ is the user-topic distribution, and ϕ is the topic-artist distribution. We assume that θ and ϕ have Dirichlet priors of α and β, respectively. The generative process of the session model is as follows: * For each topic k ∈{1, ⋯, K}, draw ϕ_k from Dirichlet(β).* For each user u in U, * Draw θ_u from Dirichlet(α). * For each session S_ur in D_u, * Draw a topic z_ur from Categorical(θ_u). * For each song in S_ur, observe an artist a_urj from Categorical(ϕ_z_ur). In the generative process, a_urj represents the jth song's artist in the rth session of u. §.§ Session with Addiction (SWA) Model Although Zheleva <cit.> reported the usefulness of generating played songs based on a session's topic, we hypothesize that users can choose a song independently of topic. For example, after a user buys an artist's album or temporarily falls in love with an artist, s/he will repeatedly listen to the artist's songs regardless of the topic. In other words, the user can be addicted to some artists. In such an addiction mode, we assume that the user directly chooses a song without going through the topic.In light of the above, our model takes both session-topic-based and addiction-based choices of songs. Figure <ref>(b) shows the graphical model of our proposed model. Each user has a Bernoulli distribution λ that controls the weights of influence for a session topic and addiction. To be more specific, when user u chooses a song in a session, we assume that the choice is influenced by the session topic with probability λ_u0 (x=0) and by u's addiction to the artist with probability λ_u1 (x=1), where λ_u0 + λ_u1 = 1. When x=0, a song is generated through the same process of the session model, while when x=1, a song is directly generated from a user-artist distribution ψ. The generative process of the SWA model is as follows: * For each topic k ∈{1, ⋯, K}, draw ϕ_k from Dirichlet(β).* For each user u in U, * Draw θ_u from Dirichlet(α). * Draw ψ_u from Dirichlet(γ). * Draw λ_u from Beta(ρ). * For each session S_ur in D_u, * Draw a topic z_ur from Categorical(θ_u). * For each song in S_ur, * Sample x from Bernoulli(λ_u). * If x=0, observe an artist a_urj from Categorical(ϕ_z_ur). * If x=1, observe an artist a_urj from Categorical(ψ_u). § INFERENCE To learn the parameters of our proposed model, we use collapsed Gibbs sampling <cit.> to obtain samples of hidden variable assignment. Since we use a Dirichlet prior for θ, ϕ, and ψ and a Beta prior for λ, we can analytically calculate the marginalization over the parameters. The marginalized joint distribution of D, latent variables Z={{z_ur}_r=1^R_u}_u ∈ U, and latent variables X={{{x_urj}_j=1^V_ur}_r=1^R_u}_u ∈ U is computed as follows:P(D,Z,X|α,β,γ,ρ) = P(D,Z,X|Θ, Φ,Ψ,Λ) P(Θ|α) P(Φ|β) P(Ψ|γ) P(Λ|ρ) dΘ dΦ dΨ dΛ,where Θ = {θ_u}_u ∈ U, Φ = {ϕ_k}_k=1^K, Ψ = {ψ_u}_u ∈ U, and Λ = {λ_u}_u ∈ U. By integrating out those parameters, we can compute Equation (<ref>) as follows:P(D,Z,X|α,β,γ,ρ) = ( Γ (2ρ)/Γ(ρ)^2)^|U|∏_u ∈ UΓ(ρ + N_u0) Γ(ρ + N_u1)/Γ(2ρ + N_u)( Γ(γ |A|)/Γ(γ)^|A|)^|U|∏_u ∈ U∏_a ∈ AΓ(N_u1a + γ)/Γ(N_u1 + γ |A|)×( Γ(β |A|)/Γ(β)^|A|)^K∏_k = 1^K∏_a ∈ AΓ(N_ka + β)/Γ(N_k + β |A|)( Γ(α K)/Γ(α)^K)^|U|∏_u ∈ U∏_k=1^KΓ(R_uk + α)/Γ(R_u + α K).Here, N_u0 and N_u1 are the number of u's logs such that x=0 and x=1, respectively, and N_u = N_u0 + N_u1. The term N_u1a represents the number of times that user u chooses artist a's song under the condition of x=1, and N_u1 = ∑_a ∈ A N_u1a. Furthermore, N_k = ∑_a ∈ A N_ka where N_ka is the number of times artist a is assigned to topic k under the condition of x=0. Finally, R_uk is the number of times u's session is assigned to topic k, and R_u = ∑_k=1^K R_uk.For the Gibbs sampler, given the current state of all but one variable z_ur, the new latent assignment of z_ur is sampled from the following probability:P(z_ur = k | D, X, Z_∖ ur, α, β, γ, ρ) ∝R_uk ∖ ur + α/R_u -1 + α KΓ(N_k ∖ ur + β |A|)/Γ(N_k ∖ ur + N_ur + β |A|)∏_a ∈ AΓ(N_ka ∖ ur + N_ura + β)/Γ(N_ka ∖ ur + β),where ∖ ur represents the procedure excluding the rth session of u. Moreover, N_ur and N_ura represent the number of logs in rth session of u and the number of a's logs in rth session of u, respectively.In addition, given the current state of all but one variable x_urj, the probability at which x_urj = 0 is computed as follows:P(x_urj = 0 | D, X_∖ urj, Z, α, β, γ, ρ) ∝ρ + N_u0 ∖ urj/2ρ + N_u - 1N_z_ura_urj∖ urj + β/N_z_ur∖ urj + β |A|,where ∖ urj represents the procedure excluding the jth song in the rth session of u. Similarly, the probability at which x_urj = 1 is computed as follows:P(x_urj = 1 | D, X_∖ urj, Z, α, β, γ, ρ) ∝ρ + N_u1 ∖ urj/2ρ + N_u - 1N_u1a_urj∖ urj + γ/N_u1 ∖ urj + γ |A|.Finally, we can make the point estimates of the integrated out parameters as follows:θ_uk = R_uk + α/R_u + α K,ϕ_ka = N_ka + β/N_k + β |A|,ψ_ua = N_u1a + γ/N_u1 + γ |A|. λ_u0 = N_u0 + ρ/N_u + 2ρ, λ_u1 = N_u1 + ρ/N_u + 2ρ,where remind that λ_u0 and λ_u1 represent the ratio of usual taste in music and addiction when u chooses songs, respectively.§ QUANTITATIVE EXPERIMENTS In this section, we answer the following research question based on our quantitative experimental results: is adopting two factors, which are users' daily taste in music and addiction to artists, effective to model music listening behavior? §.§ Dataset To examine the effectiveness of the proposed model, we constructed two datasets. The first one is created from music play logs on a music download service in Japan. On the service, users can buy a single song and an album and listen to them.For this evaluation, we obtained 10 weeks of log data between 1/1/2016 and 10/3/2016. We call this dataset JPD. The second one consists of logs on Last.fm. To guarantee the repeatability, we used a publicly available music play log data on Last.fm provided by Schedl <cit.>. Similar with JPD, we extracted 10 weeks of log data between 1/1/2013 and 11/3/2013; we call the dataset LFMD.From the 10 weeks of data of JPD, we created two pairs of training and test datasets as follows. In the first/second dataset, the training dataset consists of logs of the first four/eight weeks and the test dataset consists of the next two weeks. For each dataset, we excluded artists whose songs were played by ≤ 3 users and created session data as described in Section <ref>. Let the first and second dataset be 4WJPD (4W means four weeks) and 8WJPD, respectively. As for LFMD, we also created two pairs of training and test datasets 4WLFMD and 8WLFMD in the same manner as we created the 4WJPD and 8WJPD datasets. Table <ref> shows the statistics of the four datasets. §.§ Settings In terms of hyperparameters, in line with other topic modeling work, we set α = 1/K and β = 50/|A| in the session model and the session with addiction (SWA) model. In addition, in the SWA model, we set γ = 50/|A| and ρ = 0.5.To compare the performance of the session model andthe SWA model, we use the perplexities of the two models. Perplexity is a widely used measure to compare the performance of statistical models <cit.> and the lower value represents the better performance. The perplexity of each model on the test data is given by:perplexity(D_ test) = exp( - ∑_u ∈ U∑_r=1^R_u^ test∑_j=1^V_ur^ test p(a_urj)/∑_u ∈ U∑_r=1^R_u^ test |V_ur^ test|),where R_u^ test and V_ur^ test represent the number of u's sessions and the number of logs in rth session of u in the test data, respectively. The p(a_urj) is computed based on the estimated parameters obtained by Equation (<ref>) and (<ref>) as follows:p(a_urj) = λ_u0∑_k=1^Kθ_ukϕ_ka_urj + λ_u1ψ_ua_urj.In terms of the number of topics, we compute the perplexity for K= 5, 10, 20, 30, 40, 50, 100, 200, and 300. §.§ Results Figure <ref> shows the perplexity for each dataset. In any dataset, regardless of the amount of training data and the number of topics, the SWA model outperformed the session model. If we set the number of topics to be larger than 300, the session model might outperform the SWA model; but we set the maximum value of K to 300 for the following two reasons. The first reason is due to the expended hours for the learning process. For example, when the session model learns parameters for K=300 using 8WJPD, it takes 9.8 times longer than the SWA model does for K=30 using 8WJPD (1,713 minutes for the session model and 175 minutes for the SWA model). In data analysis, the expended hours is an important factor; if it takes a long time to learn the parameters for a model, the model is inappropriate for data analysis. The second reason is due to the understandability of topics. When the number of topics becomes too large, it is difficult to understand the difference between topics because there are many similar topics. As we will show in Section <ref>, analyzing the characteristics of each topic is useful to understand people's music listening behavior. Hence, it is undesirable to set K to a large value. For these reasons, we conclude that the SWA model is a better model than the session model. § QUALITATIVE EXPERIMENTS In this section, we report on the qualitative analysis results in terms of user characteristics, artist characteristics, and topic characteristics. Due to the space limitation, we only show the results for the training data of 8WJPD with K=30. We not only analyze people's music listening behavior but discuss how we can apply the analysis results. §.§ User Characteristics As we mentioned in Section <ref>, each user has a parameter λ that controls the degree of usual taste in music and addiction when s/he chooses songs. Given a user u, we can obtain the ratio of these two factors from Equation (<ref>), where λ_u0 + λ_u1 = 1. Figure <ref> (a) shows a histogram based on the degree of addiction. Although most people put a high priority on their usual taste in music (ratio ≤ 0.1), the second highest histogram peak is for those who put the greatest weight on addiction to artists (ratio > 0.9). The result where so many users lie somewhere between these two extremes of behavior further indicates the usefulness of considering the addiction mode in music listening behavior.By using the posterior distribution of latent variables in Equation (<ref>) and (<ref>), we can analyze the relationship between the degree of addiction and the time. We first analyzed the transition of the degree of addiction on a per-hour basis. For example, to analyze the degree between 9:00:00 and 9:59:59, we collected all play logs during the time period in the training data. By summing p(x=0) of all logs, we can obtain the strength of usual taste in music during the time period. Similarly, by summing p(x=1) of all logs, we can obtain the strength of addiction during the time period. Finally, we normalize their sum to 1 so that we can see the ratio of the degree of the two factors. The left line chart in Figure <ref> shows the results. It can be observed that the degree of addiction is high in the early morning (, at 5, 6, and 7 am), while it is low at night (, at 9, 10, and 11 pm). We can estimate that people tend to be short on time in the morning, and as a result, they listen to a specific artist's songs rather than choosing various songs according to a topic. On the other hand, at night, people have time to spare and tend to listen to various artists' songs by choosing from a topic. These results indicate that the transition of the degree of addiction on a per-hour basis enables us to analyze people's music listening behavior from a new viewpoint. In addition, we propose applying the knowledge to music recommendation. For example, it would be more appropriate to recommend unknown songs to the user at night rather than in the morning because s/he would have time to try listening to new songs.In the same manner as the above analysis, we also analyzed the transition of the degree of addiction on a day of the week basis. The right line chart in Figure <ref> shows the result. It can be observed that the degree of addiction is high on weekdays, while it is low on weekends. We can also estimate that the degree of addiction is high on weekdays because people are busy working on weekdays, while the degree is low on weekends because people have more time. These results would also be useful to recommend music. §.§ Artist Characteristics In the same way as Section <ref>, given an artist, by summing p(x=0) and p(x=1) of all the artist's logs, we can obtain the strength of usual taste and addiction during the time period, respectively. Then their sum is normalized to 1 to compute the ratio of each factor of the artist. Figure <ref> (b) shows a histogram based on the degree of addiction. It can be observed that most artists have a high degree of addiction. From these results, we can estimate whether the artist's songs are repeatedly played by users who are enthusiastic admirers of the artist or by various users who listen to the artist's songs with other artists' songs. In addition, we believe that the results could be used as one of the features to compute the similarity between artists by assuming that similar artists have similar degrees of addiction. §.§ Topic Characteristics Finally, we show that our model can also be used for topic analysis. Given a topic k, we collected representative artists in the category. To be more specific, the top 20 artists in terms of ϕ_k were extracted. For each of the 20 artists, we collected all logs in the training data and computed the ratio of the degree of taste in music and addiction as described in Section <ref>. We then computed the average values of each degree over 20 artists and normalized their sum to 1. Figure <ref> shows the ratio of 30 topics, where topics are sorted in ascending order of addiction ratio. As can be seen, the ratio between two factors is largely different from one topic to another: the addiction ratio ranged from 0.297 (10th topic) to 0.620 (17th topic). As for the low addiction topics, the 10th topic has the lowest value of 0.297. This topic is related to songs created by using VOCALOID <cit.>, which is popular singing synthesizer software in Japan. The 8th topic has the second lowest value of 0.334 and its topic is related to anime songs. From these results, we can estimate that when people listen to music related to popular culture, they tend to listen to various artists' songs in the topic. As for the high addiction topic, the 17th topic, which is related to Western artists, and the 28th topic, which is related to old Japanese artists, have the highest values of 0.620 and 0.592, respectively. These results indicate the possibility of applying the knowledge to playlist generation. In topics with a high addiction degree, it would be useful to generate a playlist that consists of songs of a specific artist; while in topics with a low addiction degree, it would be useful to generate a playlist that consists of various artists' songs.§ CONCLUSIONIn this paper we proposed a probabilistic model for analyzing people's music listening behavior. The model incorporates the user's usual taste in music and addiction to artists. Our experimental results using real-world music play logs showed that our model outperformed an existing model that considers only the user's taste in terms of perplexity. In our qualitative experiments, we showed the usefulness of our model in various aspects: time-dependent play log analysis (, the degree of addiction is high in the early morning and on weekdays), topic-dependent play log analysis (, the degree of addiction is low in an anime song topic), etc.For future work, we are interested in applying the knowledge obtained from log analysis to applications such as artist similarity computation and song recommendation as discussed in Section <ref>. We are also interested in extending our model by considering the time transition of addiction. For example, a user who is addicted to some artists in summer may be addicted to largely different artists in autumn. Considering such time dependency by using the topic tracking model <cit.> is one possible direction to take to extend our model.§ ACKNOWLEDGEMENTSThis work was supported in part by ACCEL, JST.abbrv | http://arxiv.org/abs/1705.09439v1 | {
"authors": [
"Kosetsu Tsukuda",
"Masataka Goto"
],
"categories": [
"cs.AI",
"cs.LG"
],
"primary_category": "cs.AI",
"published": "20170526055420",
"title": "Taste or Addiction?: Using Play Logs to Infer Song Selection Motivation"
} |
http://arxiv.org/abs/1705.09676v1 | {
"authors": [
"Jeremie Choquette"
],
"categories": [
"astro-ph.CO",
"astro-ph.HE",
"hep-ph"
],
"primary_category": "astro-ph.CO",
"published": "20170526182759",
"title": "Constraining Dwarf Spheroidal Dark Matter Halos With The Galactic Center Excess"
} |
|
Together We Know How to Achieve:An Epistemic Logic of Know-How Pavel Naumov Jia Tao================================================================== The existence of a coalition strategy to achieve a goal does not necessarily mean that the coalition has enough information to know how to follow the strategy. Neither does it mean that the coalition knows that such a strategy exists. The article studies an interplay between the distributed knowledge, coalition strategies, and coalition “know-how" strategies. The main technical result is a sound and complete trimodal logical system that describes the properties of this interplay. Together We Know How to Achieve:An Epistemic Logic of Know-How Pavel Naumov Jia Tao==================================================================§ INTRODUCTION An agent a comes to a fork in a road. There is a sign that says that one of the two roads leads to prosperity, another to death. The agent must take the fork, but she does not know which road leads where. Does the agent have a strategy to get to prosperity? On one hand, since one of the roads leads to prosperity, such a strategy clearly exists. We denote this fact by modal formula _a p, where statement p is a claim of future prosperity. Furthermore, agent a knows that such a strategy exists. We write this as _a_a p. Yet, the agent does not know what the strategy is and, thus, does not know how to use the strategy. We denote this by _a p, where know-how modality _a expresses the fact that agent a knows how to achieve the goal based on the information available to her.In this article we study the interplay between modality , representing knowledge, modality , representing the existence of a strategy, and modality , representing the existence of a know-how strategy. Our main result is a complete trimodal axiomatic system capturing properties of this interplay. §.§ Epistemic Transition Systems In this article we use epistemic transition systems to capture knowledge and strategic behavior. Informally, epistemic transition system is a directed labeled graph supplemented by an indistinguishability relation on vertices. For instance, our motivational example above can be captured by epistemic transition system T_1 depicted in Figure <ref>. In this system state w represents the prosperity and state w' represents death. The original state is u, but it is indistinguishable by the agent a from state v. Arrows on the diagram represent possible transitions between the states. Labels on the arrows represent the choices that the agents make during the transition. For example, if in state u agent chooses left (L) road, she will transition to the prosperity state w and if she chooses right (R) road, she will transition to the death state w'. In another epistemic state v, these roads lead the other way around. States u and v are not distinguishable by agent a, which is shown by the dashed line between these two states. In state u as well as state v the agent has a strategy to transition to the state of prosperity: u⊩_a p and v⊩_a p. In the case of state u this strategy is L, in the case of state v the strategy is R. Since the agent cannot distinguish states u and v, in both of these states she does not have a know-how strategy to reach prosperity: u⊮_a p and v⊮_a p. At the same time, since formula _a p is satisfied in all statesindistinguishable to agent a from state u, we can claim that u⊩_a_a p and, similarly, v⊩_a_a p. As our second example, let us consider the epistemic transition system T_2 obtained from T_1 by swapping labels on transitions from v to w and from v to w', see Figure <ref>. Although in system T_2 agent a still cannot distinguish states u and v, she has a know-how strategy from either of these states to reach state w. We write this as u⊩_a p and v⊩_a p. The strategy is to choose L. This strategy is know-how because it does not require to make different choices in the states that the agent cannot distinguish.§.§ Imperfect RecallFor the next example, we consider a transition system T_3 obtained from system T_1 by adding a new epistemic state s. From state s, agent a can choose label L to reach state u or choose label R to reach state v. Since proposition q is satisfied in state u, agent a has a know-how strategy to transition from state s to a state (namely, state u) where q is satisfied. Therefore, s⊩_a q. A more interesting question is whether s⊩_a_a p is true. In other words, does agent a know how to transition from state s to a state in which she knows how to transition to another state in which p is satisfied? One might think that such a strategy indeed exists: in state s agent a chooseslabel L to transition to state u. Since there is no transition labeled by L that leads from state s to state v, upon ending the first transition the agent would know that she is in state u, where she needs to choose label L to transition to state w. This argument, however, is based on the assumption that agent a has a perfect recall. Namely, agent a in state u remembers the choice that she made in the previous state. We assume that the agents do not have a perfect recall and that an epistemic state description captures whatever memories the agent has in this state. In other words, in this article we assume that the only knowledge that an agent possesses is the knowledge captured by the indistinguishability relation on the epistemic states. Given this assumption, upon reaching the state u (indistinguishable from state v) agent a knows that there exists a choice that she can make to transition to state in which p is satisfied: s⊩_a_a p. However, she does not know which choice (L or R) it is: s⊮_a_a p.§.§ Multiagent Setting So far, we have assumed that only agent a has an influence on which transition the system takes. In transition system T_4 depicted in Figure <ref>, we introduce another agent b and assume both agents a and b have influence on the transitions. In each state, the system takes the transition labeled D by default unless there is a consensus of agents a and b to take the transition labeled C. In such a setting, each agent has a strategy to transition system from state u into state w by voting D, but neither of them alone has a strategy to transition from state u to state w' because such a transition requires the consensus of both agents. Thus, u⊩_a p∧_b p∧_a q∧_b q. Additionally, both agents know how to transition the system from state u into state w, they just need to vote D. Therefore, u⊩_a p∧_b p.In Figure <ref>, we show a more complicated transition system obtained from T_1 by renaming label L to D and renaming label R to C. Same as in transition system T_4, we assume that there are two agents a and b voting on the system transition. We also assume that agent a cannot distinguish states u and v while agent b can. By default, the system takes the transition labeled D unless there is a consensus to take transition labeled C. As a result, agent a has a strategy (namely, vote D) in state u to transition system to state w, but because agent a cannot distinguish state u from state v, not only does she not know how to do this, but she is not aware that such a strategy exists: u⊩_a p∧_a p ∧_a_a p. Agent b, however, not only has a strategy to transition the system from state u to state w, but also knows how to achieve this: u⊩_b p.§.§ Coalitions We have talked about strategies, know-hows, and knowledge of individual agents. In this article we consider knowledge, strategies, and know-how strategies of coalitions. There are several forms of group knowledge that have been studied before. The two most popular of them are common knowledge and distributed knowledge <cit.>. Different contexts call for different forms of group knowledge.As illustrated in the famous Two Generals' Problem <cit.> where communication channels between the agents are unreliable, establishing a common knowledge between agents might be essential for having a strategy. In some settings, the distinction between common and distributed knowledge is insignificant. For example, if members of a political fraction get together to share all their information and to develop a common strategy, then the distributed knowledge of the members becomes the common knowledge of the fraction during the in-person meeting.Finally, in some other situations the distributed knowledge makes more sense than the common knowledge. For example, if a panel of experts is formed to develop a strategy, then this panel achieves the best result if it relies on the combined knowledge of its members rather than on their common knowledge.In this article we focus on distributed coalition knowledge and distributed-know-how strategies. We leave the common knowledge for the future research.To illustrate how distributed knowledge of coalitions interacts with strategies and know-hows, consider epistemic transition system T_6 depicted in Figure <ref>. In this system, agents a and b cannot distinguish states u and v while agents b and c cannot distinguish states v and u'. In every state, each of agents a, b and c votes either L or R, and the system transitions according to the majority vote. In such a setting, any coalition of two agents can fully control the transitions of the system. For example, by both voting L, agents a and b form a coalition {a,b} that forces the system to transition from state u to state w no matter how agent c votes. Since proposition p is satisfied in state w, we write u⊩_{a,b} p, or simply u⊩_a,b p. Similarly, coalition {a,b} can vote R to force the system to transition from state v to state w. Therefore, coalition {a,b} has strategies to achieve p in states u and v, but the strategies are different. Since they cannot distinguish states u and v, agents a and b know that they have a strategy to achieve p, but they do not know how to achieve p. In our notations, v⊩ S_a,bp∧_a,bS_a,bp ∧_a,b p. On the other hand, although agents b and c cannot distinguish states v and u', by both voting R in either of states v and u', they form a coalition {b, c} that forces the system to transition to state w where p is satisfied. Therefore, in any of states v and u', they not only have a strategy to achieve p, but also know that they have such a strategy, and more importantly, they know how to achieve p, that is, v⊩_b,c p. §.§ Nondeterministic Transitions In all the examples that we have discussed so far, given any state in a system, agents' votes uniquely determine the transition of the system. Our framework also allows nondeterministic transitions. Consider transition system T_7 depicted in Figure <ref>. In this system, there are two agents a and b who can vote either C or D. If both agents vote C, then the system takes one of the consensus transitions labeled with C. Otherwise, the system takes the transition labeled with D. Note that there are two consensus transitions starting from state u. Therefore, even if both agents vote C, they do not have a strategy to achieve p, i.e., u⊮_a,bp. However, they can achieve p∨ q. Moreover, since all agents can distinguish all states, we have u ⊩_a,b(p∨ q).§.§ Universal Principles In the examples above we focused on specific properties that were either satisfied or not satisfied in particular states of epistemic transition systems T_1 through T_7. In this article, we study properties that are satisfied in all states of all epistemic transition systems. Our main result is a sound and complete axiomatization of all such properties. We finish the introduction with an informal discussion of these properties.Properties of Single Modalities Knowledge modality K_C satisfies the axioms of epistemic logic S5 with distributed knowledge. Both strategic modality S_C and know-how modality _C satisfy cooperation properties <cit.>:_C(ϕ→ψ)→(_Dϕ→_C∪ Dψ),C∩ D=∅,_C(ϕ→ψ)→(_Dϕ→_C∪ Dψ),C∩ D=∅.They also satisfy monotonicity properties_Cϕ→_Dϕ,C⊆ D, _Cϕ→_Dϕ,C⊆ D.The two monotonicity properties are not among the axioms of our logical system because, as we show in Lemma <ref> and Lemma <ref>, they are derivable. Properties of InterplayNote that w⊩_Cϕ means that coalition C has the same strategy to achieve ϕ in all epistemic states indistinguishable by the coalition from state w. Hence, the following principle is universally true:_Cϕ→ K_C_Cϕ.Similarly, w⊩_Cϕ means that coalition C does not have the same strategy to achieve ϕ in all epistemic states indistinguishable by the coalition from state w. Thus,_Cϕ→ K_C_Cϕ.We call properties (<ref>) and (<ref>) strategic positive introspection and strategic negative introspection, respectively.The strategic negative introspection is one of our axioms. Just as how the positive introspection principle follows from the rest of the axioms in S5 (see Lemma <ref>), the strategic positive introspection principle is also derivable (see Lemma <ref>).Whenever a coalition knows how to achieve something, there should exist a strategy for the coalition to achieve. In our notation,_Cϕ→_Cϕ.We call this formula strategic truth property and it is one of the axioms of our logical system.The last two axioms of our logical system deal with empty coalitions. First of all, if formula _∅ϕ is satisfied in an epistemic state of our transition system, then formula ϕ must be satisfied in every state of this system. Thus, even empty coalition has a trivial strategy to achieve ϕ:_∅ϕ→_∅ϕ.We call this property empty coalition principle. In this article we assume that an epistemic transition system never halts. That is, in every state of the system no matter what the outcome of the vote is, there is always a next state for this vote. This restriction on the transition systems yields property _C.that we call nontermination principle.Let us now turn to the most interesting and perhaps most unexpected property of interplay. Note that _∅ϕ means that an empty coalition has a strategy to achieve ϕ. Since the empty coalition has no members, nobody has to vote in a particular way. Statement ϕ is guaranteed to happen anyway. Thus, statement _∅ϕ simply means that statement ϕ is unavoidably satisfied after any single transition.For example, consider an epistemic transition system depicted in Figure <ref>. As in some of our earlier examples, this system has agents a and b who vote either C or D. If both agents vote C, then the system takes one of the consensus transitions labeled with C. Otherwise, the system takes the default transition labeled with D. Note that in state v it is guaranteed that statement p will happen after a single transition. Thus, v⊩_∅ p. At the same time, neither agent a nor agent b knows about this because they cannot distinguish state v from states u and u' respectively. Thus, v⊩_a_∅ p ∧_b_∅ p. In the same transition system T_8, agents a and b together can distinguish state v from states u and u'. Thus, v⊩_a,b_∅ p.In general,statement _C_∅ϕ means that not only ϕ is unavoidable, but coalition C knows about it. Thus, coalition C has a know-how strategy to achieve ϕ:_C_∅ϕ→_Cϕ.In fact, the coalition would achieve the result no matter which strategy it uses. Coalition C can even use a strategy that simultaneously achieves another result in addition to ϕ: _C_∅ϕ∧_Cψ→_C(ϕ∧ψ).In our logical system we use an equivalent form of the above principle that is stated using only implication:_C(ϕ→ψ)→(_C_∅ϕ→_Cψ).We call this property epistemic determinicity principle. Properties (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), together with axioms of epistemic logic S5 with distributed knowledge and propositional tautologies constitute the axioms of our sound and complete logical system.§.§ Literature Review Logics of coalition power were developed by Marc Pauly <cit.>, who also proved the completeness of the basic logic of coalition power.Pauly's approach has been widely studied in the literature <cit.>. An alternative logical systemwas proposed by More and Naumov <cit.>. Alur, Henzinger, and Kupferman introduced Alternating-Time Temporal Logic (ATL) that combines temporal and coalition modalities <cit.>. Van der Hoek and Wooldridge proposed to combine ATL with epistemic modality to form Alternating-Time Temporal Epistemic Logic <cit.>. They did not prove the completeness theorem for the proposed logical system. Ågotnes and Alechina proposed a complete logical system that combines the coalition power and epistemic modalities <cit.>. Since this system does not have epistemic requirements on strategies, it does not contain any axioms describing the interplay of these modalities.Know-how strategies were studied before under different names. While Jamroga and Ågotnes talked about “knowledge to identify and execute a strategy" <cit.>,Jamroga and van der Hoek discussed “difference between an agent knowing that he has a suitable strategy and knowing the strategy itself" <cit.>. Van Benthem called such strategies “uniform" <cit.>. Wang gave a complete axiomatization of “knowing how" as a binary modality <cit.>, but his logical system does not include the knowledge modality.In our AAMAS paper, we investigated coalition strategies to enforce a condition indefinitely <cit.>. Such strategies are similar to “goal maintenance" strategies in Pauly's “extended coalition logic" <cit.>. We focused on “executable" and “verifiable" strategies. Using the language of the current article, executability means that a coalition remains “in the know-how" throughout the execution of the strategy. Verifiability means that the coalition can verify that the enforced condition remains true. In the notations of the current article, the existence of a verifiable strategy could be expressed as _C_Cϕ.In <cit.>, we provided a complete logical system that describes the interplay between the modality representing the existence of an “executable" and “verifiable" coalition strategy to enforce and the modality representing knowledge. This system can prove principles similar to the strategic positive introspection (<ref>) and the strategic negative introspection (<ref>) mentioned above. A similar complete logical system in a single-agent setting for strategies to achieve a goal in multiple steps rather than to maintain a goal is developed by Fervari, Herzig,Li,andWang <cit.>. In the current article, we combine know-how modalitywith strategic modalityand epistemic modality . The proof of the completeness theorem is significantly more challenging than in <cit.>. It employs new techniques that construct pairs of maximal consistent sets in “harmony" and in “complete harmony". See Section <ref> and Section <ref> for details. An extended abstract of this article, without proofs, appeared as <cit.>. §.§ Outline This article is organized as follows. In Section <ref> we introduce formal syntax and semantics of our logical system. In Section <ref> we list axioms and inference rules of the system. Section <ref> provides examples of formal proofs in our logical systems.Proofs of the soundness and the completeness are given in Section <ref> and Section <ref> respectively. Section <ref> concludes the article.The key part of the proof of the completeness is the construction of a pair of sets in complete harmony. We discuss the intuition behind this construction and introduce the notion of harmony in Section <ref>. The notion of complete harmony is introduced in Section <ref>.§ SYNTAX AND SEMANTICS In this section we present the formal syntax and semantics of our logical system given a fixed finite set of agents 𝒜. Epistemic transition system could be thought of as a Kripke model of modal logic S5 with distributed knowledge to which we add transitions controlled by a vote aggregation mechanism. Examples of vote aggregation mechanisms that we have considered in the introduction are the consensus/default mechanism and the majority vote mechanism. Unlike the introductory examples, in the general definition below we assume that at different states the mechanism might use different rules for vote aggregation. The only restriction on the mechanism that we introduce is that there should be at least one possible transition that the system can take no matter what the votes are. In other words, we assume that the system can never halt.For any set of votes V, by V^𝒜 we mean the set of all functions from set 𝒜 to set V. Alternatively, the set V^𝒜 could be thought of as a set of tuples of elements of V indexed by elements of 𝒜. A tuple (W,{∼_a}_a∈𝒜,V,M,π) is called an epistemic transition system, where * W is a set of epistemic states,* ∼_a is an indistinguishability equivalence relation on W for each a∈𝒜,* V is a nonempty set called “domain of choices", * M⊆ W× V^𝒜× W is an aggregation mechanism where for each w∈ W and each 𝐬∈ V^𝒜, there is w'∈ Wsuch that (w,𝐬,w')∈ M,* π is a function that maps propositional variables into subsets of W. A coalition is a subset of 𝒜. Note that a coalition is always finite due to our assumption that the set of all agents 𝒜 is finite. Informally, we say that two epistemic states are indistinguishable by a coalition C if they are indistinguishable by every member of the coalition. Formally, coalition indistinguishability is defined as follows:For any epistemic states w_1,w_2∈ W and any coalition C, let w_1∼_C w_2 if w_1∼_a w_2 for each agent a∈ C.Relation ∼_C is an equivalence relation on the set of states W for each coalition C.By a strategy profile {s_a}_a∈ C of a coalition C we mean a tuple that specifies vote s_a∈ V of each member a∈ C. Since such a tuple can also be viewed as a function from set C to set V, we denote the set of all strategy profiles of a coalition C by V^C: Any tuple {s_a}_a∈ C∈ V^C is called a strategy profile of coalition C. In addition to a fixed finite set of agents 𝒜 wealso assume a fixed countable set of propositional variables. We use the assumption that this set is countable in the proof of Lemma <ref>. The language Φ of our formal logical system is specified in the next definition.Let Φ be the minimal set of formulae such that * p∈Φ for each propositional variable p,* ϕ,ϕ→ψ∈Φ for all formulae ϕ,ψ∈Φ,* _Cϕ,_Cϕ,_Cϕ∈Φ for each coalition C and each ϕ∈Φ.In other words, language Φ is defined by the following grammar:ϕ := p | ϕ | ϕ→ϕ | _Cϕ | _Cϕ | _Cϕ. Bywe denote the negation of a tautology. For example, we can assume thatis (p→ p) for some fixed propositional variable p. According to Definition <ref>, a mechanism specifies the transition that a system might take for any strategy profile of the set of all agents 𝒜. It is sometimes convenient to consider transitions that are consistent with a given strategy profile 𝐬 of a give coalition C⊆𝒜. We write w→_𝐬u if a transition from state w to state u is consistent with strategy profile 𝐬. The formal definition is below. For any epistemic states w,u∈ W, any coalition C, and any strategy profile 𝐬={s_a}_a∈ C∈ V^C, we write w→_𝐬u if (w,𝐬',u)∈ M for some strategy profile 𝐬'={s'_a}_a∈𝒜∈ V^𝒜such that s'_a=s_a for each a∈ C. For any strategy profile 𝐬 of the empty coalition ∅, if there are a coalition C and a strategy profile 𝐬' of coalition C such that w→_𝐬' u, then w→_𝐬u. The next definition is the key definition of this article. It formally specifies the meaning of the three modalities in our logical system. For any epistemic state w∈ W of a transition system (W,{∼_a}_a∈𝒜,V,M,π) and any formula ϕ∈Φ, let relation w⊩ϕ be defined as follows * w⊩ p if w∈π(p) where p is a propositional variable,* w⊩ϕ if w⊮ϕ,* w⊩ϕ→ψ if w⊮ϕ or w⊩ψ,* w⊩_Cϕ if w'⊩ϕ for each w'∈ W such that w∼_C w',* w⊩_Cϕ if there is a strategy profile 𝐬∈ V^C such thatw→_𝐬 w' implies w'⊩ϕ for every w'∈ W,* w⊩_Cϕ if there is a strategy profile 𝐬∈ V^C such that w∼_C w' and w'→_𝐬 w” imply w”⊩ϕ for all w',w”∈ W.§ AXIOMS In additional to propositional tautologies in language Φ, our logical system consists of the following axioms.* Truth: _Cϕ→ϕ,* Negative Introspection: _Cϕ→_C_Cϕ,* Distributivity: _C(ϕ→ψ)→(_Cϕ→_Cψ),* Monotonicity: _Cϕ→_Dϕ, if C⊆ D,* Cooperation: _C(ϕ→ψ)→(_Dϕ→_C∪ Dψ), where C∩ D=∅.* Strategic Negative Introspection: _Cϕ→_C_Cϕ,* Epistemic Cooperation: _C(ϕ→ψ)→(_Dϕ→_C∪ Dψ),where C∩ D=∅,* Strategic Truth: _Cϕ→_Cϕ,* Epistemic Determinicity: _C(ϕ→ψ)→(_C_∅ϕ→_Cψ),* Empty Coalition: _∅ϕ→_∅ϕ,* Nontermination: _C.We have discussed the informal meaning of these axioms in the introduction. In Section <ref> we formally prove the soundness of these axioms with respect to the semantics from Definition <ref>.We write ⊢ϕ if formula ϕ is provable from the axioms of our logical system usingNecessitation, Strategic Necessitation, and Modus Ponens inference rules:ϕ_Cϕϕ_Cϕϕ,ϕ→ψψ.We write X⊢ϕ if formula ϕ is provable from the theorems of our logical system and a set of additional axioms X using only Modus Ponens inference rule.§ DERIVATION EXAMPLES In this section we give examples of formal derivations in our logical system. In Lemma <ref> we prove the strategic positive introspection principle (<ref>) discussed in the introduction. The proof is similar to the proof of the epistemic positive introspection principle in Lemma <ref>. ⊢_Cϕ→_C_Cϕ. Note that formula _Cϕ→_C_Cϕ is an instance of Strategic Negative Introspection axiom. Thus, ⊢_C_Cϕ→_Cϕ by the law of contrapositive in the propositional logic. Hence, ⊢_C(_C_Cϕ→_Cϕ) byNecessitation inference rule. Thus, byDistributivity axiom and Modus Ponens inference rule, ⊢_C_C_Cϕ→_C_Cϕ. At the same time, _C_Cϕ→_Cϕ is an instance of Truth axiom. Thus, ⊢_Cϕ→_C_Cϕ by contraposition. Hence, taking into account the following instance of Negative Introspection axiom _C_Cϕ→_C_C_Cϕ, one can conclude that ⊢_Cϕ→_C_C_Cϕ. The latter, together with statement (<ref>), implies the statement of the lemma by the laws of propositional reasoning. In the next example, we show that the existence of a know-how strategy by a coalition implies that the coalition has a distributed knowledge of the existence of a strategy. ⊢_Cϕ→_C_Cϕ. By Strategic Truth axiom, ⊢_Cϕ→_Cϕ. Hence, ⊢_C(_Cϕ→_Cϕ) by Necessitation inference rule. Thus, ⊢_C_Cϕ→_C_Cϕ by Distributivity axiom and Modus Ponens inference rule. At the same time, ⊢_Cϕ→_C_Cϕ by Lemma <ref>. Therefore, ⊢_Cϕ→_C_Cϕ by the laws of propositional reasoning. The next lemma shows that the existence of a know-how strategy by a sub-coalition implies the existence of a know-how strategy by the entire coalition. ⊢_Cϕ→_Dϕ, where C⊆ D. Note that ϕ→ϕ is a propositional tautology. Thus, ⊢ϕ→ϕ. Hence, ⊢_D∖ C(ϕ→ϕ) by Strategic Necessitation inference rule. At the same time, by Epistemic Cooperation axiom, ⊢_D∖ C(ϕ→ϕ)→(_Cϕ→_Dϕ) due to the assumption C⊆ D.Therefore, ⊢_Cϕ→_Dϕ by Modus Ponens inference rule. Although our logical system has three modalities, the system contains necessitation inference rulesonly for two of them. The lemma below shows that the necessitation rule for the third modality is admissible.For each finite C⊆𝒜, inference rule ϕ_Cϕ is admissible in our logical system. Assumption ⊢ϕ implies ⊢_Cϕ by Strategic Necessitation inference rule. Hence, ⊢_Cϕ by Strategic Truth axiom and Modus Ponens inference rule. The next result is a counterpart of Lemma <ref>. It states that the existence of a strategy by a sub-coalition implies the existence of a strategy by the entire coalition. ⊢_Cϕ→_Dϕ, where C⊆ D. Note that ϕ→ϕ is a propositional tautology. Thus, ⊢ϕ→ϕ. Hence,⊢_D∖ C(ϕ→ϕ) by Lemma <ref>. At the same time, by Cooperation axiom, ⊢_D∖ C(ϕ→ϕ)→(_Cϕ→_Dϕ) due to the assumption C⊆ D.Therefore, ⊢_Cϕ→_Dϕ by Modus Ponens inference rule.§ SOUNDNESS In this section we prove the soundness of our logical system. The proof of the soundness of multiagent S5 axioms andinference rules is standard. Below we show the soundness of each of the remaining axioms and the Strategic Necessitation inference rule as a separate lemma. The soundness theorem for the whole logical system is stated at the end of this section as Theorem <ref>. If w⊩_C(ϕ→ψ), w⊩_Dϕ, and C∩ D=∅, then w⊩_C∪ Dψ. Suppose that w⊩_C(ϕ→ψ). Then, by Definition <ref>, there is a strategy profile 𝐬^1={s^1_a}_a∈ C∈ V^C such that w'⊩ϕ→ψ for each w'∈ W where w→_𝐬^1w'. Similarly, assumption w⊩_Dϕ implies that there is a strategy 𝐬^2={s^2_a}_a∈ D∈ V^D such that w'⊩ϕ for each w'∈ W where w→_𝐬^2w'. Let strategy profile 𝐬={s_a}_a∈ C∪ D be defined as follows:s_a=s^1_a, a∈ C,s^2_a, a∈ D.Strategy profile 𝐬 is well-defined due to the assumption C∩ D=∅ of the lemma.Consider any epistemic state w'∈ W such that w→_𝐬w'. By Definition <ref>, it suffices to show that w'⊩ψ. Indeed, assumption w→_𝐬w', by Definition <ref>, implies that w→_𝐬^1w' and w→_𝐬^2w'. Thus, w'⊩ϕ→ψ and w'⊩ϕ by the choice of strategies 𝐬^1 and 𝐬^2. Therefore, w'⊩ψ by Definition <ref>. If w⊩_Cϕ, then w⊩_C_Cϕ. Consider any epistemic state u∈ W such that w∼_C u. By Definition <ref>, it suffices to show that u⊮_Cϕ. Assume the opposite. Thus, u⊩_Cϕ. Then, again by Definition <ref>, there is a strategy profile 𝐬∈ V^C where u”⊩ϕ for all u',u”∈ W such that u∼_C u' and u'→_𝐬 u”. Recall thatw∼_C u. Thus, by Corollary <ref>, u”⊩ϕ for all u',u”∈ W such that w∼_C u' and u'→_𝐬 u”. Therefore, w⊩_Cϕ, by Definition <ref>. The latter contradicts the assumption of the lemma.If w⊩_C(ϕ→ψ), w⊩_Dϕ, and C∩ D=∅, then w⊩_C∪ Dψ. Suppose that w⊩_C(ϕ→ψ). Thus, by Definition <ref>, there is a strategy profile 𝐬^1={s^1_a}_a∈ C∈ V^C such that w”⊩ϕ→ψ for all epistemic states w',w” where w∼_C w' and w'→_𝐬^1w”. Similarly, assumption w⊩_Dϕ implies that there is a strategy 𝐬^2={s^2_a}_a∈ D∈ V^D such that w”⊩ϕ for all w',w” where w∼_D w' and w'→_𝐬^2w”. Let strategy profile 𝐬={s_a}_a∈ C∪ D be defined as follows:s_a=s^1_a, a∈ C,s^2_a, a∈ D.Strategy profile 𝐬 is well-defined due to the assumption C∩ D=∅ of the lemma. Consider any epistemic states w',w”∈ W such that w∼_C∪ D w' and w'→_𝐬w”. By Definition <ref>, it suffices to show that w”⊩ψ. Indeed, by Definition <ref> assumption w∼_C∪ D w' implies that w∼_C w' and w∼_D w'. At the same time, by Definition <ref>, assumption w'→_𝐬w” implies that w'→_𝐬^1w” and w'→_𝐬^2w”. Thus, w”⊩ϕ→ψ and w”⊩ϕ by the choice of strategies 𝐬^1 and 𝐬^2. Therefore, w”⊩ψ by Definition <ref>. If w⊩_Cϕ, then w⊩_Cϕ. Suppose that w⊩_Cϕ. Thus, by Definition <ref>, there is a strategy profile 𝐬∈ V^C such that w”⊩ϕ for all epistemic states w',w”∈ W, where w∼_C w' and w'→_𝐬 w”. By Corollary <ref>, w∼_C w. Hence, w”⊩ϕ for each epistemic state w”∈ W, where w→_𝐬 w”. Therefore, w⊩_Cϕ by Definition <ref>.If w⊩_C(ϕ→ψ) and w⊩_C_∅ϕ, then w⊩_Cψ. Suppose that w⊩_C(ϕ→ψ). Thus, by Definition <ref>, there is a strategy profile 𝐬∈ V^C such that w”⊩ϕ→ψ for all epistemic states w',w”∈ W where w∼_C w' and w'→_𝐬 w”.Consider any epistemic states w'_0,w”_0∈ W such that w∼_C w'_0 and w'_0→_𝐬 w”_0. By Definition <ref>, it suffices to show that w”_0⊩ψ. Indeed, by Definition <ref>, the assumption w⊩_C_∅ϕ together with w∼_C w'_0 imply that w'_0⊩_∅ϕ. Hence, by Definition <ref>, there is a strategy profile 𝐬' of empty coalition ∅ such that w”⊩ϕ for each w” where w'_0→_𝐬'w”. Thus, w”_0⊩ϕ due to Corollary <ref> and w'_0→_𝐬 w”_0. By the choice of strategy profile 𝐬, statements w∼_C w'_0 and w'_0→_𝐬 w”_0 imply w”_0⊩ϕ→ψ.Finally, by Definition <ref>, statements w”_0⊩ϕ→ψ and w”_0⊩ϕ imply that w”_0⊩ψ.If w⊩_∅ϕ, then w⊩_∅ϕ. Let 𝐬={s_a}_a∈∅ be the empty strategy profile. Consider any epistemic states w',w”∈ W such that w∼_∅ w' and w'→_𝐬 w”. By Definition <ref>, it suffices to show that w”⊩ϕ. Indeed w∼_∅ w” by Definition <ref>. Therefore, w”⊩ϕ by assumption w⊩_∅ϕ and Definition <ref>.w⊮_C.Suppose that w⊩_C. Thus, by Definition <ref>, there is a strategy profile 𝐬={s_a}_a∈𝒜∈ V^C such that u⊩ for each u∈ W where w→_𝐬u. Note that by Definition <ref>, the domain of choices V is not empty. Thus, strategy profile 𝐬 can be extended to a strategy profile 𝐬'={s'_a}_a∈𝒜∈ V^𝒜 such that s'_a=s_a for each a∈ C. By Definition <ref>, there must exist a state w'∈ W such that (w,𝐬',w')∈ M. Hence, w→_𝐬w' by Definition <ref>. Therefore, w'⊩ by the choice of strategy 𝐬, which contradicts Definition <ref>.If w⊩ϕ for any epistemic state w∈ W of an epistemic transition system (W,{∼_a}_a∈𝒜,V,M,π), then w⊩_Cϕ for every epistemic state w∈ W. By Definition <ref>, set V is not empty. Let v∈ V. Consider strategy profile 𝐬={s_a}_a∈ C of coalition C such that s_a=v for each s∈ C. Note that w'⊩ϕ for each w'∈ W due to the assumption of the lemma. Therefore, w⊩_Cϕ by Definition <ref>.Taken together, the lemmas above imply the soundness theorem for our logical system stated below.If ⊢ϕ, then w⊩ϕ for each epistemic state w∈ W of each epistemic transition system (W,{∼_a}_a∈𝒜,V,M,π). § COMPLETENESS This section is dedicated to the proof of the following completeness theorem for our logical system. If w⊩ϕ for each epistemic state w of each epistemic transition system, then ⊢ϕ. §.§ Positive Introspection The proof of Theorem <ref> is divided into several parts. In this section we prove the positive introspection principle for distributed knowledge modality from the rest of modalityaxiomsin our logical system.This is a well-known result that we reproduce to keep the presentation self-sufficient. The positive introspection principle is used later in the proof of the completeness. ⊢_Cϕ→_C_Cϕ. Formula _Cϕ→_C_Cϕ is an instance of Negative Introspection axiom. Thus, ⊢_C_Cϕ→_Cϕ by the law of contrapositive in the propositional logic. Hence, ⊢_C(_C_Cϕ→_Cϕ) by Necessitation inference rule. Thus, byDistributivity axiom and Modus Ponens inference rule, ⊢_C_C_Cϕ→_C_Cϕ. At the same time, _C_Cϕ→_Cϕ is an instance of Truth axiom. Thus, ⊢_Cϕ→_C_Cϕ by contraposition. Hence, taking into account the following instance ofNegative Introspection axiom _C_Cϕ→_C_C_Cϕ, one can conclude that ⊢_Cϕ→_C_C_Cϕ. The latter, together with statement (<ref>), implies the statement of the lemma by the laws of propositional reasoning.§.§ Consistent Sets of Formulae The proof of the completeness consists in constructing a canonical model in which states are maximal consistent sets of formulae. This is a standard technique in modal logic that we modified significantly to work in the setting of our logical system. The standard way to apply this technique to a modal operatoris to create a “child" state w' such that ψ∈ w' for each “parent" state w where ψ∈ w. In the simplest case whenis a distributed knowledge modality _C, the standard technique requires no modification and the construction of a “child" state is based on the following lemma:For any consistent set of formulae X, any formula _Cψ∈ X, and any formulae _Cϕ_1,…,_Cϕ_n∈ X, the set of formulae {ψ,ϕ_1,…,ϕ_n} is consistent. Assume the opposite. Then,ϕ_1,…,ϕ_n⊢ψ. Thus, by the deduction theorem for propositional logic applied n times,⊢ϕ_1→(ϕ_2→…(ϕ_n→ψ)…).Hence, by Necessitation inference rule,⊢_C(ϕ_1→(ϕ_2→…(ϕ_n→ψ)…)). By Distributivity axiom and Modus Ponens inference rule,_Cϕ_1⊢_C(ϕ_2→…(ϕ_n→ψ)…). By repeating the last step (n-1) times,_Cϕ_1,…,_Cϕ_n⊢_Cψ.Hence, X⊢_Cψ by the choice of formula _Cϕ_1,…,_Cϕ_n, which contradicts the consistency of the set X due to the assumption _Cψ∈ X. Ifis the modality _C, then the standard technique needs to be modified. Namely, while _Cψ∈ w means that coalition C can not achieve goal ψ, its pairwise disjoint sub-coalitions D_1,…, D_n⊆ C might still achieve their own goals ϕ_1,…,ϕ_n. An equivalent of Lemma <ref> for modality _C is the following statement. For any consistent set of formulae X, and any subsets D_1,…,D_nof a coalition C, any formula _Cψ∈ X, and any _D_1ϕ_1,…,_D_nϕ_n∈ X, if D_i∩ D_j=∅ for all integers i,j≤ n such that i≠ j, then the set of formulae {ψ,ϕ_1,…,ϕ_n} is consistent. Suppose that ϕ_1,ϕ_2,…,ϕ_n⊢ψ. Hence, by the deduction theorem for propositional logic applied n times,⊢ϕ_1→(ϕ_2→(…(ϕ_n→ψ)…)).Then,⊢_D_1(ϕ_1→(ϕ_2→(…(ϕ_n→ψ)…))) by Lemma <ref>. Hence, by Cooperation axiom and Modus Ponens inference rule,⊢_D_1ϕ_1→_∅∪ D_1(ϕ_2→(…(ϕ_n→ψ)…)).In other words,⊢_D_1ϕ_1→_D_1(ϕ_2→(…(ϕ_n→ψ)…)).Then, by Modus Ponens inference rule,_D_1ϕ_1⊢_D_1(ϕ_2→(…(ϕ_n→ψ)…)).By Cooperation axiom and Modus Ponens inference rule, _D_1ϕ_1⊢_D_2ϕ_2→_D_1∪ D_2(…(ϕ_n→ψ)…).Again, by Modus Ponens inference rule, _D_1ϕ_1,_D_2ϕ_2 ⊢_D_1∪ D_2(…(ϕ_n→ψ)…).By repeating the previous steps n-2 times,_D_1ϕ_1,_D_2ϕ_2,…, _D_nϕ_n⊢_D_1∪ D_2∪…∪ D_nψ. Recall that _D_1ϕ_1,_D_2ϕ_2,…, _D_nϕ_n∈ X by the assumption of the lemma. Thus, X⊢_D_1∪ D_2∪…∪ D_nψ. Therefore, X⊢_Cψ by Lemma <ref>. Since the set X is consistent, the latter contradicts the assumption _Cψ∈ X of the lemma.§.§ Harmony Ifis the modality _C, then the standard technique needs even more significant modification. Namely, as it follows from Definition <ref>, assumption _Cψ∈ w requires us to create not a single child of parent w, but two different children referred in Definition <ref> as states w' and w”, see Figure <ref>. Child w' is a state of the system indistinguishable from state w by coalition C. Child w” is a state such that ψ∈ w” and coalition C cannot prevent the system to transition from w' to w”. One might think that states w' and w” could be constructed in order: first state w' and then state w”. It appears, however, that such an approach does not work because it does not guarantee that ψ∈ w”. To solve the issue, we construct states w' and w”simultaneously. While constructing states w' and w” as maximal consistent sets of formulae, it is important to maintain two relations between sets w' and w” that we call “to be in harmony" and “to be in complete harmony". In this section we define harmony relation and prove its basic properties. The next section is dedicated to the complete harmony relation.Even though according to Definition <ref> the language of our logical system only includes propositional connectivesand →, other connectives, including conjunction ∧, can be defined in the standard way. By ∧ Y we mean the conjunction of a finite set of formulae Y. If set Y is a singleton, then ∧ Y represents the single element of set Y. If set Y is empty, then ∧ Y is defined to be any propositional tautology. Pair (X,Y) of sets of formulae is in harmony if X⊬_∅∧ Y' for each finite set Y'⊆ Y.If pair (X,Y) is in harmony, then set X is consistent. If set X is not consistent, then any formula can be derived from it. In particular, X⊢_∅∧∅. Therefore, pair (X,Y) is not in harmony by Definition <ref>.If pair (X,Y) is in harmony, then set Y is consistent. Suppose that Y is inconsistent. Then, there is a finite set Y'⊆ Y such that ⊢∧ Y'. Hence, ⊢_∅∧ Y' by Lemma <ref>. Thus, X⊢_∅∧ Y'. Therefore, by Definition <ref>, pair (X,Y) is not in harmony.For any ϕ∈Φ, if pair (X,Y) is in harmony, then either pair (X∪{_∅ϕ},Y) or pair (X,Y∪{ϕ}) is in harmony. Suppose that neither pair (X∪{_∅ϕ},Y) nor pair (X,Y∪{ϕ}) is in harmony. Then, by Definition <ref>, there are finite sets Y_1⊆ Y and Y_2⊆ Y∪{ϕ} such thatX,_∅ϕ⊢_∅∧ Y_1and X⊢_∅∧ Y_2. Formula ∧ Y_1→((∧ Y_1)∧(∧ (Y_2∖{ϕ}))) is a propositional tautology. Thus, ⊢_∅(∧ Y_1→((∧ Y_1)∧(∧ (Y_2∖{ϕ})))) by Lemma <ref>. Then, by Cooperation axiom, statement (<ref>), and Modus Ponens inference rule, X,_∅ϕ⊢_∅∪∅((∧ Y_1)∧(∧ (Y_2∖{ϕ}))). In other words, X,_∅ϕ⊢_∅((∧ Y_1)∧(∧ (Y_2∖{ϕ}))). Finally, formula ∧ Y_2→ (ϕ→((∧ Y_1)∧(∧ (Y_2∖{ϕ})))) is also a propositional tautology. Thus, by Lemma <ref>, ⊢_∅(∧ Y_2→ (ϕ→((∧ Y_1)∧(∧ (Y_2∖{ϕ}))))). Then, by Cooperation axiom, statement (<ref>), and Modus Ponens inference rule, X⊢_∅(ϕ→((∧ Y_1)∧(∧ (Y_2∖{ϕ})))). Thus, by Cooperation axiom and Modus Ponens inference rule, X⊢_∅ϕ→_∅((∧ Y_1)∧(∧ (Y_2∖{ϕ}))).By Modus Ponens inference rule,X, _∅ϕ⊢_∅((∧ Y_1)∧(∧ (Y_2∖{ϕ}))).Hence, X⊢_∅((∧ Y_1)∧(∧ (Y_2∖{ϕ}))) by statement (<ref>) and the laws of propositional reasoning. Recall that Y_1 and Y_2∖{ϕ} are subsets of Y. Therefore, pair (X,Y) is not in harmony by Definition <ref>.The next lemma is an equivalent of Lemma <ref> and Lemma <ref> for modality _C. For any consistent set of formulae X, any formula _Cψ∈ X, and any function f:C→Φ, pair (Y,Z) is in harmony, whereY={ϕ | _Cϕ∈ X}, Z={ψ}∪{χ | ∃ D⊆ C (_Dχ∈ X ∧∀ a∈ D (f(a)=χ))}.Suppose that pair (Y,Z) is not in harmony. Thus, by Definition <ref>, there is a finite Z'⊆ Z such thatY⊢_∅∧ Z'. Since a derivation uses only finitely many assumptions, there are formulae K_Cϕ_1,_Cϕ_2…,_Cϕ_n∈ X such thatϕ_1,ϕ_2…,ϕ_n⊢_∅∧ Z'.Then, by the deduction theorem for propositional logic applied n times,⊢ϕ_1→(ϕ_2→(…→(ϕ_n→_∅∧ Z')…)).Hence, by Necessitation inference rule,⊢_C(ϕ_1→(ϕ_2→(…→(ϕ_n→_∅∧ Z')…))).Then, by Distributivity axiom and Modus Ponens inference rule,⊢_Cϕ_1→_C(ϕ_2→(…→(ϕ_n→_∅∧ Z')…)).Thus, by Modus Ponens inference rule,_Cϕ_1⊢_C(ϕ_2→(…→(ϕ_n→_∅∧ Z')…)).By repeating the previous two steps (n-1) times,_Cϕ_1,_Cϕ_2…, _Cϕ_n⊢_C_∅∧ Z'.Hence, by the choice of formulae K_Cϕ_1,_Cϕ_2,…,_Cϕ_n,X⊢_C_∅∧ Z'.Since set Z' is a subset of set Z, by the choice of set Z, there must exist formulae _D_1χ_1,…,_D_nχ_n∈ X such that D_1,…,D_n⊆ C,∀ i≤ n ∀ a∈ D_i (f(a)=χ_i),and the following formula is a tautology, even if ψ∉ Z':χ_1→(χ_2→…(χ_n→(ψ→∧ Z'))…).Without loss of generality, we can assume that formulae χ_1,…,χ_n are pairwise distinct.D_i∩ D_j=∅ for each i,j≤ n such that i≠ j.Proof of Claim. Suppose the opposite. Then, there is a∈ D_i∩ D_j. Thus, χ_i=f(a)=χ_j by statement (<ref>). This contradicts the assumption that formulae χ_1,…,χ_n are pairwise distinct. Since formula (<ref>) is a propositional tautology, by the law of contrapositive, the following formula is also a propositional tautology:χ_1→(χ_2→…(χ_n→(∧ Z'→ψ))…).Thus, by Strategic Necessitation inference rule,⊢_∅(χ_1→(χ_2→…(χ_n→(∧ Z'→ψ))…)).Hence, by Epistemic Cooperation axiom and Modus Ponens inference rule,⊢_D_1χ_1→_∅∪ D_1(χ_2→…(χ_n→(∧ Z'→ψ))…).Then, by Modus Ponens inference rule,_D_1χ_1⊢_ D_1(χ_2→…(χ_n→(∧ Z'→ψ))…).By Epistemic Cooperation axiom, Claim <ref>, and Modus Ponens inference rule,_D_1χ_1⊢_ D_2χ_2→_ D_1∪ D_2(…(χ_n→(∧ Z'→ψ))…).By Modus Ponens inference rule,_D_1χ_1,_ D_2χ_2⊢_ D_1∪ D_2(…(χ_n→(∧ Z'→ψ))…).By repeating the previous two steps (n-2) times,_D_1χ_1,_ D_2χ_2,…,_ D_nχ_n⊢_ D_1∪ D_2∪…∪ D_n(∧ Z'→ψ).Recall that _D_1χ_1,_D_2χ_2,…, _D_nχ_n∈ X by the choice of _D_1χ_1, …, _D_nχ_n. Thus, X⊢_D_1∪ D_2∪…∪ D_n(∧ Z'→ψ).Hence, because D_1,…,D_n⊆ C, by Lemma <ref>, X⊢_C(∧ Z'→ψ). Then, X⊢_Cψ by Epistemic Determinicity axiom and statement (<ref>). Since the set X is consistent, this contradicts the assumption _Cψ∈ X of the lemma.§.§ Complete HarmonyA pair in harmony (X,Y) is in complete harmony if for each ϕ∈Φ either _∅ϕ∈ X or ϕ∈ Y. For each pair in harmony (X,Y), there is a pair in complete harmony (X',Y') such that X⊆ X' and Y⊆ Y'. Recall that the set of agent 𝒜 is finite and the set of propositional variables is countable. Thus, the set of all formulae Φ is also countable. Let ϕ_1,ϕ_2,… be an enumeration of all formulae in Φ. We define two chains of sets X_1⊆ X_2⊆… and Y_1⊆ Y_2⊆… such that pair (X_n,Y_n) is in harmony for each n≥ 1. These two chains are defined recursively as follows:* X_1=X and Y_1=Y,* if pair (X_n,Y_n) is in harmony, then, by Lemma <ref>, either pair (X_n∪{_∅ϕ_n},Y_n) or pair (X_n,Y_n∪{ϕ_n}) is in harmony. Let (X_n+1,Y_n+1) be (X_n∪{_∅ϕ_n},Y_n) in the former case and (X_n,Y_n∪{ϕ_n}) in the latter case.Let X'=⋃_nX_n and Y'=⋃_n Y_n. Note that X=X_1⊆ X' and Y=Y_1⊆ Y'.We next show that pair (X',Y') is in harmony. Suppose the opposite. Then, by Definition <ref>, there is a finite set Y”⊆ Y' such that X'⊢_∅∧ Y”. Since a deduction uses only finitely many assumptions, there must exist n_1≥ 1 such that X_n_1⊢_∅∧ Y”.At the same time, since set Y” is finite, there must exist n_2≥ 1 such that Y”⊆ Y_n_2. Let n=max{n_1,n_2}. Note that ∧ Y”→∧ Y_n is a tautology because Y”⊆ Y_n_2⊆ Y_n. Thus,⊢_∅(∧ Y”→∧ Y_n) by Lemma <ref>. Then, ⊢_∅∧ Y”→_∅∧ Y_n byCooperation axiom and Modus Ponens inference rule. Hence, X_n_1⊢_∅∧ Y_n due to statement (<ref>).Thus, X_n⊢_∅∧ Y_n, because X_n_1⊆ X_n. Then, pair (X_n,Y_n) is not in harmony, which contradicts the choice of pair (X_n,Y_n). Therefore, pair (X',Y') is in harmony.We finally show that pair (X',Y') is in complete harmony. Indeed, consider any ϕ∈Φ. Since ϕ_1,ϕ_2,… is an enumeration of all formulae in Φ, there must exist k≥ 1 such that ϕ=ϕ_k. Then, by the choice of pair (X_k+1,Y_k+1), either _∅ϕ=_∅ϕ_k∈ X_k+1⊆ X' or ϕ=ϕ_k∈ Y_k+1⊆ Y'. Therefore, pair (X',Y') is in complete harmony. §.§ Canonical Epistemic Transition System The construction of a canonical model, called the canonical epistemic transition system, for the proof of the completeness is based on the “unravelling" technique <cit.>. Informally, epistemic states in this system are nodes in a tree. In this tree, each node is labeled with a maximal consistent set of formulae and each edge is labeled with a coalition. Formally, epistemic states are defined as sequences representing paths in such a tree. In the rest of this section we fix a maximal consistent set of formulae X_0 and define a canonical epistemic transition system ETS(X_0)=(W,{∼_a}_a∈𝒜,V,M,π). The set of epistemic states W consists of all finite sequences X_0,C_1,X_1,C_2,…,C_n,X_n, such that * n≥ 0,* X_i is a maximal consistent subset of Φ for each i≥ 1,* C_i is a coalition for each i≥ 1,* {ϕ | _C_iϕ∈ X_i-1}⊆ X_i for each i≥ 1.We say that two nodes of the tree are indistinguishable to an agent a if every edge along the unique path connecting these two nodes is labeled with a coalition containing agent a. For any state w=X_0,C_1,X_1,C_2,…,C_n,X_n and any state w'=X_0,C'_1,X'_1,C'_2,…,C'_m,X'_m, let w∼_a w' if there is an integer k such that * 0≤ k≤min{n,m},* X_i=X'_i for each i such that 1≤ i≤ k,* C_i=C'_i for each i such that 1≤ i≤ k,* a∈ C_i for each i such that k<i≤ n,* a∈ C'_i for each i such that k<i≤ m.For any state w=X_0,C_1,X_1,C_2,…,C_n,X_n, by hd(w) we denote the set X_n. The abbreviation hd stands for “head". For any w=X_0,C_1,X_1,C_2,…,C_n,X_n∈ W and any integer k≤ n, if _Cϕ∈ X_n and C⊆ C_i for each integer i such that k<i≤ n, then _Cϕ∈ X_k. Suppose that there is k≤ n such that _Cϕ∉ X_k. Let m be the maximal such k. Note that m<n due to the assumption _Cϕ∈ X_n of the lemma. Thus, m< m+1≤ n.Assumption _Cϕ∉ X_m implies _Cϕ∈ X_m due to the maximality of the set X_m. Hence, X_m⊢_C_Cϕ by Negative Introspection axiom. Thus, X_m⊢_C_m+1_Cϕ by Monotonicity axiom and the assumption C⊆ C_m+1 of the lemma (recall that m+1≤ n).Then, _C_m+1_Cϕ∈ X_m due to the maximality of the set X_m.Hence, _Cϕ∈ X_m+1 by Definition <ref>. Thus, _Cϕ∉ X_m+1 due to the consistency of the set X_m+1, which is a contradiction with the choice of integer m.For any w=X_0,C_1,X_1,C_2,…,C_n,X_n∈ W and any integer k≤ n, if _Cϕ∈ X_k and C⊆ C_i for each integer i such that k<i≤ n, then ϕ∈ X_n. We prove the lemma by induction on the distance between n and k. In the base case n=k. Then the assumption _Cϕ∈ X_n implies X_n⊢ϕ by Truth axiom. Therefore, ϕ∈ X_n due to the maximality of set X_n.Suppose that k<n. Assumption _Cϕ∈ X_k implies X_k⊢_C_Cϕ by Lemma <ref>. Thus, X_k⊢_C_k+1_Cϕ by Monotonicity axiom, the condition k<n of the inductive step, and the assumption C⊆ C_k+1 of the lemma. Then, _C_k+1_Cϕ∈ X_k by the maximality of set X_k.Hence, _Cϕ∈ X_k+1 by Definition <ref>. Therefore, ϕ∈ X_n by the induction hypothesis. If _Cϕ∈ hd(w) and w∼_Cw', then ϕ∈ hd(w'). The statement follows from Lemma <ref>, Lemma <ref>, and Definition <ref> because there is a unique path between any two nodes in a tree. At the beginning of Section <ref>, we discussed that if a parent node contains a modal formula ψ, then it must have a child node containing formula ψ. Lemma <ref> in Section <ref> provides a foundation for constructing such a child node for modality _C. The proof of the next lemma describes the construction of the child node for this modality. If _Cϕ∉ hd(w), then there is an epistemic state w'∈ W such that w∼_C w' and ϕ∉ hd(w'). Assumption _Cϕ∉ hd(w) implies that _Cϕ∈ hd(w) due to the maximality of the set hd(w). Thus, by Lemma <ref>, setY_0={ϕ}∪{ψ | _Cψ∈ hd(w)} is consistent. Let Y be a maximal consistent extension of set Y_0 and w' be sequence w,C,Y. In other words, sequence w' is an extension of sequence w by two additional elements: C and Y. Note that w'∈ W due to Definition <ref> and the choice of set Y_0. Furthermore, w∼_C w' by Definition <ref>. To finish the proof, we need to show that ϕ∉ hd(w'). Indeed, ϕ∈ Y_0⊆ Y=hd(w') by the choice of Y_0. Therefore, ϕ∉ hd(w') due to the consistency of the set hd(w'). In the next two definitions we specify the domain of votes and the vote aggregation mechanism of the canonical transition system. Informally, a vote (ϕ,w) of each agent consists of two components: the actual vote ϕ and a key w. The actual vote ϕ is a formula from Φin support of what the agent votes. Recall that the agent does not know in which exact state the system is, she only knows the equivalence class of this state with respect to the indistinguishability relation. The key w is the agent's guess of the epistemic state where the system is. Informally, agent's vote has more power to force the formula to be satisfied in the next state if she guesses the current state correctly.Although each agent is free to vote for any formula she likes, the vote aggregation mechanism would grant agent's wish only under certain circumstances. Namely, if the system is in state w and set hd(w) contains formula _Cϕ, then the mechanism guarantees that formula ϕ is satisfied in the next state as long as each member of coalition C votes for formula ϕ and correctly guesses the current epistemic state. In other words, in order for formula ϕ to be guaranteed in the next state all members of the coalition C must cast vote (ϕ,w). This means that if _Cϕ∈ hd(w), then coalition C has a strategy to force ϕ in the next state. Since the strategy requires each member of the coalition to guess correctly the current state, such a strategy is not a know-how strategy.The vote aggregation mechanism is more forgiving if the epistemic state w contains formula _Cϕ. In this case the mechanism guarantees that formula ϕ is satisfied in the next state if all members of the coalition vote for formula ϕ; it does not matter if they guess the current state correctly or not. This means that if _Cϕ∈ hd(w), then coalition C has a know-how strategy to force ϕ in the next state. The strategy consists in each member of the coalition voting for formula ϕ and specifying an arbitrary epistemic state as the key. Formal definitions of the domain of choices and of the vote aggregation mechanism in the canonical epistemic transition system are given below. The domain of choices V is Φ× W. For any pair u=(x,y), let pr_1(u)=x and pr_2(u)=y. The mechanism M of the canonical model is the set of all tuples (w,{s_a}_a∈𝒜,w') such that for each formula ϕ∈Φ and each coalition C, * if _Cϕ∈ hd(w) and s_a=(ϕ,w) for each a∈ C, then ϕ∈ hd(w'), and* if _Cϕ∈ hd(w) and pr_1(s_a)=ϕ for each a∈ C, then ϕ∈ hd(w').The next two lemmas prove that the vote aggregation mechanism specified in Definition <ref> acts as discussed in the informal description given earlier. Let w,w'∈ W be epistemic states, _Cϕ∈ hd(w) be a formula, and 𝐬={s_a}_a∈ C be a strategy profile of coalition C. If w→_𝐬 w' and s_a=(ϕ,w) for each a∈ C, then ϕ∈ hd(w'). Suppose that w→_𝐬 w'. Thus, by Definition <ref>, there is a strategy profile 𝐬'={s'_a}_a∈𝒜∈ V^𝒜 such that s'_a=s_a for each a∈ C and (w,𝐬',w')∈ M. Therefore, ϕ∈ hd(w') by Definition <ref> and the assumption s_a=(ϕ,w) for each a∈ C. Let w,w',w”∈ W be epistemic states, _Cϕ∈ hd(w) be a formula, and 𝐬={s_a}_a∈ C be a strategy profile of coalition C. If w∼_C w',w'→_𝐬 w”, and pr_1(s_a)=ϕ for each a∈ C, then ϕ∈ hd(w”). Suppose that _Cϕ∈ hd(w). Thus, hd(w)⊢_C_Cϕ by Lemma <ref>. Hence, _C_Cϕ∈ hd(w) due to the maximality of the set hd(w). Thus, _Cϕ∈ hd(w') by Lemma <ref> and the assumption w∼_C w'. By Definition <ref>, assumption w'→_𝐬 w”implies that there is a strategy profile 𝐬'={s'_a}_a∈𝒜 such that s'_a=s_a for each a∈ C and (w',𝐬',w”)∈ M.Since _Cϕ∈ hd(w'), pr_1(s'_a)=pr_1(s_a)=ϕ for each a∈ C, and (w',𝐬',w”)∈ M, we have ϕ∈ hd(w”) by Definition <ref>.The lemma below provides a construction of a child node for modality _C. Although the proof follows the outline of the proof of Lemma <ref> for modality _C, it is significantly more involved because of the need to show that a transition from a parent node to a child node satisfies the constraints of the vote aggregation mechanism from Definition <ref>. For any epistemic state w∈ W, any formula _Cψ∈ hd(w), and any strategy profile 𝐬={s_a}_a∈ C∈ V^C, there is a state w'∈ W such that w→_𝐬 w' and ψ∉ hd(w'). Let Y_0 be the following set of formulae {ψ}∪{ϕ | ∃ D⊆ C(_Dϕ∈ hd(w) ∧∀ a∈ D (pr_1(s_a)=ϕ))}.We first show that set Y_0 is consistent. Suppose the opposite. Thus, there must exist formulaeϕ_1,…,ϕ_n∈ Y_0 and subsets D_1,…,D_n⊆ C such that (i) _D_iϕ_i∈ hd(w) for each integer i≤ n, (ii) pr_1(s_a)=ϕ_i for each i≤ n and each a∈ D_i, and (iii) set {ψ,ϕ_1,…,ϕ_n} is inconsistent. Without loss of generality we can assume that formulaeϕ_1,…,ϕ_n are pairwise distinct.Sets D_i and D_j are disjoint for each i≠ j.Proof of Claim. Assume that d∈ D_i∩ D_j, then pr_1(s_d)=ϕ_i and pr_1(s_d)=ϕ_j. Hence, ϕ_i=ϕ_j, which contradicts the assumption that formulae ϕ_1,…,ϕ_n are pairwise distinct. Therefore, sets D_i and D_j are disjoint for each i≠ j. By Lemma <ref>, it follows from Claim <ref> that set Y_0 is consistent. Let Y be any maximal consistent extension of Y_0 and w' be the sequence w,∅,Y. In other words, w' is an extension of sequence w by two additional elements: ∅ and Y. w'∈ W.Proof of Claim. By Definition <ref>, it suffices to show that, for each formula ϕ∈Φ, if _∅ϕ∈ hd(w), then ϕ∈ Y. Indeed, suppose that _∅ϕ∈ hd(w). Thus, hd(w)⊢_∅ϕ by Empty Coalition axiom. Hence, hd(w)⊢_∅ϕ by Strategic Truth axiom. Then, _∅ϕ∈ hd(w) due to the maximality of set hd(w). Therefore, ϕ∈ Y_0⊆ Y by the choice of sets Y_0 and Y.Let ⊤ be any propositional tautology. For example, ⊤ could be formula ψ→ψ. Define strategy profile 𝐬'={s'_a}_a∈𝒜 as followss'_a=s_a, a∈ C,(⊤,w),.For any formula ϕ∈Φ and any D⊆𝒜, if _Dϕ∈ hd(w) and s'_a=(ϕ,w) for each a∈ D, then ϕ∈ hd(w').Proof of Claim.Consider any formula ϕ∈Φ and any set D⊆𝒜 such that _Dϕ∈ hd(w) and s'_a=(ϕ,w) for each agent a∈ D. We need to show that ϕ∈ hd(w').Case 1: D⊆ C. In this case, s_a=s'_a=(ϕ,w) for each a∈ D by definition (<ref>). Thus, ϕ∈ Y_0⊆ Y=hd(w') by the choice of set Y_0.Case 2: There is a_0∈ D such that a_0∉ C. Then, s'_a_0=(⊤,w) by definition (<ref>). Note that s'_a_0=(ϕ,w) by the choice of the set D. Thus, (⊤,w)=(ϕ,w). Hence, formula ϕ is the tautology ⊤. Therefore, ϕ∈ hd(w') because set hd(w') is maximal.For any formula ϕ∈Φ and any D⊆𝒜, if _Dϕ∈ hd(w) and pr_1(s'_a)=ϕ for each a∈ D, then ϕ∈ hd(w'). Proof of Claim.Consider any formula ϕ∈Φ and any set D⊆𝒜 such that _Dϕ∈ hd(w) and pr_1(s'_a)=ϕ for each agent a∈ D. We need to show that ϕ∈ hd(w').Case 1: D⊆ C. In this case, pr_1(s_a)=pr_1(s'_a)=ϕ for each agent a∈ D by definition (<ref>) and the choice of set D. Thus, ϕ∈ Y_0⊆ Y=hd(w') by the choice of set Y_0.Case 2: There is agent a_0∈ D such that a_0∉ C. Then, s'_a_0=(⊤,w) by definition (<ref>). Note that pr_1(s'_a_0)=ϕ by the choice of set D. Thus, ⊤=ϕ. Hence, formula ϕ is the tautology ⊤. Therefore, ϕ∈ hd(w') because set hd(w') is maximal.By Definition <ref>, Claim <ref> and Claim <ref> together imply that (w,𝐬',w')∈ M. Hence, w→_𝐬 w' by Definition <ref> and definition (<ref>). To finish the proof of the lemma, note that ψ∉ hd(w') because set hd(w') is consistent and ψ∈ Y_0⊆ Y=hd(w'). The next lemma shows the construction of a child node for modality _C. The proof is similar to the proof of Lemma <ref> except that, instead of constructing a single child node, we construct two sibling nodes that are in complete harmony. The intuition was discussed at the beginning of Section <ref>.For any state w∈ W, any formula _Cψ∈ hd(w), and any strategy profile 𝐬={s_a}_a∈ C∈ V^C, there are epistemic states w',w”∈ W such that ψ∉ hd(w”), w∼_C w', and w'→_𝐬 w”. By Definition <ref>, for each a∈ C, vote s_a is a pair. Let Y = {ϕ | _Cϕ∈ hd(w)},Z = {ψ}∪{ϕ | ∃ D⊆ C (_Dϕ∈ hd(w) ∧∀ a∈ D (pr_1(s_a)=ϕ))}.By Lemma <ref> where f(x)=pr_1(s_x), pair (Y,Z) is in harmony. By Lemma <ref>, there is a pair (Y',Z') in complete harmony such that Y⊆ Y' and Z⊆ Z'. By Lemma <ref> and Lemma <ref>, sets Y' and Z' are consistent. Let Y” and Z” be maximal consistent extensions of sets Y' and Z', respectively. Recall that set 𝒜 is finite. Thus, set C⊆𝒜 is also finite. Let integer n be the cardinality of set C. Consider (n+1) sequences w_1,w_2,…,w_n+1, where sequence w_k is an extension of sequence w that adds 2k additional elements:w_1=w,C,Y” w_2=w,C,Y”,C,Y” w_3=w,C,Y”,C,Y”,C,Y” … w_n+1=w,C,Y”,…,C,Y”_2(n+1).w_k∈ W for each k≤ n+1.Proof of Claim. We prove the claim by induction on integer k. Base Case: By Definition <ref>, it suffices to show that if _Cϕ∈ hd(w), then ϕ∈ hd(w_1). Indeed, if _Cϕ∈ hd(w), then ϕ∈ Y by the choice of set Y. Therefore, ϕ∈ Y⊆ Y'⊆ Y”=hd(w_1).Induction Step: By Definition <ref>, it suffices to show that if _Cϕ∈ hd(w_k), then ϕ∈ hd(w_k+1) for each k≥ 1. In other words, we need to prove that if _Cϕ∈ Y”, then ϕ∈ Y”, which follows from Truth axiom and the maximality of set Y”. By the pigeonhole principle, there is i_0≤ n such that pr_2(s_a)≠ w_i_0 for all a∈ C. Let w' be epistemic state w_i_0. Thus,pr_2(s_a)≠ w' . Let w” be the sequence w,∅, Z”. In other words, sequence w” is an extension of sequence w by two additional elements: ∅ and Z”.Finally, let strategy profile 𝐬'={s'_a}_a∈𝒜 be defined as followss'_a=s_a, a∈ C,(⊤,w'),.w”∈ W.Proof of Claim. By Definition <ref>, it suffices to show that if _∅ϕ∈ hd(w), then ϕ∈ hd(w”) for each formula ϕ∈Φ. Indeed, by Empty Coalition axiom, assumption _∅ϕ∈ hd(w) implies that hd(w)⊢_∅ϕ. Hence, _∅ϕ∈ hd(w) by the maximality of the set hd(w). Thus, ϕ∈ Z by the choice of set Z. Therefore, ϕ∈ Z⊆ Z'⊆ Z”=hd(w”).w∼_C w'.Proof of Claim. By Definition <ref>, w∼_C w_i for each integer i≤ n+1. In particular, w∼_C w_i_0=w'.ψ∉ hd(w”).Proof of Claim. Note that ψ∈ Z by the choice of set Z. Thus, ψ∈ Z⊆ Z'⊆ Z” = hd(w”). Therefore, ψ∉ hd(w”) due to the consistency of the set hd(w”).Let ϕ be a formula in Φ and D be a subset of 𝒜. If _Dϕ∈ hd(w') and s'_a=(ϕ,w') for each a∈ D, then ϕ∈ hd(w”). Proof of Claim. Note that either set D is empty or it contains an element a_0. In the latter case, element a_0 either belongs or does not belong to set C.Case I: D=∅. Recall that pair (Y',Z') is in complete harmony. Thus, by Definition <ref>, either _∅ϕ∈ Y'⊆ Y”=hd(w') or ϕ∈ Z'⊆ Z”=hd(w”). Assumption _Dϕ∈ hd(w') implies that _∅ϕ∉ hd(w') due to the consistency of the set hd(w') and the assumption D=∅ of the case. Therefore, ϕ∈ hd(w”).Case II: there is an element a_0∈ C∩ D. Thus, a_0∈ C. Hence, pr_2(s_a_0)≠ w' by inequality (<ref>). Then, s_a_0≠ (ϕ,w'). Thus, s'_a_0≠ (ϕ,w') by definition (<ref>). Recall that a_0∈ C∩ D⊆ D. This contradicts the assumption that s'_a=(ϕ,w') for each a∈ D.Case III: there is an element a_0∈ D∖ C. Thus, s'_a_0=(⊤,w') by definition (<ref>). At the same time, s'_a_0=(ϕ,w') by the second assumption of the claim. Hence, formula ϕ is the propositional tautology ⊤. Therefore, ϕ∈ hd(w”) due to the maximality of the set hd(w”).Let ϕ be a formula in Φ and D be a subset of 𝒜. If _Dϕ∈ hd(w') and pr_1(s'_a)=ϕ for each a∈ D, then ϕ∈ hd(w”).Proof of Claim.Case I: D⊆ C. Suppose that pr_1(s'_a)=ϕ for each a∈ D and _Dϕ∈ hd(w'). Thus, ϕ∈ Z by the choice of set Z. Therefore, ϕ∈ Z⊆ Z'⊆ Z”=hd(w”). Case II: D⊈ C. Consider any a_0∈ D∖ C. Note that s'_a_0=(⊤,w') by definition (<ref>). At the same time, pr_1(s'_a_a)=ϕ by the second assumption of the claim. Hence, formula ϕ is the propositional tautology ⊤. Therefore, ϕ∈ hd(w”) due to the maximality of the set hd(w”). Claim <ref> and Claim <ref>, by Definition <ref>, imply that (w',{s'_a}_a∈𝒜,w”)∈ M. Thus, w'→_𝐬 w” by Definition <ref> and definition (<ref>).This together with Claim <ref>, Claim <ref>, Claim <ref>, and Claim <ref> completes the proof of the lemma.π(p)={w∈ W |p∈ hd(w)}. This concludes the definition of tuple(W,{∼_a}_a∈𝒜,V,M,π). Tuple (W,{∼_a}_a∈𝒜,V,M,π) is an epistemic transition system. By Definition <ref>, it suffices to show that for each w∈ W and each 𝐬∈ V^𝒜 there is w'∈ W such that (w,𝐬,w')∈ M. Recall that set 𝒜 is finite. Thus, ⊢_𝒜 by Nontermination axiom. Hence, _𝒜∈ hd(w). By Lemma <ref>, there is w'∈ W such that w→_𝐬w'. Therefore, (w,𝐬,w')∈ M by Definition <ref>.w⊩ϕ iff ϕ∈ hd(w) for each epistemic state w∈ W and each formula ϕ∈Φ. We prove the lemma by induction on the structural complexity of formula ϕ. If formula ϕ is a propositional variable, then the required follows from Definition <ref> and Definition <ref>. The cases of formula ϕ being a negation or an implication follow from Definition <ref>, and the maximality and the consistency of the set hd(w) in the standard way.Let formula ϕ have the form _Cψ. (⇒) Suppose that _Cψ∉ hd(w). Then, by Lemma <ref>, there is w'∈ W such that w∼_C w' and ψ∉ hd(w'). Hence, w'⊮ψ by the induction hypothesis. Therefore, w⊮_Cψ by Definition <ref>.(⇐) Assume that _Cψ∈ hd(w). Consider any w'∈ W such that w∼_C w'. By Definition <ref>, it suffices to show that w'⊩ψ. Indeed, ψ∈ hd(w') by Lemma <ref>. Therefore, by the induction hypothesis, w'⊩ψ.Let formula ϕ have the form _Cψ.(⇒) Suppose that _Cψ∉ hd(w). Then, _Cψ∈ hd(w) due to the maximality of the set hd(w). Hence, by Lemma <ref>, for any strategy profile 𝐬∈ V^C, there is an epistemic state w'∈ W such that w→_𝐬w' and ψ∉ hd(w'). Thus, by the induction hypothesis, for any strategy profile 𝐬∈ V^C, there is a state w'∈ W such that w→_𝐬w' and w'⊮ψ. Then, w⊮_Cψ by Definition <ref>.(⇐) Assume that _Cψ∈ hd(w). Consider strategy profile 𝐬={s_a}_a∈ C∈ V^C such that s_a=(ψ,w) for each a∈ C. By Lemma <ref>, for any epistemic state w'∈ W, ifw→_𝐬 w', then ψ∈ hd(w'). Hence, by the induction hypothesis, for any epistemic state w'∈ W, ifw→_𝐬 w', then w'⊩ψ. Therefore, w⊩_Cψ by Definition <ref>.Finally, let formula ϕ have the form _Cψ. (⇒) Suppose that _Cψ∉ hd(w). Then, _Cψ∈ hd(w) due to the maximality of the set hd(w). Hence, by Lemma <ref>, for any strategy profile 𝐬∈ V^C, there are epistemic states w',w”∈ W such that w∼_C w', w'→_𝐬w”, and ψ∉ hd(w”). Thus, w”⊮ψ by the induction hypothesis. Therefore, w⊮_Cψ by Definition <ref>.(⇐) Assume that _Cψ∈ hd(w). Consider a strategy profile 𝐬={s_a}_a∈ C∈ V^C such that s_a=(ψ,w) for each a∈ C. By Lemma <ref>, for all epistemic states w',w”∈ W, ifw∼_C w', and w'→_𝐬 w”, then ψ∈ hd(w”). Hence, by the induction hypothesis, w”⊩ψ. Therefore, w⊩_Cψ by Definition <ref>.§.§ Completeness: the Final Step To finish the proof of Theorem <ref> stated at the beginning of Section <ref>, suppose that ⊬ϕ. Let X_0 be any maximal consistent subset of set Φ such that ϕ∈ X_0. Consider the canonical epistemic transition system ETS(X_0) defined in Section <ref>. Let w be the single-element sequence X_0. Note that w∈ W by Definition <ref>. Thus, w⊩ϕ by Lemma <ref>. Therefore, w⊮ϕ by Definition <ref>.§ CONCLUSION In this article we proposed a sound and complete logic system that captures an interplay between thedistributed knowledge, coalition strategies, and how-to strategies. In the future work we hope to explore know-how strategies of non-homogeneous coalitions in which different members contribute differently to the goals of the coalition. For example, “incognito" members of a coalition might contribute only by sharing information, while “open" members also contribute by voting. | http://arxiv.org/abs/1705.09349v2 | {
"authors": [
"Pavel Naumov",
"Jia Tao"
],
"categories": [
"cs.AI",
"cs.GT",
"cs.LO",
"cs.MA"
],
"primary_category": "cs.AI",
"published": "20170525202216",
"title": "Together We Know How to Achieve: An Epistemic Logic of Know-How"
} |
Direct Multitype Cardiac Indices Estimation via Joint Representation and Regression Learning Wufeng Xue, Ali Islam, Mousumi Bhaduri, and Shuo Li* Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. W. Xue, A. Islam, M. Bhaduri and S. Li are with the Department of Medical Imaging, Western University, London, ON N6A 3K7, Canada. W. Xue and S. Li are also with the Digital Imaging Group of London, London, ON N6A 3K7, Canada. * Corresponding author. (E-mail: [email protected]) =======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION AND SUMMARYDualities are a key element of string theory and play an essential role in our understanding of it. This explains the recent interest in Exception Field Theory (EFT) <cit.>. It aims at making the U-duality <cit.> groups in d dimensions a manifest symmetry in the low-energy effective description of string/M-theory. By doing so, it is an excellent tool to study maximal supergravities in lower dimensions which for example arise from compactifications of eleven-dimensional supergravity on a d-torus. Intriguingly, their global symmetry is captured by the corresponding duality group (see e.g. <cit.>). In order to appreciate the power of the EFT formalism, we try to understand these distinguished global symmetries from the eleven-dimensional perspective. All of them admit a GL(d) subgroup which originates from diffeomorphisms on the torus. If we study the generators of the global symmetry group, we observe that d^2 of them generate this subgroup. Furthermore, there exist additional generators from internal gauge transformations of p-form fields. However, they are not sufficient to enhance GL(d) to the full duality groups listed in table <ref>. In addition, there have to be hidden symmetries without obvious explanation from an eleven-dimensional point of view <cit.>. EFT succeeds in making the full duality group manifest by considering an extended spacetime. There are no hidden symmetries. It is very important to keep in mind that the additional directions added to the eleven-dimensional spacetime of M-theory are not physical[In the d-torus compactification outlined above, these additional coordinates allow for the interpretation of being conjugate to certain brane wrapping modes. But taking the section condition into account, EFT is defined for more general backgrounds. Take for instance a d-dimensional sphere instead of a torus. For this case there do not exist non-contractible cycles and thus no winding modes.]. Nevertheless, they are a powerful book keeping device. But at the end of the day, one has to get rid of them by imposing the section condition (SC). An example where this extra book keeping pays off are generalized Scherk-Schwarz reductions[For results in Double Field Theory see for example <cit.>.]<cit.> which result in maximal gauged supergravities. In conventional Scherk-Schwarz reductions <cit.>, the compactification ansatz for the metric is given by the left- or right-invariant Maurer-Cartan form on a group manifold G. More specifically, the Maurer-Cartan form gives rise to a vielbein e_a from which the metric follows. Here, a marks an index in the adjoint representation of the Lie algebra 𝔤 of G and we suppress vector indices. Considering the Lie derivativeL_e_a e_b = f_ab^c e_c ,we observe that the vielbein implements 𝔤 with the structure constants f_ab^c on every point of G. As a result, the lower dimensional theory after the compactification has G as a gauge symmetry. In EFT one chooses a generalized vielbein ℰ_A where A is an index in the coordinate irrep of the corresponding duality group (see table <ref>). Again, we suppress the vector index of the extended space. Under the generalized Lie derivative ℒ, which mediates infinitesimal gauge transformations in EFT, an analogous relation to (<ref>),ℒ_ℰ_Aℰ_B = X_AB^C ℰ_C ,holds. The compactified theory is a maximal gauged supergravity. Its gauge algebra 𝔤 is restricted by the embedding tensor formalism <cit.>. We denote the EFT analogue to the structure constants f_ab^c as X_AB^C. Even if (<ref>) and (<ref>) look very similar, there are three important differences. First, the generalized Lie derivative is not the conventional Lie derivative L on the extended space. Moreover, the generalized frame field is not the left- or right invariant Maurer-Cartan form on a group manifold G associated to the Lie algebra 𝔤. Finally, the generalized frame is constrained by the SC[There are exceptions which give rise to non-geometric backgrounds. But here, we are only concerned with the simplest case where the SC has to hold.]. Thus, ℒ in EFT reduces to the generalized Lie derivative of exceptional Generalized Geometry (GG) <cit.> on the physical space M. If one is able to find a generalized frame field ℰ_A which fulfills (<ref>), M is called a generalized parallelizable space <cit.>. There exists no algorithm yet to construct those spaces. Still there are some known examples such as spheres <cit.>, twisted tori, and hyperboloides <cit.>. In combination with the generalized Scherk-Schwarz ansatz, they are crucial to show that dimensional reductions on certain coset spaces are consistent <cit.>. Hence, presenting a methodical way to construct the generalized frame field ℰ_A in (<ref>) is the objective of this work.In this paper we follow a different approach to EFT which makes the group G manifest. It is based on geometric Exceptional Field Theory (gEFT) introduced in <cit.>. This theory treats the extended space as a conventional manifold. Compared to the conventional formulation, it has a modified SC and an additional linear constraint. Following <cit.>, we mainly consider manifolds with GL(5)^+-structure. More specifically, we study group manifolds G which arise as a solution of the SL(5) embedding tensor <cit.>. They represent an explicit example for an extended manifold whose structure group is a subgroup of GL(5)^+. In this setup, the background vielbein of gEFT is the left-invariant Maurer-Cartan form on G. The resulting theory is closely related to<cit.>, a version of Double Field Theory (DFT) <cit.>, derived from the worldsheet theory of a Wess-Zumino-Witten model. Perhaps most remarkably, it allows us to give a direct construction of a large class of generalized parallelizable spaces.Our main results can be summarized as follows. We start by following the ideas of <cit.> to implement generalized diffeomorphisms that are compatible with standard diffeomorphisms (mediated by the Lie derivative L). To this end, we introduce a covariant derivative ∇ on G and use it to write the generalized Lie derivativeℒ_ξ V^A = ξ^B ∇_B V^A - V^B ∇_B ξ^A + Y^AB_CD∇_B ξ^C V^D .In this equation we use flat indices A, B, C for the group manifold and Y^AB_CD denotes the Y-tensor measuring the deviance from Riemann geometry for the corresponding EFT <cit.>. The left-invariant Maurer-Cartan form E_A^I connects flat indices with tangent space indices on G. Additionally, we need to impose the modified SCY^CD_ABD_C · D_D· = 0for the algebra of infinitesimal generalized diffeomorphisms to close <cit.>. It involves flat (curvature free but torsionful) derivatives D which are connected to ∇ by ∇_A V^B = D_A V^B + Γ_AC^B V^C. By itself, the SC is not sufficient for (<ref>) to close. Furthermore, we have to impose two linear and a quadratic constraint. At this point, the results are quite general and do not require to explicitly fix the T-/U-duality group. However, solving the linear constraints heavily depends on the representation theory of the chosen duality group. A linear constraint is known in the context of the embedding tensor formalism as well. It reduces the irreps resulting from the tensor product coordinate irrep × adjoint to the embedding tensor irreps given in table <ref>. Considering SL(5) as a specific duality group, the linear constraints we find result in the same restriction. On top of that, they come with an additional subtlety. For gaugings in the 40 the dimension of the resulting group manifold is between nine and six. Thus, we are not able to identify the coordinates on G with the irrep 10 stated in table <ref>. In order to still obtain well-defined generalized diffeomorphisms on these group manifolds, we break SL(5) into smaller U-duality subgroups whose irreps can be chosen such that they add up to G. This situation is special to gEFT, it does not occur for the T-duality subgroup O(3,3) for which we reproduce the gauge algebra of<cit.>. Finally, the quadratic constraint is equivalent to the Jacobi identity on the Lie algebra of G and therefore automatically fulfilled in our setup.Additionally, the SC in (<ref>) has to be solved. It acts on fluctuations (denoted by ·) around the background group manifold G. A trivial solution is given by constant fluctuations. They are sufficient to capture the lightest modes in the generalized Scherk-Schwarz reduction. If we want to incorporate heavier modes, we have to find the most general SC solutions. They depend on all d physical coordinates of the extended space. We apply a technique introduced in <cit.> forto construct them. It interprets the group manifold as a H-principal bundle over the physical manifold M=G/H. For DFT, H is a maximally isotropic subgroup of G and the embedding of H in G is parameterized by the irrep 2^d-2 of the T-duality group O(d-1,d-1). We show that for gEFT the subgroup H is fixed by the SC irrep in table <ref>. Following <cit.>, the data selecting d physical directions out of the G coordinates on G are encoded in a connection one-form on the principle bundle. By pulling this one-form back to the physical manifold M, one obtains a gauge connection. If it vanishes, which implies that the corresponding field strength is zero, the SC is solved. In this case, we show how the generalized Lie derivative (<ref>) on G is connected to GG on M. From the data of the H-principal bundle and its connection a generalized frame field Ê_A^Î on the generalized tangent bundle T M ⊕Λ^2 T^* M (indices Î) is constructed. It allows us to rewrite (<ref>), if we restrict it to M⊂ G, asℒ_ξ V^Î = ℒ_ξ V^Î + ℱ_ĴK̂^Îξ^Ĵ V^K̂ ,the ℱ-twisted generalized Lie derivative of GG. Moreover, the generalized frame field fulfills the relationℒ_Ê_AÊ_B = X_AB^C Ê_Csimilar to (<ref>). However, we are still left with a twist term in (<ref>) and hence Ê_A is not equivalent to the generalized frame ℰ_A in (<ref>). But it is now possible to construct ℰ_A. Subsequently, we use the splitting of group elements G∋ g = m h (h∈ H) induced by the H-principal bundle construction. Especially, each point of M is in one-to-one correspondence with a m∈ G which gives rise to the SL(5) transformationM_A^B t_B = m^-1 t_A mfor t_A ∈𝔤 .If the structure constants X_AB^C satisfy an additional linear constraint which is required for the embedding tensor solution to describe a geometric background, the generalized frame fieldℰ_A^Î = - M_A^B [ E_β^i E_β^k 𝒞_kij; 0 η_δϵ,β̃ E^δ_i E^ϵ_j ]_B^Îwith𝒞 = 1/3!𝒞_ijkd x^i ∧ d x^j ∧ d x^k = λ volrepresents a generalized parallelization (<ref>) of M. The constant factor λ follows directly from the embedding tensor and vol is the volume form on M induced by the frame E_α^i. It is the inverse transpose oft_α E^α_i = m^-1∂_i m ,where the splitting of the 𝔤=𝔪⊕𝔥 generators t_A=(t_α, t_α̃) into the subalgebra t_α̃∈𝔥 and a complement coset part t_α∈𝔪 is used. For more details on the η-tensor see (<ref>) in section <ref>.In general, the choice of H for a given G is not unique. To show that different choices result in backgrounds related by a duality transformation, we take a closer look at the duality chain for the four-torus with four-form G-flux in M-theory. Its extended space corresponds to the ten-dimensional group manifold CSO(1,0,4) resulting from the 15 of the embedding tensor. It is a priori not compact and requires the modding out of the discrete subgroup CSO(1,0,4,ℤ) from the left. There are two choices of subgroups H which reproduce the duality chain four-torus with G- ↔ Q-flux <cit.>. Another T-duality transformation results in a type IIB background with f- ↔ R-flux. This chain is captured by an embedding tensor solution in the 40. It gives rise to a nine-dimensional group manifold with an unique subgroup H. This subgroup realizes the background with geometric flux only. We do not find a SC solution for the R-flux background which is in agreement with the fact that there exists no GG description for the locally non-geometric flux R-flux <cit.>. As an example for a physical manifold M without any non-contractible cycles, we discuss the four-sphere with G-flux. For all these backgrounds we construct the generalized frame ℰ_A.The rest of this paper is organized into three main parts. First, we show in section <ref> how to implement generalized diffeomorphisms on group manifolds. We approach this question from a slightly different perspective than <cit.> by keeping the discussion as general as possible and only fix specific duality groups if needed. Additionally, we emphasis the similarities and differences to . At the same time, the notation is set and a short review of relevant results in DFT and EFT is given. Section <ref> presents the derivation of the two linear and the quadratic constraints from demanding closure of the generalized Lie derivative (<ref>) under the SC (<ref>). In order to solve the linear constraints, we consider the U-duality group SL(5) in section <ref>. Moreover, a detailed picture of the SL(5) breaking for group manifolds with G<10, originating from embedding tensor solutions in the 40, is given. The second part is covered by section <ref>. It shows how the techniques to solve the SC in<cit.> carry over to gEFT. The presented SC solutions admit a description in terms of GG which is discussed in section <ref>. Based on these results, section <ref> explains the construction of the generalized frame field ℰ_A and the additional linear constraint it requires. Finally, the four-torus with G-flux, the backgrounds contained in its duality chain and the four-sphere with G-flux are worked out as explicit examples in section <ref>. Section <ref> concludes this work.§ GENERALIZED DIFFEOMORPHISMS ON GROUP MANIFOLDSCovariance with respect to diffeomorphisms plays an essential role in general relativity. In EFT, diffeomorphisms are replaced by generalized diffeomorphisms. They combine the former with gauge transformations on the physical subspace, which emerge after solving the SC. It is important to distinguish between these generalized and standard diffeomorphisms on the extended space. They are not identical, but we show in this section that they can be modified to become compatible with each other on a group manifold G. By compatible, we mean that the generalized Lie derivative transforms covariantly on G in the sense known from general relativity. A similar approach, which works for arbitrary Riemannian manifolds, has been suggested by Cederwall for DFT <cit.>. Cederwall introduces a torsion free, covariant derivative with curvature to obtain a closing algebra of infinitesimal generalized diffeomorphisms. Here, we use a different approach. Our covariant derivative has both torsion and curvature. It is motivated by<cit.> whose gauge transformations we review in subsection <ref> (for a complete review see <cit.>). In the following subsections <ref> and <ref>, we extend the structure fromto EFT. Doing so, we see that closure requires two linear and a quadratic constraint in addition to the SC. For the U-duality group SL(5), we present the solution to the linear constraints in subsection <ref> and discuss the quadratic one in subsection <ref>. For this particular U-duality group, the constraints found agree with the ones in <cit.>.§.§ From Double Field Theory to Exceptional Field TheoryFirst, we want to review the most important features of generalized and standard diffeomorphisms which we want to combine. In DFT, the infinitesimal version of the former is mediated by the generalized Lie derivative <cit.>ℒ_ξ V^I = ξ^J ∂_J V^I + ( ∂^I ξ_J - ∂_J ξ^I ) V^J .It closes according to[ ℒ_ξ_1, ℒ_ξ_2 ] = ℒ_[ξ_1, ξ_2]_C ,if the SC∂_I ·∂^I · = 0is fulfilled <cit.>. This constraint applies to arbitrary combinations of fields V^I and parameters of gauge transformations ξ^I, represented by the placeholder · . Furthermore, we make use of the C-bracket[ξ_1, ξ_2]_C = 1/2 ( ℒ_ξ_1ξ_2 - ℒ_ξ_2ξ_1 )in (<ref>). For the canonical solution of the SC where all · only depend on the momentum coordinates x^i it reduces to the Courant bracket of GG <cit.>. As can be easily checked, the generalized Lie derivativeℒ_ξ η_IJ = 0leaves the coordinate independent O(d-1,d-1) metric η_IJ invariant. This metric is used to raise and lower indices I,J,K, … running from 1, …, 2D. This completes the short review of the relevant DFT structures. On the other hand, infinitesimal diffeomorphisms are mediated by the Lie derivativeL_ξ V^I = ξ^J ∂_J V^I - V^J ∂_J ξ^Iwhich closes according to[ L_ξ_1, L_ξ_2 ] = L_[ξ_1, ξ_2] .The Lie bracket[ξ_1, ξ_2] = L_ξ_1ξ_2 = 1/2 (L_ξ_1ξ_2 - L_ξ_2ξ_1 )is defined analogous to the C-bracket in (<ref>). In contrast to generalized diffeomorphisms, neither does its closure require an additional constraint, nor is η_IJ invariant under the Lie derivative.In order to make (<ref>) and (<ref>) compatible with each other, we require that the generalized Lie derivative transforms covariantly under standard diffeomorphisms. In this caseL_ξ_1ℒ_ξ_2 = ℒ_L_ξ_1ξ_2 + ℒ_ξ_2 L_ξ_1or equivalently [L_ξ_1, ℒ_ξ_2 ] = ℒ_[ξ_1, ξ_2]holds. Assume that V^I and ξ^I in the definition of the generalized Lie derivative transform covariantly, namelyδ_λ V^I = L_λ V^I andδ_λξ^I = L_λξ^I .Then, the partial derivative in (<ref>) spoil (<ref>). We fix this problem by replacing all partial derivatives with covariant derivatives∇_I V^J = ∂_I V^J+ Γ_IL^J V^L .In this case, we obtain the generalized Lie derivativeℒ_ξ V^I = ξ^J ∇_J V^I + ( ∇^I ξ_J - ∇_J ξ^I ) V^J .Before we study it in more detail we have to choose the connection Γ. In order to fix it, we impose some additional constraints. First of all, the covariant derivative has to be compatible with the metric η_IJ. Thus, it has to fulfill∇_Iη_JK = 0 .Otherwise (<ref>) would be violated. With the SC (<ref>) expressed in terms of a covariant derivative as well, the new generalized Lie derivative still has to close. As shown in <cit.>, these two constraints are sufficient to identify Γ with the torsion-free Levi-Civita connection. From a purely symmetry oriented point of view, this is the most straightforward approach to merge generalized and standard diffeomorphisms. However, string theory on group manifolds leads to an interesting subtlety in this construction providing additional structure.In order to explain the additional ingredient for the construction, consider a group manifold G and identify η_IJ with a bi-invariant metric of split signature on it. Then, G is parallelizable and comes with the torsion-free (flat) derivativeD_A = E_A^I ∂_I ,where E_A^I (generalized background vielbein) denotes the left-invariant Maurer-Cartan form on G. D_A carries a flat index, like A, B, C, …, running from one to 2D and is compatible with the flat metricη_AB = E_A^I η_IJ E_B^J.Its torsion[D_A, D_B] = F_AB^C D_Cis given by the structure constants of the Lie algebra 𝔤 associated to G. Hence, on a group manifold it appears more natural to use the flat derivative D_A instead of ∇_I with a torsion-free Levi-Civita connection. Indeed, Closed String Field Theory (CSFT) calculations for bosonic strings on G suggest that we write the SC <cit.>D_A · D^A · = 0with flat derivatives. In CSFT, D_A has a very clear interpretation as a zero mode in the Kač-Moody current algebra on the world sheet and the SC arises as a direct consequence of level matching. On the other hand, the generalized Lie derivative (<ref>) does not close with only flat derivatives D_A anymore. The only way out is to accept the presence of two different covariant derivatives. The flat derivatives needed for the SC and the covariant derivative ∇_A for everything else. This approach completely fixes∇_A V^B = D_A V^B - 1/3 F_CA^B V^Cand reproduces exactly the results arising from CSFT <cit.>. Vectors with flat indices are formed by contracting vectors with curved indices with the generalized background vielbein, e.g. V^A = V^I E^A_I. The Christoffel symbols are obtained from the compatibility condition for the vielbein∇_A E_B^I = 0 .In order to generalize this structure to EFT, we have to * fix the Lie algebra 𝔤 of the group manifold G by specifying the torsion of the flat derivative* fix the connection of ∇ to obtain a closing generalized Lie derivativeThis outlines the steps we perform in the following two subsections. Another guiding principle is that the U-duality groups in table <ref> include the T-duality groups O(d-1,d-1) as subgroup. Thus, we are always able to consider thelimit to check our results. §.§ Section ConditionWhile in DFT all indices live in the fundamental representation of the Lie algebra 𝔬(d-1,d-1), the situation in EFT is more involved. Here, we use different indices in different representations of the U-duality group. Let us start with the coordinate irrep denoted by capital letters I, J, …. Our main example in this paper is SL(5) EFT for which this irrep is the two index anti-symmetric 10 of 𝔰𝔩(5). Moreover, a convenient way to express the SC in a uniform manner <cit.>Y^MN_LK∂_M ∂_N · = 0uses the invariant Y-tensor. It is a projector from the symmetric part of the tensor product of two coordinate irreps to the SC irrep. Both irreps are given in table <ref>. For SL(5) the SC irrep is the fundamental 5 and denoted by small lettered indices a, b, …. In this particular case, the Y-tensor reads <cit.>Y^MN_LK = 1/4ϵ^MNaϵ_LKawith the normalization Y^MN_MN = 30where ϵ is the totally anti-symmetric tensor with five fundamental indices. For the SC itself the normalization can be neglected. However, it is essential if we express the generalized Lie derivative (<ref>) in terms of the Y-tensor. With the flat indices defined in analogy to the curved ones, the flat derivativeD_A = E_A^I ∂_Ihas the same form as in . The generalized background vielbein E_A^I describes a non-degenerate frame field on the group manifold and is valued in GL(n) with n =G. At this point it is natural to ask: What happens when the dimension of G is not the same as the dimension of the coordinate irrep. We postpone the answer to section <ref>. For the moment let us assume that the dimensions match. Now n-dimensional standard diffeomorphisms act through the Lie derivative on curved indices and the SC readsY^CD_ABD_C · D_D· = 0 .Thus, we have a situation very similar todiscussed in the last subsection.Finally, the torsion of the flat derivative F_AB^C (<ref>) lives in the tensor product45×10 = 10 + 14 + 40 + 175 + 210 ,where the 45 is the anti-symmetric part of 10×10. At the moment, this is all we know about it. Discussing the closure of the generalized Lie derivative in the next section will refine this statement. §.§ Generalized Lie DerivativeIn analogy to the SC (<ref>), the generalized Lie derivative of different EFTs in table <ref> can be written in the canonical form ℒ_ξ V^M = L_ξ V^M + Y^MN_LK∂_N ξ^L V^Kby using the Y-tensor and the standard Lie derivative on the extended space. If the SC holds, the infinitesimal generalized diffeomorphisms mediated by it close to form the algebra <cit.>[ℒ_ξ_1, ℒ_ξ_2] V^M= ℒ_[ξ_1, ξ_2]_E V^M with [ξ_1, ξ_2]_E = 1/2(ℒ_ξ_1ξ_2 - ℒ_ξ_1ξ_2 ) .It should be noted that this formulation includes the DFT results for Y^MN_LK = η^MNη_LK and therefore extends naturally to EFT, e.g. for the SL(5) Y-tensor in (<ref>). Hence, it is the natural starting point for our discussion. It is instructive to keep the rough structure of the closure calculation in mind, because we have to repeat it after replacing partial derivatives with covariant ones. Evaluating[ℒ_ξ_1, ℒ_ξ_2] V^M - ℒ_[ξ_1, ξ_2]_E V^M ,one is left with sixteen different terms. All of them contain two partial derivatives. But only in four terms the partial derivatives act on the same variable. Because the Y-tensor has the propertiesδ_F^(B Y^AC)_DE - Y^(AC_FG Y^B)G_DE = 0 andδ_(F^B Y^AC_DE) - Y^AC_G(F Y^BG_DE) = 0for d≤ 6 only terms which are annihilated by the SC remain. For the other U-duality groups with d>6 the closure calculation gets more involved <cit.>. Here, we are interested in a proof of concept. Thus, we focus on the simplest cases and postpone the rest to future work. For scalars the generalized and standard Lie derivative coincideℒ_ξ s = L_ξ s .Applying the Leibniz rule we obtain the action of generalized diffeomorphisms on one-formsℒ_ξ V_M = L_ξ V_M - Y^PQ_NM∂_Q ξ^N V_P ,the objects dual to the vector representation. Finally, we remember that Y^MN_PQ has to be an invariant tensorℒ_ξ Y^MN_PQ = 0in the same fashion as η_IJ is in DFT. This completes the list of properties for the EFT generalized Lie derivative we require to make it compatible with standard diffeomorphisms.As a first step into this direction, we switch to flat indices and replace all partial derivatives in (<ref>) by covariant ones to findℒ_ξ V^A = ξ^B ∇_B V^A - V^B ∇_B ξ^A + Y^AB_CD∇_B ξ^C V^D .This expression can be rewritten in terms of flat derivatives∇_A V^B = D_A V^B + Γ_AC^B V^C and∇_A V_B = D_A V_B - Γ_AB^C V_Cby introducing the spin connection Γ_AB^C. Expanding the generalized Lie derivative yieldsℒ_ξ V^A= ξ^B D_B V^A - V^B D_B ξ^A + Y^AB_CD D_B ξ^C V^D + X_BC^A ξ^B V^C and ℒ_ξ V_A= ξ^B D_B V_A + V_B D_A ξ^B - Y^CD_BA D_D ξ^B V_C - X_BA^C ξ^B V_CwithX_AB^C = 2 Γ_[AB]^C + Y^CD_BE Γ_DA^Ecollecting all terms depending on the spin connection. We will see later that X_AB^C is closely related to the embedding tensor of gauged supergravities. Under the modified generalized Lie derivative the Y-tensor should still be invariant, which translates to the first linear constraintC1∇_C Y^AB_DE := C_1^AB_CDE = 2 Y^F(A_DEΓ_CF^B) - 2 Y^AB_(D|FΓ_C|E)^F = 0on the spin connection Γ after imposing D_A Y^BC_DE = 0. It is a direct generalization of the metric compatibility (<ref>) in .Now, we demand closure of the modified generalized Lie derivative. Equivalently, all terms (<ref>) which spoil the closure have to vanish. Let us start with the ones containing no flat derivatives. They only vanish if the quadratic constraintX_BE^A X_CD^E - X_BD^E X_CE^A + X_[CB]^E X_ED^A = 0is fulfilled. In order to analyze it, we decompose X_AB^C into a symmetric part Z^C_AB and an anti-symmetric oneX_AB^C = Z^C_AB + X_[AB]^C .Moreover, all terms with only one flat derivative acting on V^A in (<ref>) vanish, if we identify the torsion of the flat derivative with[D_A, D_B] = X_[AB]^C D_C .Note that we have used D_AX_BC^D = 0 and Y^AB_BC = Y^(AB)_(BC), which holds only for d≤ 6, in all calculations. For a consistent theory, it is essential that the Bianchi identify[D_A, [D_B, D_C]] + [D_C, [D_A, D_B]] + [D_B, [D_C, D_A]] = 0is fulfilled. Evaluating the commutator above, we find that this constraint is equivalent to the Jacobi identity(X_[AB]^E X_[CE]^D + X_[CA]^E X_[BE]^D + X_[BC]^E X_[AE]^D) D_D = 0 .But after antisymmetrizing (<ref>) with respect to B,C,D, we obtainX_[BC]^E X_[CE]^A + X_[DB]^E X_[CE]^D + X_[CD]^E X_[BE]^D = - Z^A_E[B X_CD]^Einstead of zero. Hence, we are left withZ^A_E[BX_CD]^E D_A = 0which in general does not vanish. For Z^A_BC vanishes and this problem does not occur. Thus, it is special to gEFT. As we show in section <ref>, it is solved by reducing the dimension of the group manifold representing the extended space.An important property of the generalized Lie derivative is that the Jacobiator of its E-bracket only vanishes up to trivial gauge transformations. Let us take a closer look at these transformationsξ^A = Y^AB_CD D_B χ^CDin the context of our modified generalized Lie derivative. Ultimately, this will help us to better organize terms in the closure calculation with one flat derivative acting either on ξ_1 or ξ_2. Inserting (<ref>) into the generalized Lie derivative (<ref>), we obtainℒ_ξ V^A = C_2a^AB_CDE D_B χ^CD V^E + … = 0where … denotes terms which vanish under the SC and due to the properties of the Y-tensor (<ref>). The tensor C2a C_2a^AB_CDE = Y^BF_CD X_FE^A + 1/2 Y^AF_CD X_[FE]^B + 1/2 Y^AF_EH Y^GH_CD X_[FG]^Bhas to vanish if trivial gauge transformations have the form (<ref>).Terms with two derivatives in (<ref>) vanish under the SC and due to (<ref>). All we are left with are terms with one flat derivative acting on the gauge parameters ξ_1 or ξ_2. Because (<ref>) is anti-symmetric with respect to the gauge parameters, it is sufficient to check whether all D_A ξ_1^B contributions vanish. We write them in terms of the tensorC_2b^AB_CDE = Z^A_DCδ_E^B - Z^B_DEδ_C^A - Y^BF_EC Z^A_DF + Y^AB_CF Z^F_DE+ Y^AB_EF X_[DC]^F + Y^AB_CF X_[DE]^F - 2 Y^F(A_EC X_[DF]^B) = 0 C2bas(-1/2 C_2a^AB_CDE + C_2b^AB_CDE) D_B ξ_1^C ξ_2^D V^E = 0 ,which has a contribution from trivial gauge transformations (<ref>) as well. This is reasonable since the E-bracket closes up to trivial gauge transformations. However, in general there is no reason why the two contributions have to vanish independently. Only the second linear constraintC2 -1/2 C_2a^AB_CDE + C_2b^AB_CDE = 0has to hold in conjunction with the first linear constraint (<ref>) and the quadratic constraint (<ref>) for closure of generalized diffeomorphisms under the SC. Thus, one has to restrict the connection Γ_AB^C in such a way that all these three constraints are fulfilled. This is exactly what we do in the next two subsections.Without too much effort we can already perform a first check of our results at this point. To this end, consider the O(d-1,d-1) T-duality group withY^AB_CD = η^AB η_CD , Γ_AB^C = 1/3 F_AB^C and X_AB^C = F_AB^C .In this case, we haveC_1^AB_CDE = 2/3η_DE F_C^(AB) - 2/3η^AB F_C(DE) = 0 C_2a^AB_CDE = η_CD ( F^BE^A + F^A_E^B ) = 0 C_2b^AB_CDE = η^AB ( F_DCE + F_DEC ) - 2 η_EC F_D^(AB) = 0due to the total antisymmetry of the structure constants F_ABC. Hence, this short calculation is in agreement with the closure of the gauge algebra ofpresented in <cit.>. §.§ Linear ConstraintsSolving the linear constraints for gEFT is more involved than for , which we presented as a simple example at the end of the last subsection. It requires more sophisticated tools from representation theory. Especially, we need to obtain projectors which filter out certain irreps from tensor products of the coordinate irrep in table <ref>. Here, our initial choice of SL(5) as the duality group pays off. Irreps (or more precisely projectors onto them) of SL(n) and their tensor products can be conveniently organized in terms of young tableaux making their representation theory very traceable. All required techniques are reviewed in appendix <ref>. As an explicit example, the T-duality subgroup SL(4) is discussed there. Its Lie algebra 𝔰𝔩(4) is isomorphic to 𝔰𝔬(3,3). Hence, we already know the solutions to the linear constraints, which allows us to check the machinery developed in the appendix.We start our discussion with the spin connection Γ_AB^C. The indices are in the 10 and 10 of 𝔰𝔩(5). We express said indices through the fundamental 5 indices and lower the raised indices with the totally anti-symmetric tensor to obtainΓ_a_1 a_2, b_1 b_2, c_1 c_2 c_3 = Γ_a_1 a_2, b_1 b_2^d_1 d_2ϵ_d_1 d_2 c_1 c_2 c_3 .In this form, it is manifest that the 1000 independent components of the connection are embedded in the tensor product10×10×10 = 3 (10) + 15 + 2 (40) + 2 (175) + 210 + 315which we translate into corresponding Young diagrams1,1×( 1,1×1,1,1 ) = 3 2,2,1,1,1 + 3,1,1,1,1 + 2 2,2,2,1 + 23,2,1,1 + 3,2,2 + 3,3,1 .This decomposition looks similar to (<ref>) in appendix <ref>. However, the 10 of 𝔰𝔩(5) is not self dual as the 6 of 𝔰𝔩(4). Thus, we pick up an additional box in the last irrep on the left hand side. Each of these diagrams is associated with a projector. Because some irreps appear more than once in the decomposition of the tensor product (<ref>), we label them as10× ( 10×10 ) = 10× (1 + 24 + 75) = {[10×1 = 10a; 10×24 = 10b + 15 + 40a + 175a; 10×75 = 10c + 40b + 210 + 315 ].in order to clearly distinguish all projectors.While the first linear constraint (<ref>) is straightforward to solve for 𝔰𝔩(4), things now become more involved. First, note that the constraint acts trivially on the index C. We suppress this index and rewrite (<ref>)C_1_a_1 a_2 a_3, b_1 b_2 b_3, d_1 d_2, e_1 e_2 = σ_1 Γ_a_1 a_2, a_3 b_1 b_2ϵ_b_3 d_1 d_2 e_1 e_2 = 0in terms of the permutationsσ_1 = (6 5 4 3) + (3 5 2 4 1) - (6 5 4 3 2) - (3 5 6 2 4 1) + (6 5 4 3 2 1) - (6 10 2 7 4 8 5 9 1) - (6 10 5 9 4 8 2 7 1) + (6 10 2 3 7 4 8 5 9 1) + (6 10 5 9 4 8 2 3 7 1) + (3 5 1) (4 6 2)- (3 7 1)(6 10 5 9 4 8,2) - (3 7 4 8 5 9 1)(6 10 2)which act on the ten remaining indices. This form allows for the constraint to be solved by linear algebra techniques. To this end, we choose the explicit basis (d_1 d_2),(e_1 e_2) ∈ V_10 = { (d_1 d_2)|d_1, d_2 ∈{1 …5}∧ d_1 < d_2 }(a_1 a_2 a_3),(b_1 b_2 b_3) ∈ V_10 = { (a_1 a_2 a_3)|a_1, a_2, a_3 ∈{1 …5}∧ a_1 < a_2 < a_3 }for the irreps 10 and 10 appearing in (<ref>). Keeping the properties of the totally anti-symmetric tensor in mind, we interpret σ_1 as a linear map from Γ to C_1σ_1 : V_10× V_10→ V_10× V_10×V_10× V_10 .Solutions of the first linear constraint are in the kernel of this map and can be associated to the projectors of the irreps we have discussed above. An explicit calculation shows thatσ_1 ( P_1 + P_24 ) = 0 butσ_1 P_75 0holds. Hence, the most general solution can be written in terms of the projectorP_1 = P_10a + P_10b + P_15 + P_40a + P_175a . Next, we have to check which of these irreps survive the transition from the connection Γ_AB^C to X_AB^C. In analogy to (<ref>), we express (<ref>) in terms of permutationsσ_X = () -(3 1)(4 2)+(3 5 1)(4 6 2)-(3 5 1)(4 6 7 2) + (3 5 7 2 4 6 1)throughX_a_1 a_2, b_1 b_2, c_1 c_2 c_3 = σ_X P_1 Γ_a_1 a_2, b_1 b_2, c_1 c_2 c_3 .Note that the first linear constraint is already implemented in this equation by the projector P_1. Again, we apply the techniques presented in appendix <ref> to decomposeσ_X P_1 P_10×10×10 = 12/5 P_10ab + P_10c + 4 P_15 + 3 P_40ainto orthogonal projectors on different 𝔰𝔩(5) irreps where P_10ab is defined asP_10ab = 5/12 (P_10a-P_10b) σ_X (P_10a + P_10b) .It embeds just another ten-dimensional irrep 10ab into 10a and 10b. In the following, we only focus on the 15 and 40. These are exactly the irreps which survive the linear constraint on the embedding tensor in seven-dimensional maximal gauged supergravities[In <cit.> a tree index tensor Z^ab,c represents the 40. Here we use the dual version. Both are connected by (<ref>) and capture the same data.] <cit.>. As presented in appendix A of <cit.>, the remaining two ten-dimensional irreps can be combined to one 10 capturing trombone gaugings. Since we are only interested in a proof of concept, we do not discuss trombone gaugings. They are considered in <cit.> which takes the embedding tensor irreps 10+15+40 as starting point. We a priori did not restrict the allowed groups G. But trying to implement generalized diffeomorphisms on them exactly reproduces the correct irreps of the embedding tensor. Originally, they arise from supersymmetry conditions <cit.>. Here, we do not make any direct reference to supersymmetry. Hence, it is remarkable that we still reproduce this result.Now, let us discuss the remaining linear constraint (<ref>). We proceed in the same fashion as for the first one and writeC_2_a_1 a_2 a_3, b_1 b_2 b_3, c_1 c_2, d_1 d_2, e_1 e_2 = σ_2 X_a_1 a_2, a_3 b_1, b_2 b_3 c_1ϵ_c_2 d_1 d_2 e_1 e_2in terms of a sum of permutations denoted as σ_2 which is of a similar form as (<ref>) but contains 54 different terms. Thus, we do not write it down explicitly. In the basis (<ref>), σ_2 gives rise to the linear mapσ_2 : V_10× V_10× V_10→ V_10× V_10×V_10× V_10× V_10whose kernel contains the 15, but not the 40a. However, from maximal gauged supergravities in seven dimensions <cit.>, we know that also gaugings in the dual 40 are consistent. How do we resolve this contradiction? First, we implement the components of this irrep in terms of the tensor Z^ab,c and connect it to the 40a, we discussed above, through(X_40𝐚)_a_1 a_2, b_1 b_2, c_1 c_2 c_3 = ϵ_a_1 a_2 d_1 d_2 [b_1 Z^d_1 d_2, e_1ϵ_b_2] c_1 c_2 c_3 e_1with the expected propertyP_40𝐚 (X_40𝐚)_a_1 a_2, b_1 b_2, c_1 c_2 c_3 = (X_40𝐚)_a_1 a_2, b_1 b_2, c_1 c_2 c_3 .Following the argumentation in <cit.>, we interpret Z^ab,c as a 10×5 matrix and calculate its ranks = rank ( Z^ab,c ) . The number of massless vector multiplets in the resulting seven-dimensional gauged supergravity is given by 10-s. They contain the gauge bosons of the theory and transform in the adjoint representation of the gauge group G. Thus, we immediately deducedimG = 10 - s .Inthe gauge group of the gauged supergravity which arises after a Scherk Schwarz compactification is in one-to-one correspondence with the group manifold we are considering <cit.>. There is no reason why it should be different for gEFT. So, if we switch on gaugings in the 40, we automatically reduce the dimension of the group manifold representing the extended space. Possible ranks s which are compatible with the quadratic constraint of the embedding tensor are 0 ≤ s≤ 5. For those cases we have to adapt the coordinates on the group manifold. To this end, we consider possible branching rules of SL(5) to its U-/T-duality subgroups given in table <ref>, e.g. SL(4), SL(3)×SL(2) and SL(2)×SL(2)10 →4 + 6 10 → (1, 1) + (3, 2) + (3, 1)10 → (1, 1) + (1, 1) + (2, 1) + (1, 2) + (2, 2).For the first one, we obtain a six-dimensional manifold whose coordinates are identified with the 6 of the branching rule (<ref>) after dropping the 4. In the adapted basisV_4 = {15, 25, 35, 45}V_6 = {12, 13, 14, 23, 24, 34}V_4 = {234, 134, 124, 123}V_6 = {345, 245, 235, 145, 135, 125} ,σ_2 is now restricted toσ_2 : V_6× V_6× V_6→ V_6× V_6×V_6× V_6× V_6 ,while the irreps 15 and 40 split into15 →1 + 4 + 10 40 →4 + 6 + 10 + 20 .All crossed out irreps at least partially depend on V_4 or its dual which is not available as coordinate irrep anymore. Of course, the 10 from the 15 still satisfies all linear constraints. But now only the 6 is excluded by the second linear constraint (<ref>) with (<ref>), while the 10 is in its kernel. This result is in alignment with the SL(4) case we discuss in appendix <ref>. Hence, switching on specific gaugings in the 40 breaks indeed the U-duality group into a subgroup. An alternative approach <cit.> is to keep the full SL(5) covariance of the embedding tensors by not solving the linear constraints. However this technique obscures the interpretation of the extended space as a group manifold which is crucial for constructing the generalized frame ℰ_A in the next section. Furthermore, breaking symmetries by non-trivial background expectation values for fluxes is a well-known paradigm. Only a tours with no fluxes has the maximal number of abelian isometries and should allow the full U-duality group. In case we restrict ourselves to a T-duality subgroup to solve the linear constrains, allresults are naturally embedded in the EFT formalism. For the remaining branchings (<ref>) and (<ref>), we perform the same analysis in appendix <ref>. All results are summarized in figure <ref>. §.§ Quadratic ConstraintFinally, we come to the quadratic constraint (<ref>) which simplifies drastically toX_[BC]^E X_[ED]^A + X_[DB]^E X_[EC]^A + X_[CD]^E X_[EB]^A = 0after solving the linear constraints which result in Z^C_AB = 0 for the remaining coordinates on the group manifold G. Now, it is identical to the Jacobi identity (<ref>) which is automatically fulfilled for the Lie algebra 𝔤. Thus, the flat derivative satisfies the first Bianchi identity (<ref>). For the covariant derivative (<ref>), we compute the curvature and the torsion by evaluating the commutator[ ∇_A, ∇_B ] V_C = R_ABC^D V_D - T_AB^D ∇_D V_C .Doing so, we obtain the curvatureR_ABC^D =2 Γ_[A|C^EΓ_|B]E^D + X_[AB]^EΓ_EC^D ,where we used that Γ_AB^C is constant due to (<ref>) and the torsionT_AB^C = - X_[AB]^C + 2 Γ_[AB]^C = Y^CD_[A|E Γ_D|B]^E ,for which we used (<ref>) and (<ref>). In general both are non-vanishing. Using these equations, we can compute the first Bianchi identityR_[ABC]^D +∇_[A T_BC]^D - T_[AB^E T_C]E^D =2 X_[AB]^E X_[CE]^D + 2 X_[CA]^E X_[BE]^D + 2 X_[BC]^E X_[AE]^D = 0for ∇. Again, it is fulfilled because of the Jacobi identity (<ref>). These results are in agreement with . It is straightforward to check that all gaugings, given in table 3 of <cit.>, can be reproduced in the framework we presented in the first part of this paper. Explicit examples with with ten-dimensional groups CSO(1,0,4)/SO(5) and also a nine-dimensional group are discussed in section <ref>.§ SECTION CONDITION SOLUTIONSSo far, we implemented generalized diffeomorphisms on group manifolds G which admit an embedding in one of the U-duality groups with d≤ 4 in table <ref>. Still, they only close into a consistent gauge algebra if the SC (<ref>) is fulfilled. Hence, understanding the solutions of this constraint is very important and objective of this section. In the following, we adapt a technique for finding the most general SC solutions in<cit.> to our gEFT setup. It is based on a H-principle bundle over the physical subspace M=G/H. H is a (G-M)-dimensional subgroup of G with special properties which are explained in section <ref>. As before, the construction follows closely the steps required inand introduces generalizations whenever needed. We show in section <ref> that each SC solution gives rise to a GG on M which has two constituents: a twisted generalized Lie derivative and a generalized frame field. Both act on the generalized tangent bundle T M ⊕Λ^2 T^* M. Sometimes, the choice of the subgroup H is not unique for a given G. Different subgroups result in dual backgrounds. Section <ref> presents a way to systematically study these different possibilities. It works exactly as in <cit.>, so we keep the discussion brief. Starting from the SC solution, we construct the generalized frame field ℰ_A fulfilling (<ref>) in section <ref>. We also introduce the additional constraint on the structure constants X_AB^C which is required for this construction to work. §.§ Reformulation as H-Principal BundleFollowing the discussion in <cit.>, we first substitute the quadratic version (<ref>) of the SC by the equivalent linear constraint <cit.>v_aϵ^aBC D_B· = 0which involves a vector field v_a in the fundamental (SC irrep) of SL(5). This field can take different values on each point of G. In order to relate different points, remember that translations on G are generated by the Lie algebra 𝔤. Especially, we are interested in the action of its generators in the representations5:(t_A)_b^c = X_A,b^c and10:(t_A)_B^C = X_AB^C = 2 X_A, [b_1^[c_1δ_b_2]^c_2] = 2 (t_A)_[b_1^[c_1δ_b_2]^c_2] .Both are captured by the embedding tensor. The corresponding group elements arise after applying the exponential map. Now, assume we have a set of fields f_i with a coordinate dependence such that they solve the linear constraint (<ref>) for a specific choice of v_a. Then, there exists another set of fields f_i' with a different coordinate dependenceD_A f_i' = (Ad_g)_A^B D_B f_i and(Ad_g)_A^B t_B = gt_A g^-1which solve the linear constraint after transforming v_a according tov'_a = (g)_a^b v_b .Here, (g)_a^b represents the left action of a group element g on the vector v_b. This property of the linear constraint (<ref>) is due to the fact that a totally anti-symmetric tensor ϵ is SL(5) invariant.The situation is very similar to the one in . Only the groups and their representations are different. A minor deviation from <cit.> is the splitting of the 10 indices into two sets of subindices. In order to implement the section condition, we introduce a vector v_a^0 which gives rise tov_a^0 ϵ^aβ C t_β = 0 and v_a^0 ϵ^aβ̃C t_β̃ 0 .It splits the generators t_A of 𝔤t_A = [t_α t_α̃ ]and t_α∈𝔪 , t_α̃∈𝔥into a subalgebra 𝔥 and the complement 𝔪 with α=1, …, G/H and α̃=1̃, …, H. We make this decomposition of 𝔤 manifest bysplitting the 10 index A into two non-intersecting subindices α, α̃. The generators t_α̃ generate the stabilizer subgroup H ⊂ G. Its elements leave v^0_a invariant under the transformation (<ref>). This suggests to decompose each group element g∈ G intog = m h with h ∈ Hwhile m is a coset representative of the left coset G/H. Because the action of h is free and transitive, we can interpret G as a H-principal bundleπ: G → G/H = Mover M, the physical manifold.We now study this bundle in more detail. The discussion is closely related to the one in <cit.>. So we keep it short, but still complete. A group element g∈ G is parameterized by the coordinates X^I. In order to implement the splitting (<ref>), we assign to the coset representative m (generated by t_α) the coordinates x^i and to the elements h∈ H (generated by t_α̃) the coordinates x^ĩ. Doing so, results inX^I = [ x^i x^ĩ ]with I = 1, …, dim G ,i = 1, …, dim G/H andĩ = 1̃, …, dimH.In these adapted coordinates π acts by removing the x^ĩ part of the X^I,π(X^I) = x^i .We also note that the corresponding differential map readsπ_*( V^I ∂_I ) = V^i ∂_i .Each element of the Lie algebra 𝔤 generates a fundamental vector field on G. If we want to relate the two of them, we need to introduce the mapt_A^♯ = E_A^I ∂_Iwhich assigns a left-invariant vector field to each t_A∈𝔤. It has the important property ω_L ( t_A^♯ ) = t_A where(ω_L)_g = g^-1∂_I g d X^I = t_A E^A_I d X^Iis the left-invariant Maurer-Cartan form on G. Both (<ref>) and (<ref>) are completely fixed by the generalized background vielbein E_A^I and its inverse transposed E^A_I. After taking into account the splitting of the generators (<ref>) and the coordinates (<ref>), they readE^A_I = [E^α_i0; E^α̃_i E^α̃_ĩ ]and E_A^I = [E_α^iE_α^ĩ;0 E_α̃^ĩ ] . We further equip the principal bundle with the 𝔥-valued connection one-form ω. It splits the tangent bundle T G into a horizontal/vertical bundle H G/V G. While the horizontal partH G = { X ∈ T G| ω(X) = 0 }follows directly from the connection one-form, the vertical one is defined as the kernel of the differential map π_*. We have to impose the two consistency conditionsω( t_α̃^♯ ) = t_α̃and R_h^* ω = Ad_h^-1ωon ω, where R_g denotes right translations on G by the group element g∈ G. In analogy tothe connection one-form is chosen such that the bundle H G solves the linear version (<ref>) of the SC. Following <cit.>, we introduce the projector P_m at each point m of the coset space G/H as a mapP_m: 𝔤→𝔥,P_m = t_α̃ (P_m)^α̃_B θ^Bwhere we denote the dual one-form of the generator t_A as θ^A. P_m is not completely arbitrary. It has to have the propertyP_m t_α̃ = t_α̃ ∀ t_α̃∈𝔥 .So far, this projector is only defined for coset representatives m not for arbitrary group elements g. But, we can extend it to the full group manifold G byP_g = P_m h = Ad_h^-1 P_m Ad_h .This allows us to derive the connection-one formω_g = P_g (ω_L)_gwhere (ω_L)_g is the left-invariant Maurer-Cartan from (<ref>).As a result of (<ref>), it satisfies the constraints in (<ref>).Finally, the H-principal bundle (<ref>) has sections σ_i which are only defined in the patches U_i ⊂ M. They have the formσ_i (x^j) = [ δ^j_k x^kf_i^j̃ ]in the coordinates (<ref>) and are specified by the functions f^j̃_i. As for , we choose those functions such that the pull back of the connection one-form A_i = σ_i^* ω vanishes in every patch U_i <cit.>. This is only possible if the corresponding field strengthF_i (X,Y) = d A_i (X,Y) + [A_i(X), A_i(Y)] = 0vanishes. In this case, A_i is a pure gauge and can be locally “gauged away”. It is very important to keep in mind that this field strength is not the one that describes the tangent bundle T M. Take for example the four sphere S^4 ≅SO(5)/SO(4). It is not parallelizable and thus its tangent bundle cannot be trivial. However, this has nothing to do with the field strength defined in (<ref>). §.§ Connection and Three-Form PotentialForthe projector P_m is related to the NS/NS two-form field B_ij. In the following we show that this result generalizes to the three-form C_ijk for SL(5) gEFT. To this end, we study solutions to the linear version of the SC (<ref>) in more detail. By an appropriate SL(5) rotation, it is always possible to bring v^0_a into the canonical form v^0_a = [ 1 0 0 0 0 ] .This allows us to fix an explicit basis α = {12, 13, 14, 15}andα̃= {23, 24, 25, 34, 35, 45}for the indices appearing in our construction. Furthermore, we introduce the tensorη^αβ,γ̃ = 1/2ϵ^1 α̂β̂γ̃where β̂ labels the second fundamental index of the anti-symmetric pair (e.g. β=13 and β̂=3). The lowered version of η is defined in the same wayη_αβ,γ̃ = ϵ_1 α̂β̂γ̃and its normalization is chosen such that the relationsη^αβ, α̃η_αβ, β̃ = δ^α̃_β̃andη^αβ, α̃η_γδ, α̃ = δ^[α_[γδ^β]_δ]are satisfied. Using this tensor, we express the projector(P_m)^α̃_B = [ η^γδ,α̃ C_βγδ δ^α̃_β̃ ]in terms of the totally anti-symmetric field C_αβγ on M. As we will see by considering the SC solution's GG in the next section, this field is related to the three-from fluxC = 1/6 C_αβγ E^α_i E^β_j E^γ_kd x^i ∧ d x^j ∧ d x^kon the background. Remember that the projector (<ref>) is chosen such that its kernel contains all the solutions of the linear SC (<ref>) for a fixed v_a. It is straightforward to identifyC_αβγ = 1/v_1∑_δϵ_1 α̂β̂γ̂δ̂ v_δ̂ .However, this equation is only defined for v_10. Because (<ref>) is invariant under rescaling all values of v_a specifying a distinct solution of the section condition are elements of ℝℙ^4. This projective space has five patches U_a={ v_a ∈ℝ^5 | v_a = 1} in homogeneous coordinates. From (<ref>), we see that the projector and therewith the connection only covers the subset U_1 for possible solutions of the section condition. As explained in the last subsection, a SC solution is characterized by the vanishing connection A_i. In this case, we can use (<ref>) and (<ref>) to calculate the three-from fluxC = - 1/6η_αβ,γ̃ E^α_i E^β_j E^γ̃_k d x^i ∧ d x^j ∧ d x^kwhich appears in the GG of the theory.Again, it is a convenient crosscheck to consider the symmetry breaking from SL(5) to SL(4) which we discussed in section <ref>. Now, the index of v_a runs only from a=1,…,4 and the linear constraint readsv_a^0 ϵ^aβ c = 0with a four-dimensional totally anti-symmetric tensor ϵ and the explicit basisα = {12, 13, 14}andα̃= {23, 24, 34} ,if we take v^0_a = [ 1 0 0 0 ]. At this point, we have to restrict C from our previous discussion to the two-fromC_αβ4 = B_αβin order the describe SC solutions with v_5=0. Applying this restriction to (<ref>) and (<ref>) gives rise to(P_m)^α̃_B = [ η^γ,α̃ B_βγ δ^α̃_β̃ ]and B_αβ = 1/v_1∑_γϵ_1α̂β̂γ̂ v_γ̂withη^α,β̃ = ϵ^αβ̃ andη_α,β̃ = ϵ_αβ̃ . Here the normalization for the η-tensor is chosen such that the analog relationsη^α,α̃η_α,β̃ = δ^α̃_β̃ andη^α,α̃η_β, α̃ = δ^α_βto (<ref>) hold. Furthermore, the same comments apply as above, but this time for ℝℙ^3 instead of ℝℙ^4. These results are in agreement with the ones forin <cit.>. Especially, the η-tensor gives rise the O(3,3) invariant metricη^AB = ϵ^AB = [0 η^α,β̃; η^β,α̃0 ]with indices A, B in the coordinate irrep 6 of 𝔰𝔩(4). The only difference to <cit.> is that we use a different basis for the Lie algebra in which the off-diagonal blocks η^α,β̃ and η^β,α̃ are not diagonal.In general, it can get quite challenging to find the vanishing connection A_i=0 which is required to solve the SC. However, if 𝔪 and 𝔥 in the decomposition (<ref>) form a symmetric pair with the defining property[𝔥,𝔥] ⊂𝔥 ,[𝔥,𝔪] ⊂𝔪and [𝔪,𝔪] ⊂𝔥 ,there is an explicit construction. It was worked out forin <cit.> and we adapt it to gEFT in the following. Starting point is the observation that the connection A vanishes if C_ijk = - η_αβ,γ̃ E^α_i E^β_j E^γ̃_kis totally anti-symmetric in the indices i, j, k. We rewrite this condition as2 C_ijk - C_kij - C_jki = D_ijk = 0and study it further. To this end, it is convenient to introduce the notation ( t_A , t_B , t_C ) = 2 η_αβ,γ̃ - η_γα,β̃- η_βγ,α̃which allows us to express (<ref>) asD_ijk = ( m^-1∂_i m,m^-1∂_j m,m^-1∂_k m )after taking into account that E^α_i and E^α̃_i are certain components of the left-invariant Maurer-Cartan form (<ref>) with a section where h is the identity element of H. Following <cit.>, we use the coset representativem = exp(- f( x^i ))which gives rise to the expansionm^-1∂_i m = ∑_n=0^∞1/(n+1)! [f, ∂_i f]_n with [f,t]_n = [f […, [f_n times,t] … ]] .Thus, we are left with checking thatD_ijk = ∑_n_1=0^∞ ∑_n_2=0^∞ ∑_n_3=0^∞1/ (n_1 + 1)! (n_2 + 1)! (n_3 + 1)! ([f,∂_i f]_n_1, [f,∂_j f]_n_2,[f,∂_k f]_n_3 )is zero under the restriction (<ref>). To do so, let us first simplify the notation by the abbreviation⟨ n_1, n_2, n_3 ⟩_ijk := ([f,∂_i f]_n_1, [f,∂_j f]_n_2, [f,∂_k f]_n_3 )and rearrange the terms in (<ref>) which results inD_ijk = ∑_m=0^∞ ∑_n_1+n_2+n_3=m ⟨ n_1, n_2, n_3 ⟩_ijk/(n_1 + 1)! (n_2 + 1)! (n_3 + 1)! = ∑_m=0^∞ S^m_ijk .This expression is zero if S^m_ijk vanishes for all m. Therefore, it permits to do the calculation order by order. Let us start withS^0_ijk = ⟨ 0, 0, 0⟩_ijk = 0 .It vanishes because (t_A, t_B, t_C) only gives a contribution if two of its arguments are in 𝔪 and one is in 𝔥 as it is obvious from the definition (<ref>). Here all arguments are in 𝔪. The next order gives rise toS^m_ijk = 1/2!( ⟨ 1, 0, 0 ⟩_ijk + ⟨ 0, 1, 0 ⟩_ijk +⟨ 0, 0, 1 ⟩_ijk) = 0and implements a linear constraint on the structure constants X_AB^C. It is equivalent to( [t, 𝔪], 𝔪, 𝔪 ) +( 𝔪, [t, 𝔪], 𝔪 ) +( 𝔪, 𝔪, [t, 𝔪] ) = 0where t denotes a generator in the algebra 𝔰𝔩(5). Its components furnish the adjoint irrep 24. Note that the splitting of the flat coordinate indices A into α and α̃ singles out the direction v_a^0 in (<ref>). Thus, it break SL(5) to SL(4) with the branching24→1 + 4 + 4 + 15of the adjoint irrep. There is only one generator, corresponding to the crossed out irrep, which violates (<ref>). In quadratic order, we findS^2_ijk = 1/4( ⟨ 1, 1, 0 ⟩_ijk + ⟨ 0, 1,1 ⟩_ijk+ ⟨ 1, 0, 1 ⟩_ijk) + 1/6( ⟨ 2, 0, 0 ⟩_ijk +⟨ 0, 2, 0 ⟩_ijk + ⟨ 0, 0, 2 ⟩_ijk) = 0which represents a quadratic constraint on the structure constants. A solution is given by the symmetric pair (<ref>). It implies that the first three terms are of the form ( 𝔥, 𝔥, 𝔪) plus cyclic permutations, while the last three terms are covered by (𝔪, 𝔪, 𝔪). As noticed before, all of them vanish independently. More generally, we now have[f, ∂_i f]_n ⊂𝔥nodd 𝔪nevenwhich implies ⟨ n_1, n_2, n_3 ⟩_ijk = 0 if n_1mod2 + n_2mod2 + n_3mod2 = 1 .Take any contribution ⟨ n_1, n_2, n_3 ⟩_ijk to S^m_ijk in (<ref>) which is governed by n_1 + n_2 + n_3 = m. If m is even then either two of the integers n_1, n_2, n_3 are odd while the third one is even, or they are all even. In both cases ⟨ n_1, n_2, n_3 ⟩_ijk vanishes and so does the complete S_m for even m. In combination with (<ref>), (<ref>) becomes⟨ n_1 + 1, n_2, n_3 ⟩_ijk + ⟨ n_1, n_2 + 1, n_3 ⟩_ijk+⟨ n_1, n_2, n_3 + 1 ⟩_ijk = 0 forn_1, n_2, n_3even .We use this identity to simplify the cubic contributionS^3_ijk = - 1/4!( ⟨ 3, 0, 0⟩_ijk + ⟨ 0, 3, 0⟩_ijk + ⟨ 0, 0, 3⟩_ijk) = 0which is equivalent to (<ref>) after substituting 1 with 3. Repeating this procedure again and again for S^m_ijk with odd m, we finally obtain the conditions⟨ n_1 + 2 l + 1, n_2, n_3 ⟩_ijk +⟨ n_1 , n_2 + 2 l + 1, n_3 ⟩_ijk + ⟨ n_1, n_2, n_3 + 2 l + 1 ⟩_ijk = 0 ∀ l ∈ℕ(again with n_1, n_2, n_3 even) for the desired result (<ref>) which proves A_i=0. Proving them, requires a generalization of (<ref>) and exploits that the generator t in this equation is an element of 𝔪. As a consequence, the commutator relations of the symmetric pair (<ref>) restrict t to the 4 and 4 in the decomposition (<ref>). So we see that (<ref>) is not an independent constraint, but follows directly from having a symmetric pair. Denoting the two remaining, dual irreps as x_i and y^i, where i=1, …, 4 , the relation[t, ∂_i t]_2l + 1 = [t, ∂_i t]_1 ( x_i y^i/4)^lholds. It reduces (<ref>) to (<ref>) and completes the prove. Finally, note that there is another case[ 𝔪, 𝔪 ] ⊂𝔪for which one immediately has a flat connection. It implies that all terms in (<ref>) are of the form (𝔪,𝔪,𝔪) and vanish. §.§ Generalized GeometryAll solutions of the SC which we discussed in the last two subsections are closely related to GG. In order to make this connection manifest, we have to introduce a map between 𝔥 and the vector space of two-forms Λ^2 T_p^* M at each point p∈ M. More specifically, we use the η-tensor (<ref>) to define the bijective map η_p: 𝔥→Λ^2 T_p^* M asη_p( t_γ̃ ) = 1/2. η_αβ,γ̃ E^α_i E^β_j dx^i ∧ dx^j|_σ(p) .Its inverseη_p^-1( ν ) = . η^αβ, γ̃ t_γ̃ι_E_αι_E_βν|_σ(p)follows form the properties of the η-tensors and the vectors E_a = E_a^i ∂_i. With this map and π_* (<ref>), ω_g(X) (<ref>) from section <ref>, we are able to construct the generalized frame field <cit.>Ê_A(p) = π_* p (t_A^♯) + η_p ω_σ(p) ( t_A^♯)at each point p of the physical space M. It is a map from a Lie algebra element t_A to a vector in the generalized tangent space T_p M ⊕Λ^2 T_p^* M of M at p. Note that we suppress the index labeling the patch dependence of the section for the sake of brevity. However, the generalized frame field Ê_A depends explicitly on the section. For a non-trivial H-principal bundle, we find different frame fields in each patch and have to introduce transition functions accordingly.Using the properties of the mapsπ_*(t_α̃^♯) = 0, ωσ_* = σ^* ω = A = 0 , π_* σ_*= id_T Mandω(t_α̃^♯) = t_α̃ ,we deduce the dual frameÊ^A (p, v, ṽ) = θ^A ( η_p^-1 (ṽ) + ι_σ_* p(v) (ω_L)_σ(p)).Here, we denote elements of the generalized tangent bundle as V = v + ṽ with v ∈ T M and ṽ∈Λ^2 T^* M. Finally, let us expand the generalized frame and its dual into componentsÊ_A = [ E_α^i ∂_i + C_αβγ E^β_i E^γ_j d x^i ∧ d x^j; η_βγ, α̃ E^β_i E^γ_j d x^i∧ d x^j ]andÊ^A (v,ṽ) = [E^α_i v^i; η^βγ,α̃ ( E_β^i E_γ^j ṽ_ij - C_βγδ E^δ_i v^i ) ]where the dependence on p is understood and the indices labeling the patch are suppressed. In the calculation for the dual frame, one has to take into accountθ^α̃( ω_L ( σ_* v ) ) = - C_βγδη^γδ,α̃ E^β_i v^iwhich results from σ^* ω = 0. This result makes perfect sense, because it reproduces the canonical vielbein of a SL(5) theory <cit.>𝒱_Â^Î = [ E_α^i E_α^k C_ijk; 0 E^α_[i E^β_j] ]and its inverse transposed. The C_ijk in this expression is connected to the one we are using in (<ref>) by C_ijk = C_αβγ E^α_i E^β_j E^γ_k.With the generalized frame and its inverse fixed, we are able to transport the generalized Lie derivative (<ref>) to the generalized tangent bundle with the elementsV^Î = [v^i ṽ_ij ] = V^A Ê_A^Îand the dual V_Î = [v_i ṽ^ij ] V_A Ê^A_Î .We distinguish the tangent bundle of the group manifold from the generalized tangent bundle by using hatted indices for the latter. In this index convention, (<ref>) becomesÊ_A^Î = [ E_α^i E_α^k C_kij; 0 η_ij,α̃ ]andÊ^A_Î = [E^α_i0; - C_imnη^mn,α̃η^ij,α̃ ]withη^ij,α̃ = η^βγ,α̃ E_β^i E_γ^j andη_ij,α̃ = η_βγ,α̃ E^β_i E^γ_j .Employing the dual frame on the flat derivative, we obtain∂_Î = Ê^A_Î D_A = [ ∂_i 0 ] .For the infinitesimal parameter of a generalized diffeomorphism ξ^Ĵ, we use the same convention as for V^Î in (<ref>). It is convenient to split the generalized Lie derivative into the two parts. First, we haveℒ_ξ V^Î = ξ^Ĵ∂_Ĵ V^Î - V^Ĵ∂_Ĵξ^Î + Y^ÎĴ_K̂L̂∂_Ĵξ^K̂ V^L̂ .Second, there is the curved version ℱ_ÎĴ^K̂ = ℱ_AB^C Ê^A_ÎÊ^B_ĴÊ_C^K̂ ofℱ_AB^C = X_AB^C - ℒ_Ê_AÊ_B^ÎÊ^C_Î .Together, they form the generalized Lie derivativeℒ_ξ V^Î = ℒ_ξ V^Î + ℱ_ĴK̂^Îξ^Ĵ V^K̂ .In the following we show that ℒ is the untwisted generalized Lie derivative of GG and ℱ_ÎĴ^K̂ implements its twist with the non-vanishing form and vector componentsℱ^ijkl_mn = X_α̃β̃^γ̃η^ij,α̃η^kl,β̃η_mn,γ̃ ℱ_i^jkl = X_αβ̃^γ E^α_i η^jk,β̃ E_γ^lℱ^ij_k^l= X_α̃β^γη^ij,α̃ E^β_k E_γ^lℱ_i^jk_lm = ℱ_αβ̃^γ̃ E^α_iη^jk,β̃η_lm,γ̃ + ℱ_i^jkn C_lmn - ℱ^nojk_lm C_ino ℱ^ij_klm = ℱ_α̃β^γ̃η^ij,α̃ E^β_k η_lm,γ̃ + ℱ^ij_k^n C_lmn - ℱ^ijno_lm C_kno ℱ_ij^k= ℱ_αβ^γ E^α_i E^β_j E_γ^k - ℱ_i^lmk C_jlm - ℱ^lm_j^k C_ilm ℱ_ijkl = ℱ_αβ^γ̃ E^α_i E^β_j η_kl,γ̃ + ℱ_ij^m C_klm - ℱ_i^mn_kl C_jmn - ℱ^mn_jkl C_imn + ℱ_i^mno C_jmn C_klo + ℱ^mn_j^o C_imn C_klo - ℱ^mnop_kl C_imn C_jopwithℱ_αβ^γ = X_αβ^γ - f_αβ^γ ℱ_αβ^γ̃ = X_αβ^γ̃ - G_ijkl E_α^iE_β^j η^kl,γ̃ ℱ_αβ̃^γ̃ = X_αβ̃^γ̃+ 2 f_αβ^γη_δγ,β̃η^δβ,γ̃ ℱ_α̃β^γ̃ = - ℱ_βα̃^γ̃ + f_αγ^δη_βδ,α̃η^αγ,γ̃ .Heref_αβ^γ = 2 E_[α^i ∂_i E_β]^j E^γ_j and G = d C = 1/4! G_ijkld x^i∧ d x^j ∧ d x^k ∧ d x^lare the geometric and four-form fluxes induced by generalized frame (<ref>).To explicitly check that ℒ is equivalent to the familiar generalized Lie derivative <cit.> of exceptional GG, we calculate its components. The evaluation of the first two terms in (<ref>) is straightforward. However, the term containing the Y-tensor is more involved. Therefore, we proceed componentwise and start withY^AB_CDÊ_A^i Ê_B^j = Y^αβ_CDÊ_α^i Ê_β^j = 0 .The last step takes into account that the indices α and β are by design solutions of the SC. Thus, the vector components for the first two indices of the Y-tensor vanish. Furthermore, we know that the form part of the partial derivative ∂_Î vanishes (∂^ij = 0). Hence, the only contributing Y-tensor components are Y_ij^k_L̂M̂ which we evaluate now. To this end, we considerY_ij^k_L̂M̂ = -δ_[i^k E_j]^a5ϵ_a B C E^B_L̂ E^C_M̂and use the dual generalized frame (<ref>) to obtain the non-vanishing componentY_ij^k_l^mn = -2 δ_[i^k δ_j]^[mδ^n]_l .Due to the symmetry of the Y-tensor, we are now able to compute the third term in (<ref>) and obtainY_ij^k_L̂M̂∂_k ξ^L̂ V^M̂ =∂_i ξ^k ṽ_kj + ∂_j ξ^k ṽ_ik - ∂_i ξ̃_jk v^k - ∂_j ξ̃_ki v^k.Taking into account the first and the second term as well, we finally haveℒ_ξ V^Î = ℒ_ξ[v^i; ṽ_ij ] =[ L_ξ v^i; L_ξṽ_ij - 3v^k ∂_[kξ̃_ij] ] ,which is the generalized Lie derivative of exceptional GG <cit.>.As in subsection <ref>, we check our results by considering the restriction to the T-duality subgroup SL(4). In this case we have to modify the map η_p: 𝔥→ T_p^* M, whichis now defined asη_p(t_β̃) = . η_α, β̃ E^α_i d x^i |_σ(p) ,to take the different η-tensor (<ref>) for this duality group into account. Repeating all the steps from above, we find the generalized frameÊ_A = [ E_α^i ∂_i + B_αβ E^β_i d x^i;η_β, α̃ E^β_i d x^i ]its dualÊ^A (v,ṽ) = [ E^α_i v^i; η^β,α̃ ( E_β^i ṽ_i - B_βγ E^γ_i v^i ) ] and the generalized Lie derivative of GG. It has the form (<ref>) withℒ_ξ V^Î =ℒ_ξ[ v^i; ṽ_i ] = [L_ξ v^i; L_ξṽ_i - 2 v^k ∂_[kξ̃_i] ]and the twist in (<ref>) which now has to be evaluated for the generalized frame in (<ref>). After an appropriate change of basis this expression matches the one derived in <cit.>. §.§ Lie Algebra Cohomology and Dual BackgroundsIn general, the SC admits more than one solution. They arise from different choices of v^0_a in (<ref>) and result in a distinguished splitting of the Lie algebra 𝔤 in the coset part 𝔪 and the subalgebra 𝔥. One can always restore the canonical form of v^0_a (<ref>) by a SL(5) rotation. For this case the index assignment (<ref>) remains valid and we only have to check whether the generators t_α̃ form a Lie algebra 𝔥. This situation is very closely related to thecase discussed in <cit.>. Thus, we also use Lie algebra cohomology to explore possible subgroups of the Lie group 𝔤.Let us review the salient features of the construction. First, we only consider transformations in the coset SO(5)/SO(4)⊂ SL(5). All others, at most scale v^0_a and thus leave the subalgebra 𝔥 invariant. A coset element 𝒯_A^B = exp ( λt_A^B )is generated by applying the exponential map to a 𝔰𝔬(5) generator t acting on the coordinate irrep 10. It modifies the embedding tensor according toX'_AB^C = 𝒯_A^D 𝒯_B^E X_DE^F 𝒯_F^C .We expand this expression in λ to obtainX'_AB^C = X_AB^C + λδ X_AB^C + λ^2 δ^2 X_AB^C + …and read off the 𝔤-valued two-formsc_n = t_C ( δ^n X_AB^C ) θ^A ∧θ^B .Only transformations with δ^n X_α̃β̃^γ = 0 are allowed. Otherwise 𝔥 fails to be a subalgebra. Finally, we have to check whether the restricted formsc_n = t_γ̃ ( δ^n X_α̃β̃^γ̃ ) θ^α̃∧θ^β̃are in the Lie algebra cohomology H^2(𝔥,𝔥). If so, they give rise to a infinitesimal non-trivial deformation of 𝔥. Obstructions to the integrability of this deformation lie in H^3(𝔥, 𝔥). §.§ Generalized Frame FieldA significant application of the formalism presented in this paper is to construct the frame fields ℰ_A^Î of generalized parallelizable manifolds M. In the following, we show that ℰ_A^Î = - M_A^B Ê'_B^Îfulfills the defining equation (<ref>) in the introduction if an additional linear constraint on the structure constants X_AB^C holds. The derivation is done step by step starting with the frame Ê'_A^Î. It differs from (<ref>) by using a three-from 𝒞 instead of C (see (<ref>) in the introduction). So first, we calculateX'_AB^C = ℒ_Ê'_AÊ'_B^ÎE'^C_Îwhich has the non-trivial componentsX'_αβ^γ = f_αβ^γX'_αβ^γ̃ = 𝒢_ijkl E_α^i E_β^j η^kl,γ̃X'_αβ̃^γ̃ = 2 f_αβ^γη_δγ,β̃η^δβ,γ̃X'_α̃β^γ̃ = - X'_βα̃^γ̃ - f_αγ^δη_βδ,α̃η^αγ,γ̃ .As before f_αβ^γ denotes the geometric flux (<ref>) and𝒢 = d 𝒞 = 1/4!𝒢_ijkld x^i ∧ d x^j ∧ d x^k ∧ d x^lis the field strength corresponding to 𝒞. In (<ref>), Ê'_A^Î is twisted by the SL(5) rotationM_B^A t_A = m^-1 t_B m = (Ad_m^-1)_B^A t_Awith the inverse transposet_A M^A_B = m t_B m^-1 .Next, we combine the two of them and evaluateX”_AB^C = ℒ_M_A^D Ê'_D (M_B^E Ê'_E^Î) M^C_F Ê'̂^F_Î .It is convenient to write the result asX”_AB^C = X”'_DE^F M_A^D M_B^E M^C_F with X”'_AB^C = X'_AB^C + 2 T_[AB]^C + Y^CD_EB T_DA^EandT_AB^C = - Ê'_A^Î∂_Î M^D_B M_D^C .Taking into account the special form of M_B^A in (<ref>), this tensor can be calculated:T_AB^C = - Ê_A^i E^D_i X_DB^C =[ - X_α B^C + η^δϵ,δ̃ C_αδϵ X_δ̃B^C 0 ] .In the second step, we remember that for a SC solution the connection A vanishes. This allows us to identify E_α^i E^β̃_i = - η^γδ,β̃ C_αγδ.By plugging the solution for T_AB^C into (<ref>), we obtain the non-vanishing componentsX”'_α̃β̃^γ̃ = - X_α̃β̃^γ̃X”'_αβ̃^γ = - X_αβ̃^γX”'_α̃β^γ = - X_α̃β^γ X”'_αβ̃^γ̃ = -2 X”'_αβ^γη_δγ,β̃η^δβ,γ̃X”'_α̃β^γ̃ = - X”'_βα̃^γ̃X”'_αβ^γ = - 2 X_αβ^γ +2 X_α̃[β^γ C_α]δϵη^δϵ,α̃ + f_αβ^γandX”'_αβ^γ̃ = - 2 X_αβ^γ̃ + 2 X_γα^α̃η_δβ,α̃η^γδ,γ̃ - ( 2 X_γβ^δ C_δϵα - 4 X_γα^δ C_δϵβ ) η^γϵ,γ̃ -2 X_αβ^γ C_δϵγη^δϵ,γ̃ + 𝒢_ijkl E_α^i E_β^j η^kl,γ̃after imposing the constraintsC3 X_Aβ̃^γ̃ = -2 X_Aβ^γη_δγ,β̃η^δβ,γ̃and X_αγ^δη_βδ,α̃η^αγ,γ̃ = 0 .At this point, (<ref>) proves to be a good choice. Up to a sign, many components are already as we want them to be. This gets even better, if we take into account the explicit expressionf_αβ^γ = X_αβ^γ - 2 X_α̃[β^γ C_α]δϵη^δϵ,α̃for the geometric flux which results inX”'_αβ̃^γ̃ = - X_αβ̃^γ̃ ,X”'_α̃β^γ̃ = - X_α̃β^γ̃and X”'_αβ^γ = - X_αβ^γafter imposing the constraints (<ref>). Finally, there is the last contribution (<ref>) which should evaluate to -X_αβ^γ̃. It requires an appropriate choice for the four-form𝒢_ijkl = f(x^1, x^2, x^3, x^4) ϵ_ijkl .Being the top-form on M, it only has one degree of freedom captured by the function f. With this ansatz, the last term in (<ref>) becomes𝒢_ijkl E_α^i E_β^j η^lk,γ̃ = f(E_ρ^i) ϵ_1α̂β̂γ̂δ̂η^γδ,γ̃ .If we choose f = λ (E^ρ_i) for an appropriate, constant λ, the miracle happens and we find X”'_αβ^γ̃ = -X_αβ^γ̃. The key to this result is that the structure constants X_AB^C are not arbitrary, but severely constrained by the linear constraints (<ref>), (<ref>) and (<ref>). The first two are solved in section <ref> and we present the solutions to the remaining one at the end of this section. For the moment, we continue withX”'_AB^C = - X_AB^C under (<ref>) - (<ref>) .Structure constants of a Lie algebra are preserved under the adjoint action (<ref>). Thus, we immediately implyX”_AB^C = X”'_AB^C = - X_AB^C .Up to the minus sign, this is exactly the result we are looking for. In order to get rid of this wrong sign, we introduce an additional minus in the generalized frame field ℰ_A^Î (<ref>). The result is equivalent to (<ref>) in the introduction. As argued above, the three-from 𝒞 it contains has to be chosen such that𝒢 = d 𝒞 = λ(E^ρ_i) d x^1 ∧ d x^2 ∧ d x^3 ∧ d x^4 =λvol ,where vol is the volume form on M induces by the frame field E^α_i.Finally, we have to find the solutions of the linear constraint (<ref>). Otherwise the construction above does not apply. In order to identify these solutions, we discuss embedding tensor components in the 15 <cit.>X_abc^d = δ_[a^d Y_b]cparameterized by the symmetric matrix Y_ab and in the 40 <cit.>X_abc^d = -2 ϵ_abcef Z^ef,dgiven in terms of the tensor Z^ab,c with Z^ab,c=Z^[ab],c and Z^[ab,c]=0. The structure constants of the corresponding Lie algebra 𝔤 follow from the further embedding of them into 10×10×10 <cit.>X_AB^C = X_a_1 a_2, b_1 b_2^c_1 c_2 = 2 X_a_1 a_2[b_1^[c_1δ_b_2]^c_2] .If there are only contributions from the 15, this expression is identical to the structure constants because the corresponding group manifold is ten-dimensional.We study this case first. Splitting the indices A, B, C, … into a coset component α and a subalgebra part α̃ according to (<ref>) singles out one direction in the fundamental irrep of SL(5). It is given by v^0_a in (<ref>) and results in the branching15→1 + 4 + 10from SL(5) to SL(4). The linear constraint (<ref>) is violated by the crossed out irreps. If we only take into account the remaining ones, all terms containing C_αβγ in X”'_αβ^γ̃ vanish. For (<ref>) to hold, we further require that the relationX_αβ^γ̃ - 2 X_γα^α̃η_δβ,α̃η^γδ,γ̃ = λϵ_1α̂β̂γ̂δ̂η^γδ,γ̃is satisfied. This is the case, if we identifyλ = -3/4Y_11 .For all the remaining gaugings in (<ref>), one should in principal be able to construct a generalized parallelizable space M. However, this construction relies on finding a flat connection A in order to solve the SC in the first place. In general deriving this connection can turn out to be complicated. However, as explained at the end of <ref>, if M is a symmetric space there is a simple procedure to immediately solve this challenge. Luckily all remaining irreps in (<ref>) give rise to a symmetric pairs 𝔪 and 𝔥 so that one can immediately solve the SC. Furthermore, the solutions to the quadratic constraint (<ref>) are known as well. The resulting group manifolds depend on the eigenvalues of the symmetric, real matrix Y_ab. If p of them are positive, q are negative and r are zero, we findG = CSO(p,q,r) = SO(p,q) ⋉ℝ^(p+q) rwith p + q + r = 5 .Our construction applies to all corresponding generalized frames ℰ_A. They where also constructed in <cit.> by taking a clever ansatz in a distinguished coordinate system. Before this work, <cit.> presented the generalized frame for SO(5) (p=5, q=r=0), the four-sphere with G-flux.With gaugings in the 40, group manifold with G<10 are relevant. As discussed in section <ref>, the irreps of the embedding tensor branch into the U-duality subgroups. Again, v^0_a in (<ref>) distinguishes a direction and results in an additional branching. To see how this works, consider the SL(3)×SL(2) solutions in figure <ref>. Starting with G=9, the relevant embedding tensor components(1,3) + (3,2) + (6,1) + (1,2) → (1,3) + (1,2) + (2,2) +(1,1) + (2,1) + (3,1) + (1,2)branch from SL(3)×SL(2) to SL(2)×SL(2). All crossed out irreps decent from the 4 in (<ref>). Only the last irrep (1,2) originates from the 40. It does not admit a symmetric pair. Still, one is able to construct a generalized frame field for the four-tours with geometric flux in section <ref> which is realized by a gauging in this irrep. For the scaling factor λ in (<ref>) the relation (<ref>) still applies. One can go on and repeat this discussion for group manifolds with G<9. We do not perform it here, because all the examples we present in the next section are covered by the cases above.§ EXAMPLESIt is instructive to study some explicit examples for the construction described in the previous sections. In the following, we present the four-torus with G-flux, its dual backgrounds and the four-sphere with G-flux. While the former is well-known from conventional EFT, it illustrates how dual backgrounds arise in our formalism. Furthermore, it allows to study group manifolds G with G<10, which arise from gaugings in the 40. In this case SL(5) is broken to SL(3)×SL(2). A more sophisticated setup is the four-sphere with G flux which corresponds to the group manifolds SO(5). It is was studied in <cit.> and so permits to compare the resulting generalized frame field ℰ_A with the literature. §.§ Duality-Chain of the Four-Torus with G-FluxIn string theory there is the well-known duality chain<cit.>H_ijk↔ f_ij^k ↔ Q_i^jk↔ R^ijk ,where each adjacent background is related by a single T-duality which maps IIA ↔ IIB string theory. In this section, we show how parts of this chain result from different SC solutions on a ten- and a nine-dimensional group manifold. In order to uplift these examples to M-theory, we need to consider only IIA backgrounds and two T-dualities taking IIA ↔ IIA string theory. Thus, the above duality chain splits into the two distinct duality chainsH_ijk↔ Q_i^jkandf_ij^k ↔ R^ijk .Similarly, when considering M-theory, we apply three U-duality transformations to ensure that we map M-theory to itself. One may think of this as taking a T^3 in the limit of vanishing volume. Indeed, if we had taken only a S^1 of vanishing radius, we would have obtained weakly-coupled IIA string theory. A T^2 of vanishing volume gives IIB string theory (think of taking repeated small radius limits of the two circles of T^2). In this case, we have weakly coupled IIA compactified on a small circle. Applying T-duality to this circle results in IIB in the decompactification limit. Thus, for every two-cycle of vanishing volume we see that we open up a new dual direction. Having a T^3 of vanishing volume means we loose three directions but open up three new ones (one for each of the three two-cycles in T^3). So we finally arrived at an eleven-dimensional background again. Another way to see this is to identify two of the directions of the U-duality with the two directions of the T-duality and the third one with the M-theory circle. This also ensures the correct dilaton transformation. From this arguments it becomes clear that the M-theory T^4 duality chain is also split and we haveG_ijkl↔ Q_i^jklandf_ij^k ↔ R^i,jklm .As we will see only the former can be realized in our framework. This finding is in agreement with thecase, where the R-flux background does not posses a maximally isotropic subalgebra 𝔥 <cit.>.The splitting (<ref>) and (<ref>) of the duality chain is also manifest in the embedding tensor <cit.>. For SL(5), it has two irreducible representations (not counting the trombone which we neglect in this paper). Each chain represents one of these irreps. Duality transformations are implemented by SL(5) rotations. These clearly do not mix different irreps.§.§.§ Gaugings in the 15Let us start with the first chain. It is fully contained in the irrep 15 <cit.> which we express in terms of the symmetric tensor Y_ab. The resulting embedding tensor is (<ref>) and the corresponding structure constants arise from (<ref>). It is always possible to diagonalize the symmetric matrix Y_ab by a SO(5) transformation. For gaugings in the 15 only, the quadratic constraint is fulfilled automatically. A four-torus with 𝐠 units of G-flux is given by the explicit choiceY_ab = - 4 𝐠 ( 1, 0, 0, 0, 0 ) .This particular choice is compatible with the vector v_0^a in (<ref>) and the decomposition (<ref>) of the 10 index A = (α, α̃). It gives rise to the group manifold G =CSO(1,0,4) with an abelian subgroup H which is generated by all infinitesimal translations in ℝ^6. We use the 21-dimensional, faithful representation of 𝔤 derived in appendix <ref> to obtain the matrix representationm= exp( t_1 x^1 ) exp( t_2 x^2 ) exp( t_3 x^3 ) exp t_4 x^4andh= exp( t_1̃ x^1̃) exp( t_2̃ x^2̃) exp( t_3̃ x^3̃) exp(t_4̃ x^4̃) exp( t_5̃ x^5̃) exp(t_6̃ x^6̃)of the Lie group G. This group is not compact and therefore does not represent the background we are interested in (clearly a torus is compact). Thus, we have to quotient by the discrete subgroup CSO(1,0,4,𝐙) from the left. Doing so is equivalent to impose the coordinate identifications (<ref>) and (<ref>) which are derived in appendix <ref>.For this setup, the connection A = A^α̃ t_α̃ readsA^1̃ = [ ( 𝐠 x^2 + C_134 ) d x^1 + C_234d x^2 ], A^2̃ = [ ( 𝐠 x^3 - C_124 ) d x^1 + C_234d x^3 ], A^3̃ = [ ( 𝐠 x^3 - C_124 ) d x^2 - C_134d x^3 ], A^4̃ = [ ( 𝐠 x^4 + C_123 ) d x^1 + C_124d x^4 ], A^5̃ = [ ( 𝐠 x^4 + C_123 ) d x^2 - C_134d x^4 ], A^6̃ = [ ( 𝐠 x^4 + C_123 ) d x^3 + C_124d x^4 ]in the patch we are considering. For the three-form fieldC = 𝐠/2 ( x^1 d x^2 ∧ d x^3 ∧ d x^4 - x^2 d x^1 ∧ d x^3 d ∧ x^4 + x^3 d x^1 ∧ d x^2 ∧ d x^4 - x^4 d x^1∧ x^2 ∧ x^3 ) ,with the flux contributionG_Ê = d C = 2 𝐠d x^1∧ d x^2 ∧ d x^3 ∧ d x^4to the generalized frame field Ê_A, the field strength F = d A vanishes. In order to set A=0 in the current patch, we apply the transformation g→ g exp( t_α̃λ^α̃ ) to all group elements withλ^1̃ = - 𝐠/2 x^1 x^2 ,λ^2̃ = - 𝐠/2 x^1 x^3 ,λ^3̃ = - 𝐠/2 x^2 x^3 ,λ^4̃ = - 𝐠/2 x^1 x^4 ,λ^5̃ = - 𝐠/2 x^2 x^4 ,λ^6̃ = - 𝐠/2 x^3 x^4 .It results in the desired A=0 and the background generalized vielbeinE^α_i = [ 1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] ,E^α̃_i = 𝐠/2[ x^2 - x^1 0 0; x^3 0-x^1 0; 0 x^3-x^2 0; x^4 0 0-x^1; 0 x^4 0-x^2; 0 0 x^4-x^3 ]and E^α̃_i̅ = [ 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1 ] .This gauging gives rise to a symmetric space. Thus, we also could have used the coset representativem = exp ( t_1 x^1 + t_2 x^2 + t_3 x^3 + t_4 x^4 )instead of (<ref>) to find the same result. However, it is nice to demonstrate the full procedure at least once. With (<ref>) we calculate the generalized frame field Ê_A^Î, its dual and finally the twist ℱ_ÎĴK̂ of the generalized Lie derivative (<ref>). It has contributions (<ref>) from the four-formG_ℱ = 1/4!ℱ_ijkld x^i∧ d x^j∧ d x^k∧ d x^l = - 𝐠d x^1 ∧ d x^2 ∧ d x^3 ∧ d x^4only. In total, we obtain the expected 𝐠 units of G-fluxG = G_Ê + G_ℱ = 1/4! X_αβ^γ̃ E^α_i E^β_j η_γδ,γ̃ E^γ_k E^δ_l d x^i ∧ d x^j ∧ d x^k ∧ d x^l =𝐠d x^1 ∧ d x^2 ∧ d x^3 ∧ d x^4.on the background after combining this contribution with G_Ê from the generalized frame.This result is very similar to the one obtained for the torus with H-flux in <cit.>. Again the flux is split between the twist term and the frame field in a particular way. However, this splitting arises naturally from the principal bundle construction. To see how this works, we calculate the flux contribution from the frame fieldG_Ê = d C = - 1/6 E^α_i E^β_j d ( η_αβ,γ̃ E^γ̃_j d x^j) ∧ d x^i ∧ d x^jby using the relation (<ref>). We identify𝒜_αβ = η_αβ,γ̃ E^γ̃_i d x^iwith the connection of a T^6 bundle over the tours. Thus, each independent 𝒜_αβ, like e.g. 𝒜_12, is the connection of a circle bundle. The first Chern class of these bundles are defined asc_αβ = d 𝒜_αβ .and by plugging the result (<ref>) for E^α̃_i into this equation, we obtain the independent classesc_21 = 𝐠d x^3 ∧ d x^4, c_13 = 𝐠d x^2 ∧ d x^4, c_41 = 𝐠d x^2 ∧ d x^3, c_32 = 𝐠d x^1 ∧ d x^4, c_24 = 𝐠d x^1 ∧ d x^3, c_43 = 𝐠d x^1 ∧ d x^2 ,explicitly. Each of them represents a class in the integer valued cohomology H^2(𝒮_αβ, M)=ℤ of the circle bundle 𝒮_αβ over M=T^4. Furthermore, they are not trivial, which shows that the principal bundle we constructed is non-trivial, too. If we denote the cohomology class of a closed form ω by [ ω ], we can rewrite (<ref>) as[ G_Ê ] = 1/3 ( [c_21] + [c_13] + [c_41] + [c_32] + [c_24] + [c_43] ) .Because G_Ê is a top form on the T^4 it lives in the integer valued de Rham cohomology H^4_dR(M) which is isomorph to H^2(𝒮_αβ, M). Thus, there is no obstruction in comparing the Chern numbers with [G_Ê] and (<ref>) makes perfect sense. All different S^1 factors in the H-principal bundle are equal. So it is natural that they share the same Chern number, namely one. In this case (<ref>) forces[G_Ê] = 2 𝐠which is compatible with our result (<ref>).It is interesting to note that in this example the field strength F = d A for the H-principal bundle vanishes everywhere on M. Still it is not possible to completely gauge away the connection A. Because the gauge transformation λ^ã in (<ref>) is not globally well-defined on M. This is clearly a result of the discrete subgroup which was modded out from the left to make G compact. One could think that this effect is related to the topological non-trivial G-flux in this background. But it is not, as the four-sphere with G-flux in the next section proves. There, one can get rid of the connection everywhere. However, locally we can always solve the SC and construct the generalized frame fieldℰ_α = - E_α^i ∂_i + ι_E_α𝒞', ℰ_α̃ = - 1/2η_ij,α̃d x^i ∧ d x^jwhere we take E_α^i as the inverse transpose from the frame in (<ref>) and𝒞' = 𝐠( 2 x^4d x^1 ∧ d x^2 ∧ d x^3+ x^3d x^1 ∧ d x^2 ∧ d x^4 - x^2d x^1 ∧ d x^3 ∧ d x^4 + x^1d x^2 ∧ d x^3 ∧ d x^4 ) .withG = d 𝒞' = 𝐠d x^1 ∧ d x^2 ∧ d x^3 ∧ d x^4 .The gauging (<ref>) represents the irrep 1 in the solution (<ref>) of the third linear constraint. Thus, this frame results from the construction in section <ref> with λ = 3 𝐠 and 𝒞 = -3 𝐠x^4 d x^1 ∧ d x^2 ∧ d x^3,resulting in the required𝒢 = d 𝒞 = 3 𝐠d x^1 ∧ d x^2 ∧ d x^3 ∧ d x^4 . Now, we perform a deformation of this solution by applying a 𝒯_A^B which generates the SO(5) rotation𝒯_a^b = [01000; -10000;00100;00010;00001 ]as𝒯_a_1 a_2^b_1 b_2 =2 δ_[a_1^[b_1𝒯_a_2]^b_2] .After this rotation the subalgebra 𝔥 becomes non-abelian and is governed by the non-vanishing commutator relations[t_1̃, t_2̃ ] = 𝐠t_3̃ ,[t_1̃, t_4̃ ] = 𝐠t_5̃and [t_2̃, t_4̃ ] = 𝐠t_6̃ .In this frame, solving the SC is easier than for the one we considered above. This is because the field strength A vanishes automatically for C=0. As a result, the vielbein is trivial with E^ã_i vanishing whereas the remaining components E^α_i and E^α̃_ĩ are equivalent to the previous results in (<ref>). The generalized frame field Ê_A does not contribute to the fluxes of the background. Thus, the only contribution comes from the twist (<ref>)Q_i^jkl = ℱ_i^jkl - ℱ^jk_i^l = 2 X_αβ̃^γ E^α_i η^jk,β̃ E_γ^lwhich is totally anti-symmetric in the indices i, j, l. It is convenient to recast this quantity asQ_ij = 1/3! Q_i^klmϵ_klmj =-𝐠( 1,0, 0,0 )where ϵ_klmj is the totally anti-symmetric tensor in four dimensions. So we conclude that this background has 𝐠 units of Q-flux. As it arises by a SO(5) transformation from the previous one with 𝐠 units of G-flux, we found a direct realization of the duality chain (<ref>).This gauging is in the 10 of (<ref>). So we are able to construct the generalized frameℰ_α = - E_α^i ∂_i, ℰ_α̃ = η_ij,α̃β^ijk∂_k - 1/2η_ij,α̃d x^i ∧ d x^jwith 𝒞=0 and the totally anti-symmetric β^ijk whose non-vanishing components readβ^234 = - 𝐠/2 x^1 .It sources the Q-fluxQ_i^jkl = - 2 ∂_i β^jklin (<ref>). An alternative way to obtain a generalized frame with the same properties is to rotate the generalized frame field in the previous duality frame (<ref>) by 𝒯_A^B in (<ref>).§.§.§ Gaugings in the 40In order to realize the twisted four-torus from which the second chain (<ref>) starts, we consider the embedding tensor solution (<ref>) with the non-vanishing components <cit.>Z^23,3 = - Z^32,3 = 𝐟/2to obtain 𝐟 units of geometric flux. As before the structure constants of the Lie algebra 𝔤 arise from (<ref>). However, this algebra is not ten-dimensional anymore. As discussed in section <ref> gaugings in the 40 reduce the dimension of the group manifold according to (<ref>). Thus, the G we consider here is nine-dimensional and admits a SL(3)×SL(2) structure as shown in figure <ref>. Its coordinates decompose into the two irreps(3,2): {1, 2, 1̃, 2̃, 3̃, 4̃}and (3, 1): {3, 4, 5̃}with the adapted versionα = {12, 13, 14, 15}andα̃= {24, 25, 34, 35, 45}of the basis (<ref>) for the components of the 10 indices α and α̃. In this basis, the non-vanishing commutator relations, defining 𝔤, read[t_5̃, t_3] = 𝐟t_2̃ , [t_5̃, t_4] = 𝐟t_4̃and [t_3, t_4] = 𝐟t_2 .Together, the six generators appearing in these three relations form the algebra 𝔠𝔰𝔬(1,0,3) with the center {t_2, t_2̃, t_4̃}. While the remaining t_1, t_1̃ and t_3̃ give rise to a three-dimensional abelian factor. There is a 16-dimensional faithful representation for 𝔤 which is presented in appendix <ref>. We use it to calculate the coset elements m according to (<ref>), while elements of the subgroup H are given byh = exp( t_1̃ x^1̃) exp( t_2̃ x^2̃) exp( t_3̃ x^3̃) exp(t_4̃ x^4̃) exp( t_5̃ x^5̃) .As in the duality chain in the last subsection, the identifications (<ref>) and (<ref>) on the coordinates of the group manifold are required here, too. Otherwise, we would not obtain a compact background. It is a fibrationT^2=F ↪ M → B=T^2where a point on the fiber F is labeled by the coordinates x^1, x^2, while the base B is parameterized by the remaining coordinates x^3 and x^4. The fiber is contained in the coordinate irrep (2,3) and the base is part of (1,3). Again, the gauge potential A vanishes for C=0 automatically. Thus, there is a solution of the SC with the generalized background vielbeinE^α_i = [ 1 0 0 0; 0 1 f x^4 0; 0 0 1 0; 0 0 0 1 ] ,E^α̃_i = - 𝐟[0000;00 x^5̃0;0000;000 x^5̃;0000 ]and E^α̃_i̅ = [ 1 0 0 0 0; 0 1 0 0 0; 0 0 1 0 0; 0 0 0 1 0; 0 0 0 0 1; ] .It comes with the non-vanishing geometric fluxf^2 = ∂_[i E^2_j] d x^i ∧ d x^j = - 𝐟d x^3 ∧ d x^4along the same lines as theexample three-torus with f-flux in <cit.>. As for , the twist term in the generalized Lie derivative (<ref>) vanishes for this background.It is instructive to take a closer look at the GG of this setup. Because the group manifold does not have the full ten dimensions things are more subtle.Remember that in general the SC of SL(3)×SL(2) EFT admits two different solutions. First, there are those reproducing eleven-dimensional supergravity with three internal directions and second there are solutions resulting in ten-dimensional type IIB (only two internal directions)<cit.>. This fact is manifest from the SL(5) perspective we take. Each solution of (<ref>) is labeled by a distinct v^0_a in the 5 of SL(5) which branches as5→ (1,2) + (3, 1)to SL(3)×SL(2). The first irrep in this direct sum captures SC solutions with a eleven-dimensional SUGRA description and the second one corresponds to type IIB. The latter case is implemented on the two-dimensional fiber F. Furthermore, the splitting of M into base B and fiber F allows to distinguish between three different kinds of two-forms on Λ^2 T^* M. Those with all legs on the base or the fiber and further the ones with one leg on the base and the other leg on the fiber. Over each point p of M, Λ^2 T_p^* M is a six dimensional vector space. Nevertheless 𝔥, is only five-dimensional. Hence, the map η_p in (<ref>) is not bijective anymore. This is a problem because this property is essential to our construction in section <ref>. However, we restore it by removing two-forms whose legs are completely on the base from the codomain of η_p. They are not part of the resulting GG. Apart from that (<ref>) is still valid. Especially, we are able to construct the generalized frame field ℰ_A because the gauging for this example is the surviving (1,2) of (<ref>). For the commutator relations in (<ref>), one sees that the resulting physical manifold M is not a symmetric space because both [𝔥,𝔪]⊂𝔪 and [𝔪,𝔪]⊂𝔥 are violated. Still one is able find a SC solution (as we did) because 𝔪 is a subalgebra of 𝔤 with [𝔪,𝔪]⊂𝔪.The corresponding generalized frame field isℰ_α = E'_α^i ∂_i , ℰ_α̃ = 1/2η_βγ,α̃E'^β_i E'^γ_i d x^i ∧ d x^jwith the frameE'_α^i = [-1 0 0 0; 0-1 0 0; 0 0-1 0; 0 x^3 𝐟 0 - 1 ]and the dualE'^α^i = [-1 0 0 0; 0-1 0 - x^3 𝐟; 0 0-1 0; 0 0 0-1 ] .However, this step is redundant because the twist ℱ_ÎĴ^K̂ already vanished for Ê_A.1emLet us finally come to the dual background with R-flux in (<ref>). For our specific choice of v^0_a in (<ref>), it is completely fixed by the four independent components Z^a1,1 (a=1, …, 4) of the 40 in the embedding tensor (<ref>) <cit.>. Clearly the SO(5) transformation𝒯_a^b = [00100;01000; -10000;00010;00001 ]brings (<ref>) into this form. However, there are two problems with the resulting setup. First, the generators t_α̃ do not yield a subalgebra 𝔥 after applying 𝒯. In , we face the same situation. It is in agreement with the completely non-geometric nature of the R-flux. If we would find a SC solution with our technique for the R-flux, there would be a geometric interpretation in terms of a manifold M equipped with a GG. This is not the case. But there is also another subtlety which is absent in . Remember that SL(5) gets broken to SL(3)×SL(2) for the torus with geometric flux because the corresponding structure constants originate from the 40. But the transformation (<ref>) is not an element of this reduced symmetry group. Hence, the second background in the chain (<ref>) does not admit the most general SC solutions we discuss in this paper. There are of course still solutions, where the fluctuations are constant. §.§ Four-Sphere with G-FluxIn order to obtain a four-sphere with radius R as the physical manifold M after solving the SC, we have to consider the group manifold SO(5). It arises from a embedding tensor solution in the 15 withY_ab = -4/R (1, 1, 1, 1, 1) .In comparison to the previous examples in section <ref> it is much simpler to obtain a faithful representation of the corresponding Lie algebra 𝔤=𝔰𝔬(5). A canonical choice are the anti-symmetric matrices(t_A)_b^c = -1/2 X_Ab^cwhich arise directly from the embedding tensor (<ref>) and act on the fundamental irrep of 𝔤. In contrast to (<ref>), we now parameterize coset representatives bym = exp[ Rϕ^1 (cos(ϕ^2) t_1 + sin(ϕ^2)cos(ϕ^3) t_2 + .. sin(ϕ^2) sin(ϕ^3) cos(ϕ^4) t_3 + sin(ϕ^2) sin(ϕ^3) sin(ϕ^4) t_4) ],where the angels represent spherical coordinates withϕ^1 , ϕ^2 , ϕ^3 ∈ [0, π] andϕ^4 ∈ [0, 2π) ,while the elements of the subgroup still are calculated by (<ref>). Together 𝔪 and 𝔥 form a symmetric part. As shown at the end of section <ref>, the particular choice (<ref>) for m has the advantage that the gauge potential A automatically vanishes forC = R^3 tan(ϕ^1/2) sin^3(ϕ^1) sin^2(ϕ^2) sin(ϕ^3) d ϕ^2 ∧ d ϕ^3 ∧ d ϕ^4.The corresponding field strengthG_Ê = d C = 4 R^3 cos(ϕ^1/2)sin^3(ϕ^1/2)((1+3cos(ϕ^1)) sin^2 (ϕ^2) sin (ϕ^3 ) d ϕ^1 ∧ d ϕ^2 ∧ d ϕ^3 ∧ d ϕ^4is in the trivial cohomology class of H^4_dR(S^4) because the integral∫_S^4 G_Ê = 0is zero. At the same time C and therewith the connection A^α̃ are globally well-defined. In contrast to the four-torus with G-flux in the last section, we can gauge away the connection globally although the background has G-flux in a non-trivial cohomology class, too. Another interesting quantity one can compute is the first Pontryagin class for the connection𝒜^α̃ = E^α̃_i d x^i .This quantity is analogous to the Chern classes, we computed for the T^6-bundle in the T^4 with G-flux background. It vanishes completely.To write down the generalized frame field (<ref>), we furthermore need the vielbeinE^α_i = R [ c_2 - s_1 s_2 0 0; c_3 s_2 c_2 c_3 s_1 - s_1 s_2 s_3 0; c_4 s_2 s_3 c_2 c_4 s_1 s_3 c_3 c_4 s_1 s_2 - s_1 s_2 s_3 s_4; s_2 s_3 s_4 c_2 s_1 s_3 s_4 c_3 s_1 s_2 s_4 c_4 s_1 s_2 s_3 ]with c_i = cos(ϕ^i) and s_i = sin(ϕ^i) which is part of the left-invariant Maurer-Cartan form E^A_I in (<ref>). It gives rise to the metricd s^2 = E^α_i δ_αβ E^β_j d ϕ^i d ϕ^j = R^2 ( (d ϕ^1)^2 + s_1^2(d ϕ^2)^2 + s_1^2 s_2^2 (d ϕ^3)^2 + s_1^2 s_2^2 s_3^2 (d ϕ^3)^2 )on a round sphere with radius R. Equipped with a solution of the SC for G=SO(5), we are able to apply the construction in section <ref> and obtain the generalized frame field ℰ_A with 𝒞 such that𝒢 = d 𝒞 = 3 R^3 sin^3(ϕ^1) sin^2(ϕ^2) sin(ϕ^3)d ϕ^1 ∧ d ϕ^2 ∧ d ϕ^3 ∧ d ϕ^4 = 3/Rvol .Because the complete result is not very compact, we do not present it here. Instead, we present an alternative parameterization of the group elements m in terms of Cartesian coordinatesy^1= R cos(ϕ^1)y^2= R sin(ϕ^1) cos(ϕ^2) y^3= R sin(ϕ^1) sin(ϕ^2) cos(ϕ^3) y^4= R sin(ϕ^1) sin(ϕ^2) sin(ϕ^3) cos(ϕ^4) y^5= R sin(ϕ^1) sin(ϕ^2) sin(ϕ^3) sin(ϕ^4) .They have the advantage that they yield a very simple coset representativem = 1/R[y^1 -y^2 -y^3 -y^4 -y^5;y^2 y^22 y^23 y^24 y^25;y^3 y^23 y^33 y^34 y^35;y^4 y^24 y^34 y^44 y^45;y^5 y^25 y^35 y^45 y^55 ]with y^ij = R δ^ij - y^i y^j/R + y^1and allow a direct comparison of our results with <cit.>. On the order hand, we have to implement the additional constraint∑_i=1^5 (y^i)^2 = Rin all equations that follow. As before, we calculateE^α_i = 1/R[ -y^2 y^22 y^23 y^24 y^25; -y^3 y^23 y^33 y^34 y^35; -y^4 y^24 y^34 y^44 y^45; -y^5 y^25 y^35 y^45 y^55 ] = E_α^ifor the components of the left-invariant Maurer-Cartan form. Finding the vectors E_α^i is a bit more challenging here than before because E^α_i is not a square matrix and therefore not invertible. However, it is completely fixed by E_α^i E^β_i = δ_α^β and furthermore requiring that all vectors E_α^i are perpendicular to the radial direction r⃗=(y^1 y^2 y^3 y^4 y^5)^T. Now, we calculate the vector part ℰ_A^i of the generalized frame which we denote as V_A^i in order to permit a direct comparison of our results with the ones in <cit.>. Its components areV_A^i = 1/R( δ_a1^i y^a2- δ_a2^i y^a1)where we split the 10 index A into the two fundamental indices a_1 and a_2. One can check that they generate the algebra 𝔰𝔬(5) under the Lie derivative L, namelyL_V_A V_B = X_AB^C V_C .Furthermore, it is convenient to study the two-formsσ_A = 1/2ℰ_A_ijd y^i ∧ d y^jwhich evaluate toσ_A = - 1/Rϵ_a_1 a_2 ij d y^i ∧ d y^j .Analogous to (<ref>), they generate the Lie algebra 𝔤 under the Lie derivativeL_V_Aσ_B = X_AB^C σ_C .Finally, we need the volume formvol = 1/4!ϵ_1α̂β̂γ̂δ̂E^α_i E^β_j E^γ_k E^δ_l d y^i ∧ d y^j ∧ d y^k ∧ d y^l = 1/4! Rϵ_ijklm y^i d y^j ∧ d y^k ∧ d y^l ∧ d y^mwhich fulfills the relation<cit.>[In comparison to <cit.>, we use structure coefficients X_AB^C with the opposite sign. For example, we have X_1̃2̃^3̃=X_23,24^34=R^-1 while from (2.5) in <cit.> one gets X_23,24^34=-R^-1. So the vectors V_A and the forms σ_A, which we calculate, also have a flipped sign compared to their results. However, (<ref>) is the same.]ι_V_Avol = R/3 d σ_A .Hence, we reproduce all ingredients which were discussed in <cit.> to show that the S^4 with four-form flux is parallelizable. Following this paper, we plug the generalized frame fieldℰ_A = V_A + σ_A + ι_V_A𝒞into the generalized Lie derivativeℒ_ℰ_Aℰ_B = L_V_A V_B + L_V_Aσ_B + ι_[V_A,V_B]𝒞 - ι_V_B ( d σ_A- ι_V_A d 𝒞 )where the last term vanishes for a 𝒞 governed by (<ref>). In principal, we could scale σ_A and 𝒞 by the same constant factor to obtain another generalized frame field which still fulfills (<ref>). In <cit.> it was fixed by imposing appropriate equations of motion. It is interesting to have a closer look at these equations. They originate from eleven-dimensional SUGRA with the actionS = 1/2 κ_11^2∫ d^11 x √(-G)( ℛ - 1/2 | d 𝒞 |^2 )for the bosonic sector. G is the metric in eleven dimensions, ℛ denotes the corresponding curvature scalar and 𝒞 is a three-form gauge field. Using the Freund-Rubin ansatz <cit.> to solve the equations of motion for this action on the spacetime AdS_7 ×S^4, we findℛ_S^4 = 12/R^2 = 4/3 | d 𝒞 |^2 or |𝒢|^2 = 9/R^2 .Furthermore after applying the relations𝒢∧⋆ 𝒢 = | F |^2voland⋆vol = 1where ⋆ is the Hodge star operator on the S^4, one finds that this result is in agreement with (<ref>). Clearly, this results depends on the relative factors between the two terms in the action (<ref>) which are fixed by supersymmetry. In SL(5) EFT this particular reltaion between the gravity sector and the form-field originates from the requirement that the generalized frame field is an SL(5) element. Naively, ℰ_A^Î has 100 independent components. They can be organized according to10×10 = 1 + 24 + 75 ,but only the ones in the adjoint irrep are non-vanishing. This property is automatically implemented in our approach as can be seen from (<ref>). The generalized frame field Ê'_B^Î has the frame field E_β^i and the three-form 𝒞 as constituents. They furnish the irreps 1+15 and 4 of SL(4) which arise from the branching24→1 + 4 + 4 + 15of SL(5). Thus, Ê'_B^Î is an SL(5) element. By construction, M_A^B shares this property. So it is no surprise that ℰ_A^Î, which results from the multiplication of the two, is an SL(5) element, too. As a consequence, we find the correct scaling factor for the four-form flux. § CONCLUSIONIn this work, we present a technique to explicitly construct the generalized frame fields for generalized parallelizable coset spaces M=G/H in four dimensions. It is based on the idea of making the Lie group G in the extended space of EFT manifest. This can be done in the framework of gEFT <cit.> and is closely related to the concept underlying<cit.>. As we discuss in the first part of this paper, there are several restrictions on G. They are closely related to the embedding tensor of the U-duality group SL(5) in four dimensions. We only use the extended space as a technical tool. In the end, one has to get rid of all the unphysical directions in this space by solving the SC. It deviates in gEFT from the SC known in the conventional formulation <cit.>. Collecting clues from<cit.>, we are able to solve it by choosing a particular embedding of the physical subspace M in G. Each SC solution comes with a canonical generalized frame field Ê_A and a GG governed by a twisted generalized Lie derivative. But for a generalized parallelizable space, the frame field is defined with respect to the untwisted generalized Lie derivative. As a consequence, in a last step we need to modify Ê_A such that it adsorbs the twisted part and becomes ℰ_A for which the defining relation of a generalized parallelization (<ref>) holds. There are three linear constraints which are required for the steps outlined above to go through. After solving them, we find among other things that our construction applies to all gaugings which are purely in the 15. The corresponding generalized frames are already known from <cit.>. However, we are also able to treat groups G which originate from the 40. Their dimension is smaller than ten and the U-duality group SL(5) has to be broken. Nevertheless, our results still apply and we provide an example in section <ref>.Let us finally mention that this paper marks an important success in the /gEFT program the authors have initiated by approaching a long standing question in the DFT/EFT and GG community: How to systematically construct generalized frame fields satisfying all required consistency conditions? Of course, we did not completely answer this question. But we presented all necessary tools for SL(5) EFT. Especially, the treatment in the sections <ref>-<ref> apply to other U-duality groups, too. Hence, there does not seem to be an obstruction to extend the results in this work to other dimensions appearing in table <ref>. Studying the required linear constraints, one should be able to find a large class of generalized parallelizable spaces M with M4. Because of the very close connection between these spaces, maximal gauged supergravities and the embedding tensor formalism, one might even hope to eventually obtain a full classification of them. The significance of such a classification for the understanding of consistent coset reductions was already emphasised in the introduction of this paper.We gratefully acknowledge that Emanuel Malek was involved in the initial stages of this project. We like to thank him for many important discussions during the preparation of this paper. We also would like to thank David Berman, Martin Cederwall, Olaf Hohm, Arnau Pons Domenech, and Daniel Waldram for helpful discussions. Furthermore, FH thanks the ITS at the City University of New York and the theory group at Columbia University for their hospitality during parts of this project. The work of PB and DL is supported by the ERC Advanced Grant “Strings and Gravity" (Grant No. 320045). FH is supported by the NSF CAREER grant PHY-1452037 and also acknowledges support from the Bahnson Fund at UNC Chapel Hill as well as the R. J. Reynolds Industries, Inc. Junior Faculty Development Award from the Office of the Executive Vice Chancellor and Provost at UNC Chapel Hill and from the NSF grant PHY-1620311.§ SL(N) REPRESENTATION THEORYIn the first part of this appendix, we review how projectors on 𝔰𝔩(N) irreps can be constructed from Young symmetrizers. Further, we show how these projectors are used to explicitly decompose tensor products of irreps into direct sums. As first application of this concept, the linear constraints encountered in section <ref> are solved for the T-duality group SL(4) in the second part. §.§ Theory: Young Tableaux and Projectors on IrrepsLet us fix some convention first: A Young diagram is a set of n boxes which are arranged in rows and columns starting from the left. The number of boxes in each row may not increase while going from the top of the diagram to the bottom. An example for n=6 is3,2,1 .It is in one-to-one correspondence to the partition (3,2,1) of 6. A diagram becomes a Young tableau, if we write the numbers from one to n into the boxes. In general, there are n! different ways to do so. If the numbers in a tableau are increasing in every row and column at the same time, it is called a standard tableau. The number of standard tableaux for a given diagram can be calculated by the hook length formula: For each box in a diagram λ one counts the number of boxes in the same row i to its right and boxes in the same column j below it. For the box itself, one has to add one to the result to obtain the hook length h_λ(i,j). From this data the number of standard tableaux is calculated asd_std = n!/∏ h_λ(i,j).Take the example <ref>, here we obtain531,31,1for each box andd_std = 6!/5·3^2 = 16 .Starting from a Young tableau t, one combines all permutations from the symmetric group S_n which only shuffle elements within each row of t into the row group R_t. Similarly, all permutations which only shuffle elements in columns are assigned to the column group C_t. Together R_t and C_t give rise to the Young symmetrizere_t = ∑_π∈ R_t, σ∈ C_tsign(σ) σ∘π .An instructive example ist = 12,3and e_t = ( () - (1 3) )( () + (1 2) ) = () + (1 2) - (1 3) - (3 2 1) ,where we use cycle notation for elements of S_3. We are interested in applying e_t to tensors, such as X_a_1 … a_n, where the permutations act on the indices. For instance, with the tableau t from (<ref>) we finde_t X_a_1 a_2 a_3 = X_a_1 a_2 a_3 + X_a_2 a_1 a_3 - X_a_3 a_2 a_1 - X_a_2 a_3 a_1 .It is straightforward to check that the resulting tensor is anti-symmetric with respect to the first two indices a_1, a_2 and moreover the total antisymmetrization X_[a_1 a_2 a_3] vanishes. If the indices a_i=1,⋯,N are in the fundamental of 𝔰𝔩(N), the resulting tensor e_t X_a_1 a_2 a_3 is an irrep of the Lie algebra. Thus, the Young symmetrizer e_t is proportional to the projector from a tensor product (X_a_1 a_2 a_3 is nothing else) to this irrep. This works for all other Young tableaux as well. In order to calculate the dimension of the irrep we project onto from the tableaux t, we first have to assign the number N to the top left corner of the diagram λ corresponding to t. In each column to the right we increase the number and in each row towards the bottom we decrease it. Taking again the diagram <ref> as an instructive example, we havemathmode, boxsize=2em N N+1 N+2N-1 N N-2. boxsize=1em,aligntableaux=centerThese numbers are denoted in analogy to the hook length by f_λ(i,j). Finally, the dimension of the irrep associated to t isd_irrep = ∏ f_λ(i,j)/∏ h_λ(i,j) ,which gives rise to the dimension N(N^2-1)/3 for the Young symmetrizer (<ref>). For N=5 this yields 40, exactly one of the two irreps in the embedding tensor.As already mentioned e_t is only proportional to a projector and fulfillse_t e_t = k_t e_t ,where k_t is a constant depending on the tableau t. We use this to define the projector onto t asP_t = 1/k_t e_t with P_t^2 = P_t .Such projectors come with the following properties: * Projectors of tableaux corresponding to different diagrams are orthogonal.* Projectors of standard tableaux are linear independent. They can be combined to a system of orthogonal projects P_λ, i. Here λ labels the diagram they decent from.* The sum of all these projectors for all diagrams with n boxes is the identity of S_n.Now, assume that we have a projector P to a reducible representation and want to decompose it into a sum of orthogonal projectors P_λ, i onto irreps asP = ∑_λ∑_i P_λ, i .As mentioned above, these orthogonal projectors arise from a sumP_λ, i = ∑_t (c_λ, i)_t e_t ∘ Pover different projectors originating from standard tableaux for a specific diagram λ. However, the coefficients (c_λ, i)_t in this expansion still have to be fixed. This can be done by requiring that the commutator[P, P_λ, i] = ∑_t (c_λ, i)_t [e_t ∘ P, P] = 0of P with each of the P_λ, i vanishes. For the resulting nullspace an orthonormal basis is chosen:P_λ, i∘ P_λ, j =P_λ, ii = j0 ij .§.§ Application: Linear Constrains for SL(4)In order to solve the linear constraints from section <ref>, we first decompose the constraint quantity Γ_AB^C into irreps. Here, we work with the Lie algebra 𝔰𝔩(4), thus indices denoted by capital letters are in the irrep 6 and small letter indices label the fundamental representation 4. As explained in the first part of this appendix, Young symmetrizer act on the latter representation. Hence, we first translateΓ_AB^C →Γ_[a_1 a_2], [b_1 b_2]^[c_1 c_2] .We further have to distinguish between raised and lowered indices. While the former live in the 6, the latter are in dual 6[Note that for 𝔰𝔩(4) the six-dimensional representation is real, e.g. 6 = 6. Thus, in general we do not need to distinguish between the two of them. However, it is still a good bookkeeping device.]. Changing from an irrep to its dual is done by contraction with the totally anti-symmetric tensorΓ_a_1 a_2, b_1 b_2, c_1 c_2 = Γ_a_1 a_2, b_1 b_2^d_1 d_2ϵ_d_1 d_2 c_1 c_2 .In total, the connection has 216 independent components which are organized through the following irreps6×6×6 = 3 (6) + 10 + 10 + 50 + 2 (64) .They are in one-to-one correspondence with their Young diagrams1,1×( 1,1×1,1 ) = 3 2,2,1,1 + 3,1,1,1 + 2,2,2 + 3,3 + 2 3,2,1 .On the right hand side of this equation we identify the projectorP_6×6×6 = 1/8( () - (1 2) ) ( () - (3 4) ) ( () - (5 6) )on a reducible representation. In order to correctly decompose it into a sum (<ref>) of projectors onto irreps, we further have to take the diagrams (1,1,1,1,1,1) and (2,1,1,1,1) into account, even if they clearly vanish for 𝔰𝔩(4). Still, they contribute to the decomposition into irreps of the symmetric group S_6. While the first gives rise to one projector, the second leads to two orthogonal projectors. As in (<ref>), we suppress their contribution in the following. When a diagram appears more than once in a decomposition, there are different ways to organize the corresponding projectors. Here we use the following scheme:6× ( 6×6 ) = 6× (1 + 15 + 20') = {[6×1 = 6a; 6×15 = 6b + 10 + 10 + 64a;6×20'= 6c + 50 + 64b ]..In this convention, we can finally write down the resulting decompositionP_6×6×6 = P_6a + P_6b + P_6c + P_10 + P_10 + P_50 + P_64a + P_64b .Now we are ready to discuss the first linear constraint (<ref>). In fundamental indices, it readsC_a_1 a_2, b_1 b_2, c_1 c_2, d_1 d_2, e_1 e_2 =ϵ_a_1 a_2 b_1 b_2 ( - Γ_c_1 c_2, d_1 d_2, e_1 e_2 - Γ_c_1 c_2, e_1 e_2, d_1 d_2) + ϵ_d_1 d_2 e_1 e_2 ( Γ_c_1 c_2, b_1 b_2, a_1 a_2 - Γ_c_1 c_2, a_1 a_2, b_1 b_2)after substituting the Y-tensorY^a_1 a_2, b_1 b_2_c_1 c_2, d_1 d_2 = 1/4ϵ^a_1 a_2 b_1 b_2ϵ_c_1 c_2 d_1 d_2and lowering all indices as described in (<ref>). Clearly, this expression vanishes if the terms in the brackets next to the totally anti-symmetric tensors vanish. They are not independent. Thus, we are left with the constraintΓ_a_1 a_2, b_1 b_2, c_1 c_2 + Γ_a_1 a_2, c_1 c_2, b_1 b_2 = 0 ,which we recast in terms of a projector2 P_1 Γ_a_1 a_2, b_1 b_2, c_1 c_2 = 0 with P_1 = 1/2( () + (3 5) (4 6) ) .All irreps in the decomposition (<ref>) that are not in the nullspace of this projector and thus violate (<ref>) have to vanish. To this end, we replace (<ref>) by(1 - P_1) P_6×6×6 =P_6b + P_10 + P_10 +P_64a .Intriguingly, this equation gives us exactly the components of the embedding tensor for half-maximal, electrically gauged supergravities in seven dimensions. However, not all of these irreps survive the linear constraint applied to them <cit.>. Let us also check whether this is the case for our setup. Therefore, we calculate X_AB^C according to (<ref>). In components, this equation gives rise toX_a_1 a_2, b_1 b_2, c_1 c_2 = Γ_a_1 a_2, b_1 b_2, c_1 c_2 - Γ_b_1 b_2, a_1 a_2, c_1 c_2 + Γ_c_1 c_2,a_1 a_2, b_1 b_2or written in terms of permutationsσ_X = () - (1 3)(2 4) + (1 3 5)(2 4 6) as X_a_1 a_2, b_1 b_2, c_1 c_2 = σ_X (1 - P_1) Γ_a_1 a_2, b_1 b_2, c_1 c_2 .Again, we are able to rewrite σ_X in terms of orthogonal irrep projectors. Doing so, we finally obtainσ_X (1 - P_1) P_6×6×6 = 3 P_10 + 3 P_10 .These two irreps combine to the 20 independent components of the totally anti-symmetric tensor F_ABC. From the considerations in the last section, we already know that in this case all remaining linear constraints are solved. Furthermore, this decomposition gives us exactly the right factor of 3 between Γ_AB^C and X_AB^C.§ ADDITIONAL SOLUTIONS OF THE LINEAR CONSTRAINTIn this appendix, we give the remaining solutions for the group manifolds presented in table <ref>. First, we continue with the SL(3)×SL(2) case. The coordinates are represented by the branching (<ref>)10→ (1, 1) + (3, 2) + (3, 1)after removing the last term. Counting the dimensions of the remaining irreps, we see that there are seven independent directions on the manifold. Again, we choose a basis for the vector spaceV_(1,1) = {12}V_(3,2) = {13, 14, 15, 23, 24, 25}V_(3, 1) = {34, 35, 45}V_(1,1) = {345}V_(3,2) = {245, 235, 234, 145, 135, 134}V_(3, 1) = {125, 124, 123}and check the implications on the representations of the embedding tensor15 → (1, 3) + (3, 2) + (6,1)40 →(1, 2) + (3, 1) + (3,2) + (3, 3) + (6, 2) + (8, 1) .While there are no restriction on the irreps resulting from the branching of the 15, the second linear constraint (<ref>) only allows the (8,1) contribution from the 40. These are exactly the gaugings one would expect from the gauged supergravity point of view <cit.>. Another possible decomposition of the coordinates reads10→(1, 1) + (3, 2) + (3, 1) .It gives rise to a nine-dimensional group manifold. Again, all irreps in (<ref>) are allowed and the40→ (1, 2) + (3, 1) + (3,2) + (3, 3) + (6, 2) + (8, 1)is restricted to the (1,2) components. Of course, we could also consider the branching (<ref>) with both (1,1) and (3, 1) removed. This choice would result in a six-dimensional group manifold. However, doing the explicit calculation, we see that no irreps survives in this case.Subsequently, we continue with this procedure for the T-duality subgroup SL(2)×SL(2). First, we choose a basis for the vector spaceV_(1,1) = {12}V_(1,2) = {15, 25}V_(2,2) = {13, 14, 23, 24}V_(1, 1) = {34}V_(2,1) = {35, 45}V_(1,1) = {345}V_(1,2) = {234, 134}V_(2,2) = {245, 235, 145, 135}V_(1, 1) = {125}V_(2,1) = {124, 123}which is adapted to the branching of the coordinates10→ (1, 1) + (1, 2) + (2, 2) + (1, 1) + (2, 1) .By removing the irreps10→ (1, 1) + (1, 2) + (2, 2) + (1, 1) + (2, 1)from this decomposition, we obtain a seven-dimensional group manifold with the possible gaugings15→(1, 3) + (1, 2) + (2, 2) + (1, 1) + (2, 1) + (3, 1) ,40→(1, 2) + (1, 2) + (2, 2) +(1, 1) + (2, 1) + (1, 3) + (2, 3)+ (1, 2) + (2, 2) + (3, 2) + (1, 1) + (2, 1) + (2, 1) + (3, 1) .In order to see which irreps have to be removed, first note that the linear constraint for the 40 has an eight-dimensional solution space. In contrast to the previous cases, it is not possible to identify the crossed out irreps by their dimension alone. However, we can compare the linear constraint solutions to the ones obtained for the SL(4) case and see that they share 3 independent directions. For the branching10→ (2,2) + (3,1) + ( 1,3)of SL(4) to SL(2)×SL(2), we see that these could furnish the irreps (3,1) or (1,3). Furthermore, this solution does not overlap with the (8,1) from (<ref>) which branches as(8, 1) → (1, 1) + 2 (2, 1) + (3, 1) .Thus, (1,3) is the only possible choice. A similar argumentation follows after taking into account the(1,2) of the SL(3)×SL(2) case in (<ref>). It shares two common directions with the solution of the linear constraint. The branching from SL(3)×SL(2) to SL(2)×SL(2) of this irrep is trivial(1,2) → (1,2) .Now, only three unidentified direction are left. They can be fixed just by their dimension. Doing so, we obtain the branching (<ref>). This gauging is expected from the gauged supergravity point of view as well <cit.>.Moreover, we find two more interesting cases which do not lie completely in one of the previous cases for SL(3)×SL(2) and SL(4). The first one gives rise the eight-dimensional group manifolds with the coordinate irreps10→(1, 1) + (1, 2) + (2, 2) + (1, 1) + (2, 1) .Here, the solution space for the 40 part of the linear constraints has four independent directions. They are partially contained in the (1, 2) and (8,1) of SL(3)×SL(2). With both the solution shares two directions each. According to (<ref>) and (<ref>), we identify them with the irreps (1, 2) and (2, 1). These are the only irreps which can be switched on. There are no restrictions for the 15 part by the linear constraints. Thus, we obtain15→(1, 3) + (1, 2) + (2, 2) + (1, 1) + (2, 1) + (3, 1),40→(1, 2) + (1, 2) + (2, 2) +(1, 1) + (2, 1) + (1, 3) + (2, 3)+ (1, 2) + (2, 2) + (3, 2) + (1, 1) + (2, 1) + (2, 1) + (3, 1) . Finally, there are five-dimensional group manifolds with the coordinate irreps10→ (1, 1) + (1, 2) + (2, 2) + (1, 1) + (2, 1) .In total, the solution space of the 40 part possesses 11 independent directions. They are partially contained[There are four directions in the (8,1) of SL(3)×SL(2), but only one of them is not contained in the 10 of SL(4).] in the (8,1) of SL(3)×SL(2) and sit completely in the 10 of SL(4). Thus, we only obtain a new (1,1) from (<ref>) and the right hand side of (<ref>).In contrast to the previous cases, only ten directions of the linear constraints' 15 part can be switched on. The solution for the 15 lies entirely in the 10 of SL(4). Taking into account the branching rule (<ref>), we find15→(1, 3) + (1, 2) + (2, 2) + (1, 1) + (2, 1) + (3, 1),40→(1, 2) + (1, 2) + (2, 2) +(1, 1) + (2, 1) + (1, 3) + (2, 3)+ (1, 2) + (2, 2) + (3, 2) + (1, 1) + (2, 1) + (2, 1) + (3, 1). All other solutions the linear constraints are completely contained in one of the previously discussed SL(4) or SL(3)×SL(2) cases.§ FAITHFUL REPRESENTATIONS AND IDENTIFICATIONSWe first consider the Lie algebra of CSO(1,0,4) which is given in terms of the non-vanishing commutators[t_α̂, t_β̂] = 𝐠t_α̂β̂ ,where we assigned the generatorst_A = ( t_1,t_2,t_3,t_4,t_1̃,t_2̃,t_3̃,t_4̃,t_5̃,t_6̃) .This algebra is relevant for the first duality chain (<ref>) in section <ref> and has the lower central seriesL_0= { t_1,t_2,t_3,t_4,t_1̃,t_2̃,t_3̃,t_4̃,t_5̃,t_6̃}⊃{ t_1̃,t_2̃,t_3̃,t_4̃,t_5̃,t_6̃}⊃{0} .Following the procedure outlined in <cit.>, we construct the N=21-dimensional subspaceV^2 = { t_1^2,t_1 t_2,t_1 t_3,t_1 t_4,t_2^2,t_2 t_3,t_2 t_4,t_3^2,t_3 t_4,t_4^2,t_1̃,t_2̃,t_3̃,t_4̃,t_5̃,t_6̃,· = 2 t_1,t_2,t_3,t_4, · = 1 1 }· = 0of the universal enveloping algebra. The center of this algebra is given by { t_1̃, t_2̃, t_3̃, t_4̃, t_5̃, t_6̃}. These six generators form an abelian subalgebra 𝔥. With this data, we are able to obtain the matrix representation for the generators t_A by expanding the linear maps ϕ_t_A in the basis V^2. Finally the exponential maps (<ref>) and (<ref>) give rise to the group elementsg = m h = ( [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x^1; 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x^2; 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x^3; 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x^4; x^1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (x^1)^2/2; x^2 x^1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x^1 x^2; x^3 0 x^1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 x^1 x^3; x^4 0 0 x^1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 x^1 x^4; 0 x^2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 (x^2)^2/2; 0 x^3 x^2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 x^2 x^3; 0 x^4 0 x^2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 x^2 x^4; 0 0 x^3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 (x^3)^2; 0 0 x^4 x^3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 x^3 x^4; 0 0 0 x^4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 (x^4)^2/2;- 𝐠x^2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0x^1̃;- 𝐠x^3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0x^2̃;- 𝐠x^4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0x^3̃; 0- 𝐠x^3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0x^4̃; 0- 𝐠x^4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0x^5̃; 0 0- 𝐠x^4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1x^6̃; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1; ])with 𝐠 representing the number of G-flux units the background carries. Working with such large matrices is cumbersome. So we represent g instead by the ten tuple ( x^1, x^2, x^3, x^4, x^1̃, x^2̃, x^3̃, x^4̃, x^5̃, x^6̃ ). In this case, the group multiplication is given by(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ ,x^5̃ ,x^6̃) (y^1 ,y^2 ,y^3 , y^4 ,y^1̃ ,y^2̃ , y^3̃ , y^4̃ , y^5̃ , y^6̃) =(x^1+y^1 ,x^2+y^2 ,x^3+y^3 ,x^4+y^4 , - 𝐠 x^2 y^1 + x^1̃ + y^1̃ ,- 𝐠x^3 y^1 + x^2̃ + y^2̃- 𝐠x^3 y^2 + x^3̃ + y^3̃ , - 𝐠x^4 y^1 + x^4̃ + y^4̃ , - 𝐠x^4 y^2 + x^5̃ + y^5̃ ,- 𝐠 x^4 y^3 + x^6̃ + y^6̃ ) .Let us check that this indeed gives rise to a group. The identity element is e=( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ) and fulfillsg e = e g = g .Furthermore, there is the inverse elementg^-1=(-x^1 ,-x^2 ,-x^3 ,-x^4 ,- 𝐠x^1 x^2 -x^1̃ , - 𝐠x^1 x^3 - x^2̃ , - 𝐠x^2 x^3 - x^3̃ , - 𝐠x^1 x^4 - x^4̃ , - 𝐠 x^2 x^4 -x^5̃ , - 𝐠x^3 x^4 - x^6̃)fulfillingg^-1 g = g g^-1 = e .Because 𝐠 is an integer, the group multiplication (<ref>) does not only close over the real numbers, but also for x^i and x^ĩ being integers. Thus, CSO(1,0,4,ℤ) is a subgroup of CSO(1,0,4) and we can mod it out by considering the right coset CSO(1,0,4,ℤ)\CSO(1,0,4) which gives rise to the equivalence relationg_1 ∼ g_2 if and only if g_1 = k g_2 with g_1 , g_2 ∈CSO(1,0,4) and k ∈CSO(1,0,4,ℤ) .After substituting k=(n^1 ,n^2 ,n^3 ,n^4 ,n^1̃ ,n^2̃ ,n^3̃ ,n^4̃ ,n^5̃ ,n^6̃) with n^i, n^ĩ∈ℤ, we obtain the identifications(x^1 ,x^2 ,x^3 ,x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ ,x^5̃ , x^6̃) ∼ (x^1+n^1 ,x^2+n^2 ,x^3+n^3 ,x^4+n^4 , - 𝐠x^1 n^2 + x^1̃ + n^1̃ ,- 𝐠x^1 n^3 + x^2̃ + n^2̃- 𝐠x^2 n^3 + x^3̃ + n^3̃ ,- 𝐠 x^1 n^4 + x^4̃ +n^4̃ , - 𝐠x^2 n^4 + x^5̃ + n^5̃ ,- 𝐠x^3 n^4 + x^6̃ + n^6̃ )from (<ref>). Especially, we have(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃)∼ (x^1 + 1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃)∼ (x^1 ,x^2 + 1 ,x^3 , x^4 ,x^1̃ - 𝐠x^1,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃) ∼ (x^1 ,x^2 ,x^3 + 1 , x^4 ,x^1̃ ,x^2̃ - 𝐠x^1 , x^3̃ - 𝐠x^2 , x^4̃ , x^5̃ , x^6̃)∼ (x^1 ,x^2 ,x^3 , x^4 + 1 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ - 𝐠x^1 , x^5̃ - 𝐠x^2, x^6̃ - 𝐠x^3)for the physical coordinates and(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃) ∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ + 1 ,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃) ∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ + 1 , x^3̃ , x^4̃ , x^5̃ , x^6̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ + 1 , x^4̃ , x^5̃ , x^6̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ + 1 , x^5̃ , x^6̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃ + 1 , x^6̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃ , x^6̃ + 1)for the remaining ones. Taking into account these identifications, the left invariant Maurer-Cartan formE^A_I = [1000000000;0100000000;0010000000;0001000000; 𝐠x_2000100000; 𝐠x_3000010000;0 𝐠x_300001000; 𝐠x_4000000100;0 𝐠x_400000010;00 𝐠x_40000001 ] ,is globally well defined, namelyE_1= d x^1 E_2= d x^2E_3= d x^3E_4= d x^4 E^1= d x^1̃ + 𝐠x^2 d x^1 = d (x^1̃- 𝐠x^1) + (x^2 + 1)𝐠 d x^1 E^2= d x^2̃ + 𝐠x^3 d x^1 = d (x^2̃ - 𝐠x^1) + (x^3 + 1)𝐠d x^1 E^3= d x^3̃ + 𝐠x^3 d x^2 = d (x^3̃ - 𝐠x^2) + (x^3 + 1)𝐠d x^2 E^4= d x^4̃ + 𝐠x^4 d x^1 = d (x^4̃ - 𝐠x^1) + (x^4 + 1)𝐠d x^1 E^5= d x^5̃ + 𝐠x^4 d x^2 = d (x^5̃ - 𝐠x^2) + (x^4 + 1)𝐠d x^2 E^6= d x^6̃ + 𝐠x^4 d x^3 = d (x^6̃ - 𝐠x^3) + (x^4 + 1)𝐠d x^3 .For the second duality chain (<ref>), the nine-dimensional Lie algebra 𝔤 in (<ref>) is relevant. We perform the exponential maps (<ref>) as well as (<ref>), to obtain the group elementg = m h = ( [100000000000000 x^5̃;010000000000000x^3;001000000000000x^4;000100000000000x^1;000010000000000 x^1̃;000001000000000 x^3̃; x^5̃00000100000000(x^5̃)^2 /2;x^3 x^5̃0000010000000 x^3 x^5̃;x^40 x^5̃000001000000 x^4 x^5̃;0x^30000000100000(x^3)^2 / 2;0x^4x^3000000010000x^3 x^4;00x^4000000001000(x^4)^2 / 2;0 -𝐟 x^40000000000100x^2; -𝐟 x^400000000000010 x^4̃ -𝐟 x^4 x^5̃; -𝐟 x^300000000000001x^2̃-𝐟 x^3 x^5̃;0000000000000001;])with 𝐟 representing the number of F-flux units the background carries. Again we represent g in terms of the nine tuple ( x^1, x^2, x^3, x^4, x^1̃, x^2̃, x^3̃, x^4̃, x^5̃ ) instead of working with this big matrix. In this case, the group multiplication is given by(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃) ( y^1 ,y^2 ,y^3 , y^4 ,y^1̃ ,y^2̃ , y^3̃ , y^4̃ , y^5̃) =(x^1+y^1 ,- 𝐟x^4 y^3 + x^2+y^2,x^3+y^3 ,x^4+y^4 , x^1̃ + y^1̃ , 𝐟x^5̃ y^3 + x^2̃ + y^2̃x^3̃ + y^3̃ , 𝐟 x^5̃ y^4 + x^4̃ + y^4̃ , x^5̃ + y^5̃ ) .To verify that g is a group, first consider the identity element e=( 0, 0, 0, 0, 0, 0, 0, 0, 0 ), It satisfiesg e = e g = g .Moreover, the inverse element is given byg^-1=(-x^1 ,- 𝐟x^3 x^4 -x^2 ,-x^3 ,-x^4 ,-x^1̃ , 𝐟x^3 x^5̃ - x^2̃ , - x^3̃ , 𝐟x^4 x^5̃ - x^4̃ , -x^5̃)fulfillingg^-1 g = g g^-1 = e .Since 𝐟 is an integer, the group multiplication (<ref>) does not only close over the real numbers, but also for x^i and x^ĩ being integers. Hence, we can mod out the discrete subgroup G_ℤ formed by restricting all coordinates to integers from the left. This results in the equivalence relationg_1 ∼ g_2 if and only if g_1 = k g_2 with g_1 , g_2 ∈ G and k ∈ G_ℤ .Finally, we substitute k=(n^1 ,n^2 ,n^3 ,n^4 ,n^1̃ ,n^2̃ ,n^3̃ ,n^4̃ ,n^5̃) with n^i, n^ĩ∈ℤ and find the identifications(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ ,x^5̃) ∼ (x^1+n^1 ,-𝐟x^3 n^4 + x^2+n^2 ,x^3+n^3 ,x^4+n^4 , x^1̃ + n^1̃ , 𝐟x^3 n^5̃ + x^2̃ + n^2̃x^3̃ + n^3̃ , 𝐟x^4 n^5̃ + x^4̃ + n^4̃ , x^5̃ + n^5̃)from (<ref>). Particularly, for the physical coordinates(x^1 ,x^2 ,x^3 , x^4 , x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃) ∼ (x^1 + 1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃)∼ (x^1 ,x^2 + 1 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃) ∼ (x^1 ,x^2 ,x^3 + 1 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃)∼ (x^1 ,x^2 - 𝐟 x^3,x^3 , x^4 + 1 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃)and for the remaining coordinates(x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ , x^5̃) ∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ + 1 ,x^2̃ , x^3̃ , x^4̃ , x^5̃) ∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ + 1 , x^3̃ , x^4̃ , x^5̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ + 1 , x^4̃ , x^5̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ , x^3̃ , x^4̃ + 1 , x^5̃)∼ (x^1 ,x^2 ,x^3 , x^4 ,x^1̃ ,x^2̃ + 𝐟x^3, x^3̃ , x^4̃ + 𝐟x^4 , x^5̃ + 1).After taking these identifications into account, we compute the left-invariant Maurer-Cartan formE^A_I = [ 1 0 0 0 0 0 0 0 0; 0 1𝐟x^4 0 0 0 0 0 0; 0 0 1 0 0 0 0 0 0; 0 0 0 1 0 0 0 0 0; 0 0 0 0 1 0 0 0 0; 0 0 - 𝐟x^5̃ 0 0 1 0 0 0; 0 0 0 0 0 0 1 0 0; 0 0 0 - 𝐟x^5̃ 0 0 0 1 0; 0 0 0 0 0 0 0 0 1 ] .Taking into account the identifications (<ref>) and (<ref>), it is straightforward to check that this E^A_I is globally well defined:E_1= d x^1E_2= d x^2 + 𝐟x^4 dx^3 = d( x^2 - 𝐟x^3 ) + ( x^4 + 1 )𝐟dx^3E_3= d x^3E_4= d x^4 E^1= d x^1̃E^2= d x^2̃ - 𝐟x^5̃ dx^3 = d( x^2̃ + 𝐟x^3 ) - ( x^5̃ + 1 )𝐟dx^3 E^3= d x^3̃E^4= d x^4̃ - 𝐟x^5̃ dx^4 = d( x^4̃ + 𝐟x^4 ) - ( x^5̃ + 1 )𝐟dx^4 E^5= d x^5̃ .JHEP | http://arxiv.org/abs/1705.09304v1 | {
"authors": [
"Pascal du Bosque",
"Falk Hassler",
"Dieter Lust"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170525180026",
"title": "Generalized Parallelizable Spaces from Exceptional Field Theory"
} |
1]Michael J. Kastoryano [1]NBIA, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark 2,1]Angelo [email protected] [2]QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, DenmarkDivide and conquer method for proving gaps of frustration free Hamiltonians [ December 30, 2023 =========================================================================== Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter n is at most o(log(n)^2+ϵ/n) for any positive ϵ. § INTRODUCTION Many-body quantum systems are often described by local Hamiltonians on a lattice, in which every site interacts only with few other sites around it, and the range of the interactions is given in terms of the metric of the lattice.One of the most important properties of these Hamiltonians is the so-called spectral gap: the difference between the two lowest energy levels of the operator. The low-temperature behavior of the model (and in particular of its ground states) relies on whether the spectral gap is lower bounded by a constant which is independent on the number of particles (a situation usually referred to as gapped), or on the contrary the spectral gap tends to zero as we take the number of particle to infinity (the gapless[We are using the terminology as it is frequently used in the quantum information community. In other contexts, one could only be interested in the thermodynamic limit, and the situation we have denoted as gapless does not necessarily imply that there is a continuous spectrum above the groundstate energy in such limit.] case).Quantum phase transitions are described by points in the phase diagram were the spectral gap vanishes<cit.>, and therefore understanding the behavior of the spectral gap is required in order to classify different phases of matter. A constant spectral gap implies exponential decay of correlations in the groundstate<cit.>, and it is conjectured (and proven in 1D) that entanglement entropy will obey an area law <cit.>.Moreover, the computational complexity of preparing the groundstate via an adiabatic preparation scheme<cit.> is given by the inverse of the spectral gap, implying that groundstates of gapped models can be prepared efficiently.It is also believed that it is possible to give synthetic descriptions of such groundstates in terms of Projected Entangled Paris States (PEPS)<cit.>, and to prepare them with a quantum computer<cit.>.Because of the importance of the spectral gap, there is a large history of powerful results in mathematical physics regarding whether some systems are gapped or not, such as the Lieb-Schultz-Mattis theorem <cit.> and its higher dimensional generalization <cit.>, the so-called “martingale method” for spin chains <cit.>, the local gap thresholds by Knabe <cit.> and by Gosset and Mozgunov <cit.>. Cubitt, Perez-Garcia and Wolf have shown <cit.> that the general problem of determining, given a finite description of the local interactions, whether a 2D local Hamiltonian is gapped or not is undecidable. Nonetheless, this result does not imply that it is not possible to study the spectral gap of some specific models, and the problem can be decidable if we restrict to specific sub-classes of interactions.While these results have constituted tremendous progress, there is still a lack of practical tools for studying the gap for large classes of lattice systems, especially in dimensions greater than one.In this paper we consider frustration-free, finite range local Hamiltonian on spin lattices, and we present a technique for proving a lower bound on the spectral gap. Compared to the other methods for bounding the spectral gap that are available in the literature, the one we propose uses a recursive strategy that is more naturally targeted to spin models in dimension higher than 1, and which we hope might allow to generalize some of the results that at the moment have only been proved in 1D.The approach we present is based on a property of the groundstate space reminiscent of the “martingale method”. A description of the groundstate space might not be available in all cases, but it is easily obtained for tensor network models such as PEPS<cit.>. We are able to prove that this condition is also necessary in gapped systems, obtaining an equivalence with the spectral gap. More specifically, we will define two versions of the martingale condition, a strong and a weak one, and we will show that the spectral gap implies the strong one. The strong martingale condition implies the weak one, hence completing the loop of equivalences. This “self-improving” loop will allow us to give an upper bound on the rate at which the spectral gap vanishes in gapless systems, as any rate slower than that allows us to prove a constant spectral gap.In order to prove the equivalence between the strong martingale condition and the spectral gap, we will use a tool known as the Detectability Lemma<cit.>. We will also show that if the Detectability Lemma operator contracts the energy by a constant factor, then the system is gapped. This condition is reminiscent of the “converse Detectability Lemma”<cit.>,but we do not know whether these two conditions are equivalent.Proving gaps of Hermitian operators has a long history in the setting of (thermal)stochastic evolution of classical spin systems. In this setting, there are numerous tools for bounding the spectral gap of the stochastic generator (which in turn allows to bound the mixing time of the process) both for classical <cit.> and for quantum commuting Hamiltonians<cit.>.In the classical setting, the theorems establish an intimate link between the mixing time of a stochastic semigroup (the Glauber dynamics) and the correlation properties in the thermal state at a specified temperature: for sufficiently regular lattices and boundary conditions, correlations between two observables are exponentially decaying (as a function of the distance between their supports) if and only if the Glauber dynamics at the same temperature mixes rapidly (in a time O(log(N)), where N is thevolume of the system). All of the proofs of the classical results in some way or another rely on showing that exponential decay of correlations implies a Log-Sobolev inequality of the semi-group, and in the other direction, that the log-Sobolev inequality implies a spectral gap inequality, which in turn implies exponential decay of correlation. We will take inspiration from a weaker form of the classical theorem that shows the equivalence between spectral gap of the semigroup and exponential decay of correlation.The paper is organized as follows. In <ref>, we will describe the main assumption on the groundstate space that implies the spectral gap, and then we will state the main results.In <ref> we will recall some useful tools, namely the detectability lemma and its converse. In <ref>, we will finally prove the main theorem, together with the local gap threshold.§ MAIN RESULTS §.§ Setup and notationLet us start by fixing the notation and recalling some common terminology in quantum spin systems. We will consider a D-dimensional lattice Γ (the standard example being Γ=^D, but the same results will hold for any graph which can be isometrically embedded in ^D). At each site x∈ we associate a finite-dimensional Hilbert space _x, and for simplicity we will assume that they all have the same dimension d. For every finite subset ⊂, the associated Hilbert space _ is given by ⊗_x∈_x, and the corresponding algebra of observables is _ = (_). If ⊂^' we will identify _ as the subalgebra _⊗_^'∖⊂_^'. If P is an orthogonal projector, we will denote by P_⊥ the complementary projection 1-P. A local Hamiltonian isa map associating each finite ⊂ to a Hermitian operator H_, given byH_ = ∑_X ⊂ h(X) ,where h(X) ∈_X is Hermitian. We will denote the orthogonal projector on the groundstate space of H_ (i.e. the eigenprojector corresponding to the smallest eigenvalue of H_) as P_. We will make the following assumptions on the interactions h(X): (Finite range) there exist a positive r such that h(X) is zero whenever the diameter of X is larger than r. The quantity r will be denoted the range of h;(Frustration freeness) for every X, h(X) P_ = E_0(X)P_, where E_0(X) is the lowest eigenvalue of h(X).Note that frustration freeness implies that P_ P_^' = P_^' whenever ⊂^'. By applying a global energy shift, we can replace h(X) with h(X) - E_0(X), and we will assume that E_0(X)=0 for every X, so that H_≥ 0.For every , we will denote by λ_ the difference between the two lowest distinct eigenvalues of H_ (which, since we have assumed that 0 is the lowest eigenvalue, is the same as the smallest non-zero eigenvalue of H_). This quantity will be called the spectral gap of H_, and it can be expressed as follows:λ_= inf_|ϕ⟩H_ϕ/P_^⊥ϕ. We will interpret this as a ratio of two quadratic functionals on _:_(ϕ)= ⟨ϕ|-⟩P_ϕ = P_^⊥ϕ; _(ϕ)= H_ϕ.We will use the symbol _(ϕ) since the functional can be thought as a type of variance: it equals |ϕ⟩ - P_|ϕ⟩^2, it is always positive and vanishes only on states in P_. We can then rewrite <ref> as the following optimization problem: λ_ is the largest constant such that λ__(ϕ) ≤_(ϕ).In order to simply the proofs, we will also make the following extra assumption on the interactions h(X): (Local projections) Every h(X) is an orthogonal projection.The assumption that every h(X) is an orthogonal projection is not a fundamental restriction. Let us denote by E_1(X) (resp. E_max(X)) the second-smallest eigenvalue (resp. the largest eigenvalue) of h(X), and remember that we have assumed that the lowest eigenvalue of each h(X) is zero. If we then assume the two following conditions (Local gap) e = inf_X E_1(X) >0;(Local boundness) E=sup_X E_max(X) < ∞;then we can see that for every finite ⊂:e ∑_X⊂ P_X^⊥≤ H_≤ E ∑_X ⊂ P_X^⊥,where we have denoted by P_X the projector on the groundstate of h(X). Therefore H_ will have a non-vanishing spectral gap if and only the spectral gap of the Hamiltonian composed of projectors ∑_X P_X^⊥ is not vanishing. This shows that, as far as we are interested in the behavior of the spectral gap, requiring local gap and local boundness is equivalent to requiring that the interactions h(X) are projectors.Given a local Hamiltonian H which is finite range and frustration free it is easy to see that interactions can be partitioned intog groups, referred to as “layers”, in such a way that every layer consists of non-overlapping (and therefore commuting) terms. For a fixed ⊂, let us index the layers from 1 to g, and denoteL_i the orthogonal projector on the common groundstate space of the interactions belonging to group i. Since they are commuting, L_i can also be seen as the product of the groundstate space projectors of each interaction term.For any given ordering of {1,…, g}, we can then define the product L = ∏_i=1^g L_i (different orders of the product will in general give rise to different operators). Any operator constructed in this fashion is called an approximate ground state projector. §.§ Statement of the results We will now state the main assumptions needed in the proof of the spectral gap theorem.In order to do so, we will need to introduce some notation for the overlap between groundstate spaces of different regions. Let A, B be finite subsets of . Let P_A∪ B, P_A, and P_B be respectively the orthogonal projectors on the ground state space of H_A∪ B, H_A and H_B. Then we defineδ(A,B) = (P_A - P_A∪ B) (P_B - P_A∪ B). Because offrustration freedom, we have P_A P_A∪ B = P_A∪ B P_A = P_A∪ B and the same holds for P_B. In turn this imply that(P_A - P_A∪ B)(P_B - P_A∪ B) = P_A P_B - P_A∪ B ,so that δ(A,B) can be both seen as a measure of the overlap between (P_A-P_A∪ B) and (P_B - P_A∪ B) (the cosine of the first principal angle between the two subspaces), as well as a measure of how much P_A∪ B can be approximated by P_AP_B. The intuition behind Def. <ref> is that in a gapped system, if l is the diameter of the largest ball contained in A∩ B, then δ(A,B) should be a fast decaying function of l. In this setting we will refer to the “size” of the overlap of A and B as l (see Fig 1b). One might also hope that δ(A,B) only depends on l and not on the size of A Δ B = (A ∪ B) ∖ (A∩ B). This is captured by the following assumption: There exists a positive function δ(l) with exponential decay in l, i.e. δ(l) ≤ cα^l for some 0<α<1 and c>0, such that for every connected A and B, such that A∩ Bhas size l, the following bound holds:δ(A,B) ≤δ(l).We will now present some weaker versions of <ref>. As we will show later, they will all turn out to be equivalent, but it might be hard to verify the stronger versions in some concrete examples. The first relaxation we have is to require a slower decay of the function δ(l). There exists a positive function δ(l) with polynomial decay in l, i.e. δ(l) ≤ cl^-α for some α>0 and c>0, such that for every connected A and B, such that A∩ Bhas size l, <ref> holds. Clearly, <ref> implies <ref>. As formulated, conditions <ref> and <ref> and B require <ref> to be satisfied homogenously for all regions A and B of arbitrary size. However, in order to prove a bulk spectral gap, such a strong homogeneity assumption can be relaxed. We can allow for the size of A∩ B, of A and of B to be taken into account; intuitively, we would like to have less stringent requirement if A∩ B is very small compared to A and B.In particular, we will define classes F_k of sets, which have the property that they can be decomposed as overlapping unions of sets in F_k-1, with a sufficiently large overlap. Then we will only require <ref> to hold for this specific decomposition, and moreover we will allow the bound δ(l) to depend on k.The construction of the sets F_k we present is a generalization of the one originally proposed by Cesi <cit.> and used in the context of open quantum systems by one of the authors <cit.>. For each k ∈, let l_k = (3/2)^k/D and denoteR(k) = [0,l_k+1] ×…× [0,l_k+D] ⊂^D.Let F_k be the collection of Λ⊂ which are contained in R(k) up to translations and permutation of the coordinates. We now show thatsets in F_k can be decomposed “nicely” in terms of sets in F_k-1. For each Λ∈ F_k ∖ F_k-1 and each positive integer s ≤1/8 l_k, there exist s distinct pairs of non-empty sets (A_i,B_i)_i=1^s such that * Λ = A_i ∪ B_i and A_i, B_i ∈ F_k-1∀ i=1,…,s;* (Λ∖ A_i, Λ∖ B_i) ≥l_k/8s -2;* A_i ∩ B_i ∩ A_j ∩ B_j = ∅∀ i ≠ j.We will call a set of s distinct pairs (A_i, B_i)_i=1^s of non-empty sets satisfying the above properties an s-decomposition of Λ. The proof of this proposition – a minor variation over the one presented by Cesi<cit.> – is contained in <ref>.With this definition of F_k at hand, we can now present the weakest version of <ref>.There exists an increasing sequence of positive integers s_k, with ∑_k1/s_k < ∞, such that∑_k=1^∞δ_k := ∑_k=1^∞sup_∈ F_k ∖ F_k-1sup_A_i, B_iδ(A_i,B_i) < ∞,where the second supremum is taken over all s_k-decompositions = A_i ∪ B_i given by <ref>. It is not immediately clear from the definition that <ref> is implied by <ref>, so we show this in the next proposition.<Ref> implies <ref> with any s_k such that ∑_ks_k/l_k < ∞. Let δ(l) be as in <ref>. Since for every _k∈ F_k ∖ F_k-1 and for every s_k-decomposition _k=A_i ∪ B_i of _k, the overlap A_i ∩ B_i has size at least l_k/8s_k-2, then δ_k ≤δ(l_k/8s_k- 2) .Since δ(l) decays as l^-α for some positive α, δ_k is summable if ∑ s_k/l_k is summable.If we consider <ref> with s_k growing faster than l_k/k, then the previous proposition does not apply – note that in any case s_k has to be smaller than 1/8 l_k for the construction of <ref> to be possible. In practice we do not need to consider such situations. In Cesi<cit.>, s_k was chosen to be of order l_k^1/3. As we will see later, we will be interested in choosing s_k with slower rates than that (while still having ∑ 1/s_k finite), so the condition s_k = l_k/k will not be restrictive for our purposes. So from now on, we will only consider <ref> in the case wheres_k = l_k/k. The main result of the paper is to show that <ref> is sufficient to prove a spectral gap. In turn, this will imply <ref>, which as we have already seen in <ref> implies <ref>, showing that allthree conditions are equivalent.Let H be a finite range, frustration free, local Hamiltonian, and let F_k be as in <ref>. Then the following are equivalent * inf_k inf_∈ F_kλ_≥λ > 0 (or in other words, H is gapped);* H satisfies <ref> with δ(ℓ) = 1/(1+λ/g^2)^l/2 for some constant g;* H satisfies <ref>;* H satisfied <ref> with s_k such that ∑_ks_k/l_k < ∞.By proving the equivalence of these conditions, we are also able to show that in any gapless model, the spectral gap cannot close too slowly, since a slow enough (but still infinitesimal) gap will imply <ref> and therefore a constant gap. The threshold is expressed in the following corollary.If H is gapless, then for any ⊂ of diameter n it holds thatλ_ = o(log(n)^2+ϵ/n),for every ϵ>0. We also provide an independent condition for lower bounding the spectral gap. Consider again the construction of the detectability lemma, where L=L_1⋯ L_g is an approximate ground state projector.If there exist a constant 0<γ<1 such (Lϕ) ≤γ(ϕ),then the spectral gap of H is bounded below by λ≥1-γ/4.While similar in spirit to the Converse Detectability Lemma (see <ref>), we do not know if these are equivalent, nor whether the hypothesis of <ref> isnecessary. Nachtergaele <cit.> presented a general method for proving the spectral gap for a class of spin-lattice models, which has become known as the martingale method.Given an increasing and absorbing sequence _n →, and a fixed parameter l, it requires three conditions ((C1), (C2), (C3)) to be satisfied uniformly along the sequence to prove a lower bound to the spectral gap.Let us briefly recall what these conditions would be if we applied them to the setting we are considering, and compare them to <ref>.The first condition, denoted (C1) in the original paper, is automatically satisfied by finite range interactions, which is also the case we are considering here.If we denote A_n = _n and B_n = _n+1∖_n-l (where now l is a parameter partially controlling the size of A_n ∩ B_n = _n ∖_n-l), then condition (C2) requires that H_A_n ∩ B_n has a spectral gap of γ_l independently of n (for every n large enough). We do not need to require such assumption, since we are using a recursive proof. Condition (C3) can be restated, using our notation, as requiring that δ(A_n, B_n) ≤ϵ_l < 1/√(l+1) for all n large enough.Clearly, the big difference with <ref> is that the requirement on δ is not of asymptotic decay, but only to be bounded by a specific constant.Upon careful inspection, we see this is only a fair comparison in 1D. In higher dimensions, condition (C2) could be as hard to verify as the original problem of lower bounding the spectral gap, since the size of A_n ∩ B_n will grow with n. Condition (C3) is also clearly implied by <ref>. Therefore, one could compare the method we propose with the martingale method as a strengthening of condition (C3) in exchange of a weakening of condition (C2), a trade-off which we hope makes it more applicable in dimensions D>1.§.§ Example 1: translation invariant 1D spin chains To clarify the differences between <ref>, let us consider the case of 1D spin chains. We will consider a translational invariant model to further simplify the situation. Then we can take, without loss of generality, A=[0,n] and B=[n-d,n-d+m], with n, m, d being positive integers such that min(m,n) > d. The intersection A∩ B = [n-d,n] has length d+1, so that <ref> is equivalent to the fact that the functionδ(d) = sup_d<n,mδ([0,n], [n-d,n-d+m])has exponential decay in d. <Ref> would relax this to a polynomial decay, but both require a bound that is uniform in n and in m.We can now consider the larger interval in each F_k, namely Λ_k = [0,(3/2)^k+1]. Denoting l_k = (3/2)^k, we can write Λ_k and its s-decompositions asΛ_k= [0, 1] · l_k+1A^i_k= [0, 1/2+ i/6s ] · l_k+1B^i_k= [1/2 + i/6s - 1/12s, 1] · l_k+1,for i=1, …, s. The overlap A^i_k ∩ B^i_k has size l_k/12s for every i. If we fix for concreteness s_k = l_k^1/3, as in <cit.>, then we can definen_i,k = *1/2l_k + i/6 l_k^2/3,m_i,k = *1/2l_k - 2i-1/12 l_k^2/3,d_k = *l^2/3_k/12,so thatA^i_k = [0, n_i,k]B^i_k = [n_i,k + d_k, n_i,k + d_k + m_i,k].Note that n_i,k and m_i,k are always smaller than 24 √(3)d_k^3/2 So we then see that in order to show that the model satisfies <ref>, it would be sufficient for example to verify thatδ(d) = sup_d<n,m ≤ 24 √(3)d^3/2δ([0, n ], [n -d, n - d + m ])is decaying polynomially fast in d. Compared to <ref>, n and m are restricted given a specific d, i.e. we only have to consider the case where they are at most a constant times d^3/2. It should be clear now that this restriction on the n and m depends on the choice of the scaling of s_k. Choosing faster rates of growth for s_k leads to more restrictive conditions (and thus in principle easier to verify): the downside is that this will be reflected in the numerical bound for the spectral gap, which will become worse (although finite). §.§ Example 2: PVBS modelsOne notable model in dimension larger than 1 for which the original martingale method has been successfully applied is the Product Vacua and Boundary State (PVBS) model <cit.>, a translation invariant, finite range, frustration free spin lattice Hamiltonian, with parameters D positive real numbers (λ_1, …, λ_D ).We refer to the original paper for the precise definition of the model.The spectral gap of the PVBS Hamiltonian is amenable to be analyzed using the “1D version” of the martingale method, applied recursively in each of the dimensions, and it has been shown that in the infinite plane the Hamiltonian is gapped if and only if not all λ_j are equal to 1.In this section we show that our result recovers the same finite-size limit analysis as in the original paper: for simplicity we will only do the analysis in the case of rectangular regions, with the caveat that in that case the finite-size gap closes if only one of the λ_j is equal to 1 (even if the GNS Hamiltonian is still gapped).In Ref. <cit.> it has been shown that in the case of two connected regions A and B such that A∩ B is also connected,δ(A,B)^2 = C(A∖ B) C(B∖ A)/C(A)C(B),where C(X) = ∑_x ∈ X∏_j=1^D λ_j^2x_j is the normalization constant of the model.If we now consider ∈ F_k to be a rectangular region (so that every A_i and B_i appearing in the geometrical construction of <ref> will also be rectangles), then the normalization constant C() will be a product of different constants in each dimension independently. Assuming without loss of generality that the dimension being cut by <ref> is the D-th, we see that if λ_D = 1 thenδ(A_i,B_i) = ( A_i∖ B_i/A_iB_i∖ A_i/B_i)^1/2 = 1-1/8s_k ,which is not infinitesimal.On the other hand, if λ_D ≠ 1, thenδ(A_i, B_i) = ( ∑_x=0^l_A-lλ_D^2x/∑_x=0^l_Aλ_D^2x∑_x=l^l_Bλ_D^2x/∑_x=0^l_Bλ_D^2x)^1/2 ,have denoted by l_A (resp. l_B, l) the length of A (resp. B, A∩ B) along dimension D. Thereforeδ(A_i,B_i)≤λ_D^l+1 [(1-λ_D^2(l_A+1))(1-λ_D^2(l_B+1))]^-1/2 if λ_D < 1, λ_D^-(l+1) [(1-λ_D^-2(l_A+1))(1-λ_D^-2(l_B+1))]^-1/2 if λ_D > 1.If all λ_j are distinct from one, then the PVBS model satisfies <ref> withδ(l) = λ_*^l/1-λ^2_* , λ_* = max_i min(λ_i, λ_i^-1),and therefore it is gapped by <ref>. If at least one of them is equal to 1 then δ(l) will be lower bounded away from zero, and therefore the gap will close. Note that one could get a better estimate on the spectral gap by following the proof of <ref>, and using a different δ(l) in each of the dimensions, instead that just taking the worst case as we did here.§ DETECTABILITY LEMMA AND SPECTRAL GAP§.§ The detectability lemma and its converse The Detectability Lemma <cit.> originated in the context of the quantum PCP conjecture<cit.>.It has since then become a useful tool in many-body problems.A converse result is known as the Converse Detectability Lemma <cit.>, and will also be used later. At the same time as we recall them, we will reformulate them in terms of inequalities between some quadratic functionals. In analogy to <ref>, given L=∏_i=1^g L_i we define the following quadratic functional on _DL(ϕ) = ⟨ϕ|-⟩L^*Lϕ.Before stating the Detectability Lemma and its converse, let us make some preliminary observations regarding L and DL(ϕ).For any L given above, denote P the projector on the groundstate space of H. Then * L P = P L = P, and in particular LP = 0;* L≤ 1;(1) follows from the definition of L and frustration freedom. Since L is a product of projectors its norm is bounded by 1, so also (2) is trivial.For every ϕ∈_Λ it holds thatDL(ϕ) ≤(ϕ) ≤1/1-LP^⊥^2 DL(ϕ),and 1/(1-LP^⊥^2) is the smaller constant that makes the upper bound hold true. Let us start by observing that(L ϕ) = L^*P^⊥ Lϕ = L^*Lϕ - Pϕ = ϕ - DL(ϕ).On the one hand, sinceis a positive quadratic functional, we have that (Lϕ) ≥ 0 and therefore (ϕ) ≥ DL(ϕ).On the other hand we have the following bound(ϕ) - DL(ϕ) = L^*P^⊥ Lϕ = P^⊥ L^*L P^⊥ϕ≤LP^⊥^2 P^⊥ϕ;from which the upper bound in <ref> follows by rearranging the terms. Optimality follows by choosing a ϕ such that LP^⊥ϕ = LP^⊥P^⊥ϕ.As can be seen from <ref>, if LP^⊥ is smaller than 1, then DL is up to constants equivalent to . The Detectability lemma and its converse then relate DL to , thus allowing to connect LP^⊥ to the spectral gap, via <ref>. With the notation above, it holds that(Lϕ) ≤ g^2 DL(ϕ). The proof of this statement can be found in Ref. <cit.>. A simple corollary follows:If λ is the spectral gap of H, thenL P^⊥^2 ≤1/1+λ/g^2.In particular, for finite systems LP^⊥<1. If λ is the spectral gap of H, then λ(ϕ) ≤(ϕ). In particular, λ(Lϕ) ≤(Lϕ) ≤ g^2 DL(ϕ). But in <ref> we have seen that (Lϕ) = (ϕ) - DL(ϕ), and therefore (ϕ) ≤ (1+g^2/λ)DL(ϕ). The result follows from optimality of the constant in <ref>.With the same notation as above,DL(ϕ) ≤ 4 (ϕ). The proof of this statement can be found in Ref. <cit.>. Again, from this functional formulation we can derive the usual statement of the Converse Detectability lemmaIf λ is the spectral gap of H, thenλ≥1-LP^⊥^2/4.It follows from <ref>. We are now ready to prove <ref>.From <ref>, we have that L P^⊥ < 1, and then <ref> implies that lim_n→∞ L^n = P. Thereforelim_m→∞∑_n=0^m DL(L^nϕ)= lim_m→∞∑_n=0^m(L^n)^*L^nϕ - (L^n+1)^*L^n+1ϕ= lim_m→∞⟨ϕ|-⟩(L^m+1)^*L^m+1ϕ= ⟨ϕ|-⟩Pϕ = (ϕ).By applying <ref> to each term in the summation, we obtain that:(ϕ) = ∑_n=0^∞ DL(L^nϕ) ≤ 4 ∑_n=0^∞(L^n ϕ) ≤ 4 ∑_n=0^∞γ^n (ϕ) = 4/1-γ(ϕ). §.§ Spectral gap implies <ref>Let us start by proving the following converse relationship between spectral gap and δ(A,B). Let A, B ⊂ be finite, and let l=((A∪ B)∖ A, (A∪ B)∖ B). If H_Λ for Λ=A∪ Bis a finite range Hamiltonian with spectral gap λ_Λ, thenδ(A,B) ≤1/(1+λ_Λ/g^2)^l/2,where g is a constant depending only onand on the range of H. In order to prove this result, we will make use of the Detectability Lemma. With the same notation as in <ref>, it implies that LP_Λ^⊥^2 ≤1/1+λ_Λ/g^2. By taking q-powers of L and iterating the previous bound q times we obtainL^q P_Λ^⊥≤1/(1+λ_Λ/g^2)^q/2 = ϵ_Λ^q,since LP_Λ^⊥⊂ P_Λ^⊥, where we have denoted ϵ_Λ = (1+λ_Λ/g^2)^-1/2 < 1. Therefore, if H_Λ is gapped, L^q will be an exponentially good approximation of P_Λ, with q chosen independently of Λ.We now want to show that L^q can be split as a product of two terms L^q = M_A M_B in such away that both M_A and M_B are good approximations to P_A and P_B, using a strategy presented in Ref. <cit.>. With the notation defined above, if q≤ l, then there exist two operators M_A and M_B, respectively acting on A and on B, such that L^q=M_AM_B and the following holds:P_A - M_A≤ϵ_Λ, (P_A -M_A)M_B≤ϵ_Λ^q;and the same holds with A and B interchanged. Let us start by defining M_A and M_B as follows: we will group the projectors appearing in L^q in two disjoint groups, such that M_A will be the product (in the same order as they appear in L^q) of the projectors of one group, M_B the product of the rest, and L^q=M_A M_B. In order to do so, we will consider the layers L_1, …, L_g sequentially (following the order in which are multiplied in L), and then we will start again from L_1 up to L_g, until we have considered ⌊gq/2⌋ different layers. Each layer will be split into two parts, where terms of one of them will end up appearing in M_A and terms in the other will appear in M_B. In the first layer, we will only include in M_A terms which intersect (A∪ B)∖ B. From the second layer, we only included terms which intersect the support of the terms considered from the first. We keep doing this recursively, when at each layer we include terms which intersect the support of the selected terms of the previous step (one can see this as a sort of light-cone, defined by the layer structure, generated by (A∪ B)∖ B, as depicted in <ref>). The remaining ⌈gq/2⌉ are treated similarly, but starting instead from the end of the product, and reversing the role of B and A.At this point, it should be clear that by construction L^q=M_AM_B, since every projector appearing in L has been assigned to either M_A or M_B, and the two groups can be separated without breaking the multiplication order. If q ≤ l, then M_B will be supported in B, and M_A will be supported on A.Denote with L_A and L_B the approximate ground state projections of P_A and P_B respectively, as in <ref>. Then we have that P_A - L_A^q≤ϵ_Λ^q and the same for B. It should be clear that M_A and M_B contain strictly more projection terms than L_A and L_B, and therefore P_AM_A = M_A P_A = P_A and M_AP_A^⊥≤ϵ_Λ, and the same holds for B. Observe that we can write P_AM_B = P_A M_A M_B := P_A R L_B^q, where we have redistributed the projectors of M_A in order to “fill” the missing ones in M_B to complete it to L_B^q. What is left is put into R, which can be reabsorbed into P_A. Therefore P_AM_B = P_AL_B^q, and this implies thatP_A(P_B-M_B) = P_A(P_B-L_B^q)≤ϵ_Λ^q.The same construction (but exchanging the roles of A and B) can be done in order to bound (P_A-M_A)M_B≤ϵ_Λ^q.With this construction, we can easily prove <ref>.We observe thatP_AP_B - M_AM_B = P_A(P_B - M_B)+ (P_A - M_A)M_B .We can now apply <ref>, and choose q=l to obtainP_A P_B - P_A ∪ B≤1/(1+λ_Λ/g^2)^l/2 .In the next section, we will show that condition <ref> implies the spectral gap. Then <ref> allows us to prove the converse, therefore showing the equivalence stated in <ref>.§ <REF> IMPLIES SPECTRAL GAP §.§ Quasi-factorization of excitationsWe will start with some useful inequalities regarding orthogonal projectors in Hilbert spaces. Let P and Q be two orthogonal projections on a Hilbert space . Then it holds that-PQ≤ 1-P-Q ≤P^⊥Q^⊥where PQ = PQ +QP is the anti-commutator.We start by observing that -1 ≤ P - Q ≤ 1, since P and Q are positive and bounded by 1, and therefore 0 ≤ (P-Q)^2 ≤ 1. By observing that (P-Q)^2 = P + Q -PQ, it immediately follows the l.h.s. of <ref>:1-P-Q ≥ - PQ.By algebraic manipulation we can show that{P,Q} = (1-P^⊥)(1-Q^⊥) + (1-Q^⊥)(1-P^⊥) = = 2(1 - P^⊥ - Q^⊥) + P^⊥Q^⊥ = -2(1- P - Q) + P^⊥Q^⊥.Applying <ref> we obtain thatP^⊥Q^⊥ = PQ + 2(1-P-Q) ≥ 1-P-Q.We are now ready to prove the following quasi-factorization result.Let A, B be subsets of . Then it holds thatc P_A∪ B^⊥ϕ≤P_A^⊥ϕ + P_B^⊥ϕ,where c = 1-2δ(A,B).Notice that frustration freedom implies that P_A∪ B^⊥ P_A^⊥ = P_A^⊥ P_A∪ B^⊥ = P_A^⊥, and the same holds for P_B^⊥. Therefore if P_A∪ B^⊥|ϕ⟩ = 0, both sides of the equation reduce to 0, and we can restrict ourselves to the case in which |ϕ⟩ is contained in the image of P_A∪ B^⊥. We can then apply <ref> to P_A^⊥ and P_B^⊥ and we obtain:P_A∪ B^⊥ϕ≤P_A^⊥ϕ + P_B^⊥ϕ + P_A∪ B^⊥P_AP_BP_A∪ B^⊥ϕ.To conclude the proof, we just need to observe thatP_A∪ B^⊥ P_A P_B P_A∪ B^⊥ = (P_A - P_A∪ B)(P_B - P_A ∪ B),and that therefore by applying the Cauchy-Schwartz inequalityP_A∪ B^⊥ P_A P_B P_A∪ B^⊥ϕ ≤P_A∪ B^⊥|ϕ⟩(P_A - P_A∪ B)(P_B - P_A ∪ B)P_A∪ B^⊥|ϕ⟩ ≤(P_A - P_A∪ B)(P_B - P_A ∪ B)P_A∪ B^⊥|ϕ⟩^2 = (P_A - P_A∪ B)(P_B - P_A ∪ B)P_A∪ B^⊥ϕ.Since the same holds for P_A∪ B^⊥ P_A P_B P_A∪ B^⊥, and the operator norm is invariant under taking the adjoint, we obtain thatP_A∪ B^⊥P_AP_BP_A∪ B^⊥ϕ≤ 2 (P_A - P_A∪ B)(P_B - P_A ∪ B)P_A∪ B^⊥ϕ,which concludes the proof. A bound similar to what we have obtained in the previous lemma could also have been derived from the converse of the Detectability Lemma (<ref>). Indeed, if we apply it to the Hamiltonian P^⊥_A + P^⊥_B, we obtain the followingϕ^2 - P_AP_B |ϕ⟩^2 ≤4 P_A^⊥ + P_B^⊥ϕ.If we now choose |ϕ⟩= P_A∪ B^⊥|ϕ⟩, then a simple calculation shows thatP_AP_B|ϕ⟩^2 = P_BP_AP_Bϕ = P_A∪ B^⊥ P_B P_A P_B P_A ∪ B^⊥ϕ =P_B (P_A - P_A∪ B)(P_B - P_A ∪ B) P_Bϕ≤(P_A-P_A∪ B)(P_B-P_A ∪ B)ϕ^2.We thus obtain the following boundc^'P_A∪ B^⊥ϕ≤P_A^⊥ϕ + P_B^⊥ϕ,but now c^' =1/4(1-δ(A,B)). While very similar to <ref>, the constant c^' does not tend to 1 when δ(A,B) goes to zero: as we will see next, this is a crucial property and it is for this reason that <ref> will not be useful for our proof.For one dimensional systems, we expect the martingale condition to be implied by exponential decay of correlations, as has been shown in the commuting Gibbs sampler setting <cit.>. However, at this point we only know how to obtain this result if for any state |ψ⟩, there exists a (non- Hermitian) operator f_A^c on the complement of A⊆Λ such thatP_A|ψ⟩=f_A^c|φ⟩,and |φ⟩ is the unique ground state of H_Λ. In that case, the proof is analogous to the one in Ref. <cit.>. Eqn. (<ref>) does not hold in general, however it can be shown to hold for injective PEPS. Hence, for injective MPS correlation decay implies the martingale condition.§.§ Spectral gap via recursionAs we have mentioned in the introduction, the strategy for proving a lower bound to the spectral gap will be a recursive one: given , we will decompose it into two overlapping subsets, so that = A ∪ B and we will be able to use <ref>. This would lead to the following expression(1-2δ(A,B)) P_^⊥ϕ≤P_A^⊥ϕ+ P_B^⊥ϕ≤ ≤1/min(λ_A, λ_B)H_A + H_Bϕ =1/min(λ_A, λ_B)H_ + H_A∩ Bϕ.We now face the problem of what to do with the term H_A∩ Bϕ. Because of frustration freedom, we can bound it with H_ϕ, leading toλ_A∪ B≥1-2δ(A,B)/2min(λ_A, λ_B).Then it is clear that, even in the case of δ(A,B)=0, this strategy is going to fail: at each step of the recursion our bound on λ_ is cut in half, so in the limit of → we will obtain a vanishing lower bound. The way out of this obstacle is to observe that if we have s_k different ways of splittingas A_i ∪ B_i, and if moreover the intersections A_i ∩ B_i are disjoint for different i, then we can average <ref> and obtainP_^⊥ϕ≤1/s_k∑_i=1^s_k(1-2δ(A_i,B_i))^-1/min(λ_A_i, λ_B_i)H_A_i + H_B_iϕ≤ ≤(1-2δ_k)^-1/min{λ_A_i, λ_B_i}_iH_ + 1/s_k∑_i=1^s_k H_A_i ∩ B_iϕ≤ ≤ (1-2δ_k)^-11+1/s_k/min{λ_A_i, λ_B_i}_iH_ϕ.Then <ref> becomesλ_≥1-2δ_k/1+1/s_kmin{λ_A_i, λ_B_i}_i.Now the problem is simply to check whether we can find a right balance between the number s_k of different ways to partition(in order to make the product (1+1/s_k) convergent in the recursion), the size of A_i and B_i (if one of them is similar in size to , then we will not have gained much from the recursion), and the size of their overlaps (in order to make δ_k small). The geometrical construction presented in <ref> shows that such balance is obtainable, if we choose 1/s_k to be summable.By formalizing this idea, we can finally prove the main theorem of this section.Fix an increasing sequence of positive integers (s_k)_k such that ∑_k 1/s_k is summable. Let l_k and F_k be as in <ref>, and δ_k = δ^s_k as in <ref> andλ_k = inf_∈ F_kλ_ .Let k_0 be the smallest k such that δ_k < 1/2 for all k≥ k_0. Then there exists a constant C>0, depending onand on the sequence (s_k)_k but not on k, such thatλ_k≥λ_k_0 C ∏_j=k_0+1^k (1 -2δ_j).In particular, if <ref> is satisfied, the Hamiltonian is gapped. Fix a ∈ F_k∖ F_k-1 and let (A_i, B_i)_i=1^s_k be an s_k-decomposition ofas in<ref>. We can then apply <ref> to each pair (A_i,B_i), average over the resulting bounds, and obtain as in <ref>λ_≥1-2δ_k/1+1/s_kmin_i {λ_A_i, λ_B_i}≥1-2δ_k/1+1/s_kλ_k-1.Sincewas arbitrary, we have obtained thatλ_k ≥1-2δ_k/1+1/s_kλ_k-1.By iterating <ref> k-k_0 times, we obtainλ_k ≥∏_j=k_0+1^k 1-2δ_j/1+1/s_jλ_k_0 .We want to show now that this gives rise to the claimed expression. Notice that if we denote C^-1 := ∏_j=1^∞ (1+1/s_k) then1 ≤ C^-1≤∏_j=1^∞[ 1+1/s_k] < ∞ .This can be seen by observing that the serieslog∏_j=1^∞( 1+1/s_k) =∑_j=1^∞log(1+1/s_k) is summable, since by comparison it has the same behavior as ∑_j1/s_k, which is summable by assumption. This implies in particular that ∏_j=1^k (1+1/s_k)^-1≥ C > 0 for all k. Finally, in order to prove that the Hamiltonian is gapped, we only need to show that <ref> impliesK := ∏_j=k_0+1^∞(1 -2δ_j)> 0.This again is equivalent to the fact that (δ_j)_j=k_0+1^∞ is a summable sequence, which is imposed by <ref>. § LOCAL GAP THRESHOLDEquivalence between <ref> and <ref> can be seen as a “self-improving” condition on δ_k, where assuming that it decays faster than some threshold rate implies that it is actually decaying exponentially. This type of argument is reminiscent of “spectral gap amplification” as described in Ref.<cit.>. The same type of self-improving statement can be obtained for the spectral gap of H.Fix an increasing sequence of integers s_k such that ∑1/s_k <∞ and ∑_ks_k/l_k<∞. Let H be a local Hamiltonian, and let (as in <ref>) λ_k = inf_Λ∈ F_kλ_Λ, where λ_Λ is the spectral gap of H_Λ. If there exist a C>0 and a k_0 such thatλ_k > Ck s_k/l_k, ∀ k≥ k_0,then system is gapped (and inf_k λ_k>0). Since for every s_k-decomposition A_i, B_i of Λ∈ F_k the overlap A_i ∩ B_i has size at least l_k/8 s_k, by <ref>, we have thatδ_k ≤(1+λ_k/g^2)^-l_k/16s_k .We now need to check that δ_k is summable. By the root test, it is sufficient thatlim sup(1+λ_k/g^2)^-l_k/16 k s_k = ( lim inf(1+λ_k/g^2)^l_k/26k s_k )^-1 < 1 ,i.e. thatlim inf(1+λ_k/g^2)^l_k/16 k s_k = exp(lim infλ_k l_k/16k s_klog(1+λ_k/g^2)^1/λ_k ) > 1.If lim infλ_k = 0, then lim inf (1+λ_k/g^2)^1/λ_k = e^1/g^2 > 1 (and if lim infλ_k >0 there is nothing left to prove, since then we already know that the system is gapped ), and therefore we can reduce to check thatlim infλ_k/kl_k/s_k > 0,which is implied by <ref>. If we now read the condition of <ref> in terms of the length of the sides of the sets in F_k, we obtain a proof of <ref>.Let Λ∈ F_k: then its diameter will be at most a constant times l_k. If we denote it by n, then k ≥ q log(n) for some q>0. If we choose s_k = k^1+ϵ for some ϵ >0, we see that <ref> is satisfied if we can find ϵ and C>0 such thatλ_Λ > C log(n)^2+ϵ/nholds for all rectangles Λ with sides bounded by n. If the Hamiltonian is gapless, then necessarily λ_Λ = o(log(n)^2+ϵ/n) for every ϵ > 0. This result has to be compared with similar results obtained in Refs. <cit.> in the specific case of nearest-neighbor interactions in 1D chains and in 2D square and hexagonal lattices. In all these cases, the authors obtain a local gap threshold which implies a spectral gap in the limit in the following sense: denoting λ_n the spectral gap of a finite system defined on a subset of “side-length” n (where the exact definition depends on the dimension and the geometry of the lattice, but the general idea is that such a subset has n^D sites), there exists a sequence γ_n (the local gap threshold) such that, if for some n_0 it holds that λ_n_0 > γ_n_0, then the system is gapped in the limit. The converse is that, if the Hamiltonian is gapless, then λ_n = γ_n. The values of γ_n present in Refs. <cit.> are recalled in <ref>.The obvious downside of <ref> over the results in Refs. <cit.> is that these only require a single n_0 satisfying λ_n_0 > γ_n_0, while <ref> is a condition to be satisfied for each n. On the other hand, it can be applied in more general settings than nearest neighbor interactions, as well as in dimensions higher than 2, and can be easily generalized to regions with different shapes. The upper bound on λ_n for a gapless Hamiltonian which we derive is worse by a polynomial factor than the ones obtained in 1D and in the 2D square lattice, and it is only off by a logarithmic factor than the 2D hexagonal lattice case.While the logarithmic factor in our bound is probably just an artifact of the proof, it is an interesting open question whether the optimal scaling for the general case is 1/n^2.One should also mention the Lieb-Schultz-Mattis theorem <cit.> and its generalization to higher dimensions <cit.>, which proves that a class of half-integer spin models (not necessarily frustration free) with translational invariance, continuous symmetry and unique ground state is gapless. For this class of models the gap is bounded by log n/n (the log n factor can be removed in 1D), which is slightly better than the general upper bound we have obtained.§ ACKNOWLEDGMENTSWe acknowledge financial support from the European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Villum Young Investigator Award. A.L. would like to thank Amanda Young for the fruitful discussion about the PVBS model. § GEOMETRICAL CONSTRUCTIONLet d_k = l_k/8s. For i=1,…,s, we defineA_i= ([0,l_k+1]×…× [0,l_k+D-1] ×[0, l_k+D/2 + 2id_k]) ∩ ; B_i= ([0,l_k+1]×…× [0,l_k+D-1] ×[l_k+D/2 + (2i-1)d_k, l_k+D]) ∩ .Let us start by proving that A_i and B_i are in F_k-1. In order to do so, we need to show that up to translations and permutations of the coordinates, they are contained in R(k-1). If we look at coordinate j=1,…, D-1, then their sides are contained in [0,l_k+j], so it is enough to show that across the D-th coordinate they are not more than l_k long. A_i has a larger side than B_i, so we can focus on it only. Then we see that1/2 l_k+D + 2id_k ≤1/2(3/2)^k+D/D + 2sd_k = 3/4 l_k + 1/4 l_k = l_k.So that A_i and B_i belong to F_k-1 for every i.If either A_i or B_i were empty for a given i, thenwould be contained in a set belonging to F_k-1, and thus it would itself belong to F_k-1, but we have excluded this by assumption. So A_i and B_i are not empty.Clearly = A_i ∪ B_i, and (∖ A_i, ∖ B_i) ≥ d_k -2. Finally, we see thatA_i ∩ B_i = ([0,l_k+1]×…× [0,l_k+D-1] ×[l_k+D/2 + (2i-1)d_k,l_k+D/2 + 2id_k]) ∩,so that A_i ∩ B_i ∩ A_j ∩ B_j = ∅ for all i≠ j. | http://arxiv.org/abs/1705.09491v2 | {
"authors": [
"Michael J. Kastoryano",
"Angelo Lucia"
],
"categories": [
"math-ph",
"math.MP",
"quant-ph"
],
"primary_category": "math-ph",
"published": "20170526091725",
"title": "Divide and conquer method for proving gaps of frustration free Hamiltonians"
} |
[email protected] Research Institute, AIST, Tsukuba Central 4, Higashi 1-1-1,Tsukuba 305-8562, JapanDivision of Applied Physics, Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan Department of Science, Faculty of Education, Hirosaki University, 1 Bunkyo-cho, Hirosaki, Aomori 036-8560, Japan Nanoelectronics Research Institute, AIST, Tsukuba Central 4, Higashi 1-1-1,Tsukuba 305-8562, JapanPaul-Drude-Institut fur Festkörperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-cho, Hyogo 679-5148, JapanJapan Synchrotron Radiation Research Institute, 1-1-1 Kouto, Sayo-cho, Hyogo 679-5198, JapanRIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-cho, Hyogo 679-5148, JapanICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860, Castelldefels, Barcelona, SpainX-ray Science Division, Argonne National Laboratory, 9700 S. Cass Ave, Lemont, IL 60439, USA [email protected] Division of Applied Physics, Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, JapanPhase-change materials based on Ge-Sb-Te alloys are widely used in industrial applications such as nonvolatile memories, but reaction pathways for crystalline-to-amorphous phase-change on picosecond timescales remain unknown. Femtosecond laser excitation and an ultrashort x-ray probe is used to show the temporal separation of electronic and thermal effects in a long-lived (>100 ps) transient metastable state of Ge_2Sb_2Te_5 with muted interatomic interaction induced by a weakening of resonant bonding. Due to a specific electronic state, the lattice undergoes a reversible nondestructive modification over a nanoscale region, remaining cold for 4 ps. An independent time-resolved x-ray absorption fine structure experiment confirms the existence of an intermediate state with disordered bonds. This newly unveiled effect allows the utilization of non-thermal ultra-fast pathways enabling artificial manipulation of the switching process, ultimately leading to a redefined speed limit, and improved energy efficiency and reliability of phase-change memory technologies. Sub-nanometre resolution of atomic motion during electronic excitation in phase-change materials Muneaki Hase December 30, 2023 ================================================================================================empty§ INTRODUCTION The class of chalcogenide phase-change materials, such as Ge-Sb-Te and Ag-In-Sb-Te, has been found to be the most appropriate candidate for optical data storage media in the forms of rewritable CDs and DVDs, as well as Blu-ray discs. Ge_2Sb_2Te_5 (GST) has also been demonstrated to be a compound well suited for non-volatile memory applications, owing to a fast phase-switching process between amorphous and crystalline phases (10 ∼ 100 ns), excellent thermal and chemical stability of the end phases and good reliability allowing more than 10^9 write-erase cycles<cit.>. GST superlattice films are now being applied in a new generation of non-volatile electrical memory, interfacial phase-change memory<cit.>, surpassing current FLASH technology both in cyclability and speed<cit.>. The process of rapid phase change involved in the writing and erasing of data in conventional optical recording is presently induced by a purely thermal process using nanosecond laser pulses: heating of the material leads to the formation of a molten phase and subsequently the crystalline (SET) or amorphous (RESET) state, depending on the cooling speed. Thus in phase-change memory devices the speed of write cycles is limited to tens of nanoseconds<cit.>. Therefore in order to overcome this limitation, an alternative method of switching between SET and RESET states of GST is necessary. Avoidance of thermally based processes implies the necessity of using electronic effects, and thus the nature of the electronic structure of GST needs to be taken into account. While its amorphous phase is characterized by its covalent bonding nature, the distorted rock-salt crystalline phase is locally rhombohedral with three long and three short bonds and is usually described in terms of resonant bonds<cit.>. Recently, the possibility of an ultrafast phase transition triggered by an electronic excitation due to the breaking of resonant bonds has been proposed<cit.>. Based upon the experimental results from a time-resolved x-ray absorption fine structure measurement, the possible presence of non-thermal contributions to the amorphization of GST alloy on sub-nanosecond time scales was reported <cit.>. The existence of a solid-solid amorphization process induced via electronic excitation and subsequent lattice relaxation was further argued for by density functional calculations <cit.> and time-resolved electron diffraction studies<cit.>, in which, however, due to the polycrystalline samples used and the transmission mode applied, the direct demonstration of the existence of electronically driven effects without the significant influence of thermal processes is challenging. This reason, in addition to the fact that electronic excitation is in general immediately followed by electron-phonon coupling-induced heating of the lattice, makes the dynamics induced by an increase in lattice temperature difficult to separate from purely electronic effects.Very recently, Waldecker et al. reported on the decoupling of the electronic and lattice degrees of freedom on a several picosecond time scale after optical excitation. They interpreted the temporal separation of optical properties from the structural transition in GST alloy films to be a result of the depopulation of resonant bonds before electron-phonon energy exchange occurred<cit.>. These observations on a several picosecond time scale will open a new route to control phase change in resonantly bonded materials. However, the observation of the decoupling was limited to a narrow time window, preventing from evaluating the lifetime and exact structure, both of which are required for real device applications. Here we provide new insight into a possible solid-solid transformation process in an epitaxially-grown single-crystal film of GST based on reversible conditions through time-resolved x-ray techniques with sub-nanometre resolution allowing the temporal ranges of the lattice evolution to be discerned. The use of a much wider time window up to > 1 ns reveals the long lifetime of the transient state and associated bond angles disordering. The x-ray diffraction intensity decreases as a function of time delay without a thermal shift in peak position for several picoseconds after photo-excitation, followed by the sudden onset of a thermal-expansion-induced shift. These features coincide well with previous results<cit.>, but, being obtained from the direct observation of atomic motion as opposed to indirect information derived from the evolution of the optically excited dielectric function, strongly and unambiguously support the existence of the separation of non-thermal (electronic) effects from thermal (lattice) effects. These observations were also made possible by the use of an epitaxial sample in the rocking curve mode measurements, providing clear evidence of the movement of atoms independent from the optically-induced heating of the sample. Our x-ray absorption fine structure (XAFS) data further prove that the transient state is an intermediate state between the original crystalline and amorphous states, but different from the liquid phase that would be expected to exist for the case of switching based on a melting process.§ RESULTS We used a time-resolved x-ray diffraction (XRD) technique<cit.> (Figure 1) to directly probe the ultrafast structural dynamics in a 35-nm-thick single crystalline GST film grown on a Si (111) substrate photo-excited by 1.55 eV (800 nm) photons. The laser-induced dynamics of the GST lattice were studied by taking time-resolved XRD rocking curves of the GST (222) Bragg diffraction peak (Figure 2a,b). This peak served as an optimal peak for observations of the lattice dynamics due to the boundary conditions imposed by the pump laser; the corresponding integrated intensities and peak shift were measured as a function of the time delay (Figure 2c,d). The diffraction intensity of the (222) peak decreases immediately after excitation, and at ∼4 ps the intensity level reaches ≈ 80% of its initial value. In contrast, the (222) diffraction peak position starts to shift towards lower Q_z values only after a ∼4 ps time delay with a concomitant further decrease in intensity (Figure 2d).The integrated intensity taken at a fixed incident angle at the (222) peak position with high time resolution (Q_z=const) shows that the diffracted signal intensity has an inflection point at ≈ 4 ps, which, together with a combination of intensity and position of the diffraction peak, demonstrates the existence of two-step dynamics. The (222) peak reaches a maximum deflection ∼ 20 ps after the excitation. The shift of the position of the Bragg peak towards smaller Q_z values after excitation implies that the maximum thermal expansion of the GST film is Δd/d ≈ 1.4%<cit.>, where Δ d is the fractional change in d spacing. Subsequently, the peak position reverts to higher Q_z values, corresponding to contraction of the lattice.The unshifted position of the (222) diffraction peak, observed with better time resolution than in previous work<cit.>, implies that thermal effects leading to expansion of the lattice do not occur during the first ∼4 ps, since a diffraction peak shift toward lower Q_z values would be expected if the lattice temperature had risen due to electron-phonon interactions<cit.>.Such a delay in the lattice heating onset may be due to the presence of charge screening leading to a decrease of electron-phonon coupling and/or Auger recombination in the early stages of the carrier relaxation process<cit.>. These processes make possible the conservation of electron energy, leaving the electrons excited for a time much longer than the characteristic time of optical phonon emission. Such a long-lived excited state in GST results in a specific non-thermal lattice response in which nanoscale local order changes due to the high concentration of electrons remaining in an excited state. This premise is supported by a combination of the diffraction main peak intensity and position dynamics: while the intensity of the diffraction signal has a contribution from thermal effects, including the Debye-Waller factor (DWF), the diffraction peak position reflects solely the lattice strain (expansion or contraction) along the direction of the scattering vector. The correlation between the increase in the lattice temperature and the shift of the diffraction intensity peak is based on the fact that the excess energy transferred to the lattice from the optically excited electrons leads to an increase in the stress, which has gradients at the film interfaces. This produces a strain wave propagating into the sample at the speed of sound and an increase in the portion of the film with modified lattice spacing as the wave propagates. This results in a new Bragg condition and changes in the corresponding dynamics of the diffraction rocking curve profile, in particular, a shift of the main diffraction peak position.§ DISCUSSION Moderate decreases in diffraction intensity can be described by the Debye-Waller factor, e^-1/2Q^2⟨ u^2⟩, where ⟨ u^2⟩ denotes the mean-square displacement of collective atoms along the scattering vector Q<cit.>, but the absence of a thermally-induced diffraction peak position shift in the experimental data is an evidence of a different scenario. While one might assume that the observed changes are result of a response from a complete disordering of the part of the crystal lattice, this explanation is not valid: dramatic decreases in the diffraction intensity from crystalline samples are usually associated with structural disorder, which in turn is connected with randomization or fluctuations in atomic positions, but in tetrahedral covalently bonded semiconductors non-thermal melting has been reported to occur within several hundred femtoseconds<cit.>, a time scale much faster than seen in the current GST film. In fact, the observed recovery of the Bragg peak intensity on the nanosecond time scale demonstrates that the initial crystalline state remains. Re-crystallization from the molten state also cannot explain the observed dynamics as optically induced crystallization processes in GST require much longer times - typically more than 10 ns<cit.>. Therefore, the observed decrease in diffraction intensity for times < 4 ps cannot be interpreted to be the result of non-thermal melting in the general sense, i.e. a complete loss of long-range order. The absence of a thermal-shift during the first 4 ps, however, indicates the presence of non-thermal effects in photo-excited GST, arising due to the effects of intense optical excitation on resonant bonding. It should be noted that the term "resonant bonding" with respect to phase-change materials is used differently by different authors: while some authors consider all bonds to be resonant, others refer to only longer bonds as being so. In this manuscript, we have adopted the latter approach. The experimentally observed dynamics indicate that the diffraction peak intensity does not fully recover even 1.8 ns after excitation (Figure 2c) due to residual thermal effects, which completely disappear after 3 ∼ 5 ns depending on the excitation fluence. The corresponding return of the diffraction peak to its initial intensity demonstrates that the GST film was not amorphized, but transiently transformed to an intermediate state. The drop in the diffraction signal intensity cannot be explained soley by the DWF for the following two major reasons. First, the absence of the diffraction peak position shift until 4 ps after the excitation, which is more clearly represented by an inflection point on the time-dependent diffraction intensity curve, indicates the lack of thermal effects immediately after laser exposure. The inflection point marks the start of the shift of the diffraction signal peak position due to the expansion of the GST film, i.e. the onset of thermal effects. Second, if one assumes that the diffraction peak shift is the result of the strain in the sample due to thermal effects, then the acoustic phonon-induced contribution of the strain, corresponding to the shift at 128 ps after excitation, should be highly damped. This assumption leads to a calculated lattice temperature using the thermal expansion coefficient of 1.74x10^-5 K^-1 (ref. <cit.>) of ≈ 930 K, a temperature which is higher than the melting point, which is inconsistent with observation and thus indicates the presence of long-lived non-thermal effects<cit.>. In addition to thermal expansion, the laser-induced thermal stress in the film gives rise to the generation of acoustic phonons originating from the interfaces of the film with the capping layer and the substrate (Figure 3)<cit.>. The half-period of the acoustic phonon (T/2 is equal to the ratio of the sample thickness (l) and the sound velocity (v): T/2=l/v. The half-period of the experimentally observed strain waves originating from the interfaces was estimated to be ≈ 16 ps (Figure 3b). This allows the calculation of the sound velocity in the metastable state to be ≈ 2.19 nm ps^-1, a value which is ≈ 2/3 of the literature value for crystalline GST<cit.> and close to the value for the amorphous case<cit.>. To account for the observed strain dynamics, the propagation of strain waves in laser-excited GST as well as the resulting transient diffraction intensity change was simulated using a one-dimensional linear chain model as implemented in the package UDKM1DSIM<cit.> for two cases: using the literature value of the sound velocity for crystalline GST and the experimentally estimated sound velocity as an input parameter. In the first case the obtained time delay for the maximum deflection of the diffraction peak was about 2/3 of the half-period of the experimentally observed strain wave (Figure 3b). On the other hand, for the case of the experimentally estimated speed of sound value, the simulation results are in good agreement with the experimental data (Figure 3c), which indicates a significant change in the interatomic potentials from the ground state, a result which can be related to the local reorientation of atoms ensembles, leading to increased bonds lengths and a wider angular distribution of bonds. The results of the simulated transient diffraction pattern obtained using the UDKM1DSIM package are in a good agreement with the experimental data for the case of the experimentally estimated speed of sound in the GST film, but it should be noted that an earlier work, in which much slower optical excitation was used, the simulation data matched the experimental results for the literature value of the sound velocity<cit.>. This disparity is a consequence of the differences in excitation conditions (in the current work the pump pulse is much faster: 30 fs instead of the 700 fs (presented in ref. <cit.>), giving rise to the markedly different dynamics observed in the current work. In addition it should be pointed out that the speed of the diffraction response in the earlier work was also masked by a ≈100 ps convolution in time, due to the probe pulse width, making small changes in sound velocity difficult to distinguish.We argue that the ultrafast excitation of electrons from bonding to anti-bonding states leads to the breaking of longer (weaker) Ge/Sb-Te bonds<cit.>, i.e., the non-thermal melting of resonant bonds, leaving stronger covalent bonds intact. Based on the results presented in refs. <cit.> and <cit.>, we schematically represent the excited structure as non-connected cubes (Figure 1). The process of the formation of such building blocks is expected to occur over very short times, less than the characteristic recombination time of non-equilibrium charge carriers. The breaking of the longer Ge/Sb-Te bonds results in local structural relaxation, while the average long range order in GST crystal persists. The corresponding state can be characterized as a "disrupted crystalline" state. This transient disordering in GST is schematically visualized in Figure 1. In the ground state, building blocks are connected to each other by resonant bonds; in contrast, upon electronic excitation the building blocks locally relax due to the loss of resonant bonding, leading to the tilting and shifting of the blocks from their initial positions<cit.>. The decrease in the (222) Bragg diffraction peak intensity continues until the excited system gradually reverts to its initial state with resonant bond interaction over the course of at least 100 ps (Figure 2c). This implies that despite in the initial stage (first 4 ps) the excited state is purely electronic, it persists for much longer times, accompanied and thus hidden by thermal effects, which does not prevent its usage for ultrafast structural modification. Upon photo-excitation, local distortions can lead to a change in some Ge atomic configurations from an "octahedral" to a tetrahedral geometry with sp^3-hybridization<cit.>. Local structural relaxation back to the ground state is suppressed by a decrease in the probability of the restoration of resonant bonding due to a transient increase in mean-square atomic displacements and fluctuations in bond angles. These effects extend the long relaxation period of the excited electrons and serve as a kinetic barrier leading to the unusually long duration of structural recovery from the excited state in GST.The existence of a metastable (intermediate) state is also supported by an independent time-resolved XAFS experiment. A unique XAFSk^2 χ(k) signal was collected in which an excited state structure of GST (Figure 4a), different from both the liquid and amorphous states (Figure 4b), was transiently observed upon ultrafast optical excitation with the sample quickly reverting to its crystalline state (within ≈1 ns).A similar change could be reproduced based upon a molecular dynamics (MD) simulation using the plane wave density functional program VASP (see Experimental Section for details), in which 4% of the valence electrons were placed into the conduction band (a value equal to the theoretically estimated concentration of the excited electrons for the experimental conditions used); the MD simulations were carried out using a Nosé thermostat, which maintained the lattice temperature at approximately 300 K (the standard deviation of the temperature fluctuations was approximately 20 degrees implying that temperature fluctuations do not lead to significant changes in the XAFS spectra and can be neglected), i.e. the observed change is a consequence of a purely electronic effect. Figure 4c shows the results of the simulated time averaged XAFS signal for both the ground and excited states.In the MD simulations, we have used GeTe as an approximant for GST to avoid the configurational complexity of the vacancy distribution associated with GST.While for the ground state spectra, obtained from the experiment, there is a clear "beat" at approximately 3.7 Å^-1 that arises from quantum mechanical interference of the outgoing photoelectron wave function with itself due to backscattering from a well-defined second coordination shell, for the excited state, the interference is strongly diminished, and the spectra itself closely approximates a damped sine function, a shape characteristic of backscattering from a single coordination shell (Figure 4a). The simulations of the 300 K systems reproduce the experimental trends well with the difference between the two simulated spectra being solely due to the change in the electronic conditions, strongly supporting the non-thermal nature of the lattice perturbations. This is interpreted to be a consequence of ultrafast strong optical excitation leading to a transient increase in lattice local disorder;the same disorder is speculated to lead to the experimentally observed transient dramatic change in sound velocity and the extraordinary behavior of the diffraction signal peak position/intensity, measured in the XFEL pump-probe experiment.The existence of a disrupted crystalline state in GST can be used in phase-change memory to transiently lower the potential barrier between the SET and RESET states allowing energetically preferable switching with anticipated improvements over the current speed limit to rates in the GHz region (which corresponds to a few picoseconds for the structural relaxation to occur). The long lived excited state can be utilized in a different types of applications, like optical and electrical switches and sensors, possibly working even in the THz region, since the observed excited state appears in the first tens of femtoseconds (just after the laser excitation).The process of breaking the resonant bonds, leading to significant lattice local distortions, can serve as a precursor of non-thermal high-speed phase-switching. Moreover, it represents a new metastable state, characterized by both electronic and crystal lattice conditions, that results in optical and electric properties, different from the ground state. This difference can be utilized for the introduction of fundamentally new concepts of device operation for the recording and processing of data, including the possibility for applications in multi-level memory devices. The understanding of the transformation dynamics on ultrafast time scales over a few picoseconds provides new insight into the ultimate speed limit of phase change materials<cit.> and suggests that electronic effects may play a heretofore unrecognized important role in device switching, possibly leading to significant reductions in the energy requirements. Ultimately, resonant bonding state manipulation may allow the alteration of the crystal structure solely by the selective modification of the electronic state. In particular, the recent development of interfacial phase-change memory with spatially separated GeTe and Sb_2Te_3 layers, in which switching between the SET and RESET state is achieved without melting<cit.>, may be assisted by the presence of excited states induced by the large electric fields applied during switching. § METHODS §.§ Sample preparation:Crystalline Ge_2Sb_2Te_5 (GST) thin films were fabricated by molecular beam epitaxy (MBE) on the Si(111)-(√3x√3)R30^∘-Sb reconstructed surface as described in detail elsewhere<cit.>. According to x-ray reflectivity measurements (not shown) the film thickness of the sample was 35 nm. The GST films were investigated by x-ray diffraction (XRD) and showed a cubic phase characterized by a high degree of texture and structural order in both the out-of-plane and in-plane directions <cit.>. The lattice constant a was 6.04 Å, which is consistent with the literature value of 6.01 Å ∼ 6.04 Å<cit.>. Neglecting the slight rhombohedral distortion, GST is hence lattice-mismatched by ≈11% to Si (a = 5.431 Å). The samples were capped with 30 nm of sputtered amorphous Si_3N_4 to prevent oxidation. §.§ Pump-probe x-ray diffraction:Ultrafast time-resolved XRD measurements were conducted at the Japanese XFEL facility SACLA<cit.> using the output of the x-ray FEL in spontaneous emission mode. A monochromatic beam was generated from the XFEL beam using two channel-cut Si(111) crystals in a non-dispersive arrangement corresponding to a wavelength of 1.24 Å with a FWHM pulse of ≈ 10 fs. This beam was directed onto the sample as a probe pulse and the diffraction reflections obtained in the Bragg condition were detected by a 512 x 1024 channel multi-port CCD camera (Figure 1). Each measurement was repeated up to 250 times to improve statistics. A high-speed chopper was used to isolate individual x-ray pulses at 30 Hz. The p-polarized optical laser pulses for the pump (≈30 fs-duration at a center wavelength of 800 nm) were synchronized with the XFEL; the angle between the laser and x-ray beams was ≈5^∘. The maximum pump power was 3 mW (or 55 μJ pulse^-1) and the laser was focused into a 700 μm diameter spot, a size significantly larger than the diameter of x-ray beam spot (250 μm). The results presented here were obtained with pump fluences up to 13.9 mJ cm^-2 at the sample surface, which is below the threshold value for the irreversible changes in GST<cit.> after taking into account the reflectivity and the ellipsoidal broadening of the laser beam spot for the used angle of incidence. The jitter in the time delay between x-ray and laser pulses was ≤ 0.2 ps, supporting an effective time resolution better than 200 fs. The time delay (τ) between the pump laser and the x-ray pulse was adjusted by an optical delay line in time increments of 60 fs to 100 ps. An effective time resolution of 60 fs was obtained by time sorting x-ray pulses using a timing monitor for the diffraction intensity plot shown in Figure 2d.§.§ Time-resolved XAFS measurement:XAFS measurements were taken at sector 20 of the advanced photon source (APS). In the experiment, a 40 nm thick polycrystalline layer of GST encapsulated by 20 nm thick transparent Si_3N_4 layers with a ZnS-SiO_2 protective cap was irradiated by a 190 fs, 800 nm pump laser pulse and the resulting lattice dynamics were probed via measurement of Ge K-edge XAFS as a function of laser delay. The fluence used was 9 mJ cm^-2. The spot size of the laser was approximately 55 μm and the ∼ 5 μm Kirkpatrick-Baez focused x-ray probe beam was well contained within the pump beam spot.A quartz disk served as the substrate and was continuously rotated using an ultra-low wobble spindle motor to eliminate heat build up and reduce radiation damage. The laser system consisted of a Coherent Millenia Ti-Sapphire oscillator phase-locked to the RF signal of the storage ring and a Coherent RegA regenerative amplifier system which could provide power up to 4 μJ pulse^-1. The disk was rotated assuring that several microseconds passed before a given area on the sample was subsequently irradiated.The Ge fluorescence signal was detected via a 100,000 element gated Pilatus detector. By varying the laser trigger signal delay, the relative position of the laser pump to the x-ray probe pulse could be varied in ∼ 18 ps intervals. The x-ray probe pulse had a duration of approximately 100 ps resulting in a similar convolution in time of the experimental data. The fact that the unique signature of the excited state is visible despite the convolution resulting from the duration of the x-ray pulse, substantiates the long lifetime of the excited state.§.§ VASP simulations:The simulated XAFS spectra were generated in two steps. In the first, ab-inito molecular dynamics (AIMD) were used to follow the trajectory of a 128 atom GeTe cluster for 30 picoseconds using density functional theory and the plane wave code VASP 5.4.1<cit.>. The effects of optical excitation were simulated by constraining the band occupancies such that 4% of the valence electrons were promoted to the conduction band. TThe generated trajectories were then used as input to the real-space multiple scattering code FEFF 9.64<cit.> to generate XAFS spectra, which were then averaged to generate the final simulated XAFS spectra.A cutoff energy of 175 eV was used for the plane waves in VASP and k-point sampling was carried out using the gamma point. A convergence study showed that use of the gamma point for integration of the Brillioun zone was sufficient to reflect the the general trends of the XAFS data. The energy difference was found to be less than 10 meV/atom. Projector augmented waves were used to include the effects of the core electrons. 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B volume62, pages2437 (year2000). § ACKNOWLEDGEMENTS This work was supported by X-ray Free Electron Laser Priority Strategy Program, entitled Lattice dynamics of phase change materials by time-resolved X-ray diffraction (NO. 12013011 and 12013023), from the Ministry of Education, Science, Sports, and Culture of Japan.The XFEL experiments were performed at the BL3(EH2) of SACLA with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2012B8041, 2013A8051, 2013B8056, 2014A8039 and 2014B8061).Sector 20 facilities at the Advanced Photon Source, and research at these facilities, are supported by the US Department of Energy - Basic Energy Sciences, the Canadian Light Source and its funding partners, the University of Washington, and the Advanced Photon Source. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.§ AUTHOR CONTRIBUTIONS STATEMENT K.V.M., P.F., K.M., R.T., T.Shimada, A.V.K., A.G., T.Sato, T.K., K.O., T.T., S.W., D.B. and M.H. carried out the experiments. K.V.M., P.F. and M.H. processed the data and performed the simulations. V.B., A.G. and R.C. prepared the samples. K.V.M., P.F., A.V.K. and M.H. wrote the manuscript. All of the co-authors contributed to the discussions of the results and manuscript.§ ADDITIONAL INFORMATION Competing financial interests:The authors declare no competing financial interests. | http://arxiv.org/abs/1705.09472v1 | {
"authors": [
"Kirill V. Mitrofanov",
"Paul Fons",
"Kotaro Makino",
"Ryo Terashima",
"Toru Shimada",
"Alexander V. Kolobov",
"Junji Tominaga",
"Valeria Bragaglia",
"Alessandro Giussani",
"Raffaella Calarco",
"Henning Riechert",
"Takahiro Sato",
"Tetsuo Katayama",
"Kanade Ogawa",
"Tadashi Togashi",
"Makina Yabashi",
"Simon Wall",
"Dale Brewe",
"Muneaki Hase"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170526081945",
"title": "Sub-nanometre resolution of atomic motion during electronic excitation in phase-change materials"
} |
[email protected] Theoretical Astrophysics, Eberhard-Karls University of Tübingen, Tübingen 72076, [email protected] Theoretical Astrophysics, Eberhard-Karls University of Tübingen, Tübingen 72076, Germany InstituteofPhysicsandResearchCentreofTheoreticalPhysicsandAstrophysics, FacultyofPhilosophyandScience,SilesianUniversityinOpava,Opava,[email protected] Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC),Rua Abolição, CEP: 09210-180, Santo André, SP, Brazil Higher derivative extensions of Einstein gravity are important within the string theory approach to gravity and as alternative and effective theories of gravity. H. Lü,A. Perkins, C. Pope, K. Stelle[Phys. Rev. Lett. 114 (2015), 171601] found a numerical solution describing a spherically symmetric non-Schwarzschild asymptotically flat black hole in the Einstein gravity with added higher derivative terms. Using the general and quickly convergent parametrization in terms of the continued fractions, we represent this numerical solution in the analytical form, which is accurate not only near the event horizon or far from black hole, but in the whole space. Thereby, the obtained analytical form of the metric allows one to study easily all the further properties of the black hole, such as thermodynamics, Hawking radiation, particle motion, accretion, perturbations, stability, quasinormal spectrum, etc. Thus, the found analytical approximate representation can serve in the same way as an exact solution. 04.50.Kd,04.70.Bw,04.25.Nx,04.30.-w,04.80.CcNon-Schwarzschild black-hole metric in four dimensional higher derivative gravity: analytical approximation A. Zhidenko December 30, 2023 ===========================================================================================================§ INTRODUCTION Recent observation of gravitational waves from, apparently, the binary black holes merger <cit.> and considerable progress in observations of the galactic black hole in the electromagnetic spectrum <cit.> made black holes important objects for testing the regime of strong gravity. At the same time, the current lack of accuracy in determination the angular momentum and mass of the resultant ringing black hole leaves open the window for alternative theories of gravity, allowing for deviations from Schwarzschild and Kerr geometries <cit.>. One of such interesting alternatives is the Einstein gravity with added quadratic in curvature term for which the most general action has the form I = ∫ d^4x√(-g)(γ R - α C_μνρσC^μνρσ + β R^2) , where α, β and γ are constants, C_μνρσ is the Weyl tensor.In <cit.> it was shown that in addition to the Schwarzschild solution in the theory (<ref>), there is another spherically symmetric asymptotically flat non-Schwarzschild black-hole solution within the same theory. The numerical solution for the non-Schwarzschild case was represented in <cit.>. Numerical solution, although it can be used for further numerical analysis at fixed values of parameters, does not give a clear picture of dependence of the metric on physical parameters of the system. Therefore, the general method of parametrization of black-hole spacetimes was developed in <cit.> for spherically symmetric and in <cit.> for axially-symmetric black holes. In the spherically symmetric case, considered here, the method is based on the continued fraction expansion in terms of a compactified radial coordinate. Comparison of observables, such as position of the innermost stable circular orbit and shadows cast by black holes demonstrated that this method turned out to be rapidly convergent <cit.>, giving us an opportunity for finding relatively concise analytical approximation for a black-hole metric.Here we shall use the above mentioned continued fraction parametrization and find the analytical form for the asymptotically flat non-Schwarzschild numerical solution <cit.> in the Einstein gravity with quadratic in curvature corrections. The obtained metric satisfies the currently existing constrains on the post-Newtonian (weak field) behavior.The metric functions are represented as a ratio of polynomials of the radial coordinate with the coefficients, which depend on the coupling constant and black-hole radius. The latter re-scales the black hole mass and radial coordinate and can be fixed in further analysis. The main result of our work is the obtained analytical fourth order representation of the metric (written down explicitly in Appendix A), which is accurate in the whole space outside the black hole. This allows one to use it effectively for various studies of the black-hole properties and analysis of interactions between the black hole and surrounding matter.The paper is organized as follows. Sec. <ref> briefly discuss the theory under consideration and shows that without loss of generality it can be reduced to the Einstein-Weyl theory when considering spherically symmetric solutions. Sec. <ref> relates the deduction of the analytical expressions for the metric functions with the help of the continued fraction parametrization. Sec. <ref> is devoted to testing the accuracy of the obtained analytical metric through calculation of observable characteristics: rotational frequency on the innermost stable circular orbit and eikonal quasinormal modes. Finally, in Sec. <ref> we discuss the obtained results and spotlight the main potentially interesting applications that can be done based on the obtained here analytical form of the metric. § STATIC SOLUTIONS IN THE EINSTEIN GRAVITY WITH ADDED QUADRATIC TERMS §.§ Analytical approximation One of the coupling constants can be fixed when choosing the system of units, so we take γ=1. Then, the equations of motion take the form R_μν-1/2 R g_μν -4 α B_μν +2βR(R_μν-1/4 R g_μν)+2β(g_μν□ R-∇_μ∇_ν R)=0 , where B_μν= (∇^ρ∇^σ + 1/2 R^ρσ) C_μρνσ is the tracefree Bach tensor. It is the only conformally invariant tensor that is algebraically independent of the Weyl tensor.One can write a static metric as follows ds_4^2 = -λ^2dt^2 + h_ij dx^i dx^j, where λ and h_ij are functions of the spatial coordinates x^i. In <cit.> it was shown that, taking the trace of the field equations (<ref>) and integrating the equations of motion over the spatial domain from the event horizon to infinity, one can find that ∫√(h)d^3 x[D^i(λ R D_i R) - λ (D_i R)^2 - m_0^2 λ R^2]=0 , where D_i is the covariant derivative with respect to the spatial 3-metric h_ij. By definition, λ vanishes on the event horizon, so that if D_i R goes to zero sufficiently rapidly at spatial infinity, then the total derivative term can be discarded and any static black-hole solution of (<ref>) must have vanishing Ricci scalar R=0. The latter means that, without loss of generality, we can be constrained by the Einstein-Weyl gravity (β=0). Then, since B_μν is tracefree, the trace of (<ref>) implies the vanishing Ricci scalar (R=0). Therefore, the Schwarzschild solution is also a solution for the Einstein-Weyl gravity.Summarizing, when considering static solutions in the most general Einstein gravity with quadratic in curvature corrections given by (<ref>), one can take γ =1 and β =0 without loss of generality. § BLACK HOLES IN HIGHER-DERIVATIVE GRAVITYHere, first, we shall find a general analytical form of the non-Schwarzschild metric and then expand it in terms of the small deviation k from the Schwarzschild branch. §.§ Analytical approximation The line element of a black hole is given by ds^2=-h(r)dt^2+dr^2/f(r)+r^2(dθ^2+sin^2θ dϕ^2) , where the functions h(r) and f(r) satisfy the Einstein-Weyl equations of motion, which have the following form:h”(r) = 4 h(r)^2 (1-r f'(r)-f(r))-r h(r)(rf'(r)+4f(r)) h'(r)+r^2 f(r) h'(r)^2/2 r^2 f(r) h(r) ,f”(r) = h(r)-f(r) h(r)-r f(r) h'(r)/α f(r) (2 h(r)-r h'(r))-h(r) (3 r^2 f'(r)^2+12 r f(r) f'(r)-4 r f'(r)+12 f(r)^2-8 f(r))/2 r^2 f(r) (r h'(r)-2 h(r)) +f(r) (r^2 h'(r)^2-r h(r) h'(r)-2 h(r)^2)-r h(r) f'(r) (r h'(r)+4 h(r))/2 r^2 h(r)^2 .The metric functions f(r) and h(r) can be expanded into the Taylor series near the horizon r_0: [ h(r)=c[(r-r_0) + h_2(r-r_0)^2 + …]; f(r)= f_1(r-r_0) + f_2(r-r_0)^2 + …, ] where f_1 is the shooting parameter, which we choose in such a way that the solution is asymptotically flat, and c is the arbitrary scaling factor, which we choose such that t is the time coordinate of a remote observer, i.e.,lim_r→∞h(r)=1. Substituting (<ref>) into (<ref>) and (<ref>), one can express all the coefficients in terms of c and f_1, which we find using numerical integration as prescribed in <cit.>.It is useful to introduce the dimensionless parameter, which parameterizes the solutions up to the rescaling p=r_0/√(2α). Notice that for all p the Schwarzschild metric is the exact solution of the Einstein-Weyl equations as well, but at some minimal nonzero p_min, in addition to the Schwarzschild solution, there appears the non-Schwarzschild branch (found numerically in <cit.>) which describes the asymptotically flat black hole, whose mass is decreasing, when p grows, and vanishing at some p_max. The approximate maximal and minimal values of p are:p_min≈ 1054/1203 ≈ 0.876,p_max≈ 1.14 Following the parametrization procedure given in <cit.> we define the functions A and B through the following relations: h(r) ≡ xA(x) , h(r)/f(r) ≡ B(x)^2 , where x denotes the dimensionless compact coordinate x ≡ 1-r_0/r . We represent the above two functions as follows: A(x) = 1-ϵ (1-x)+(a_0-ϵ)(1-x)^2+Ã(x)(1-x)^3,B(x) = 1+b_0(1-x)+B̃(x)(1-x)^2, where Ã(x) and B̃(x) are introduced in terms of the continued fractions, in order to describe the metric near the event horizon x=0: Ã(x)=a_1/ 1+ a_2x/ 1+ a_3x/ 1+ a_4x/ 1+… ,B̃(x)=b_1/ 1+ b_2x/ 1+ b_3x/ 1+ b_4x/ 1+… . At the event horizon one has: Ã(0) = a_1, B̃(0) =b_1. We notice that (<ref>) and (<ref>) imply that a_0=b_0=0, i.e. the post-Newtonian parameters for the non-Schwarzschild solution coincide with those in General Relativity. We fix the asymptotic parameter ϵ as ϵ=-(1-2M/r_0) , using the value of the asymptotic mass which can be found by numerical fitting of the asymptotical behavior of the metric functions.Expanding (<ref>) and (<ref>) near the event horizon we find the parameters a_1,a_2,a_3,…,b_1,b_2,b_3,… as functions of c and f_1. In their turn, the values of c and f_1 can be found numerically for each value of the parameter p. It appears that ϵ, a_1, and b_1 approach zero forp≈1054/1203,where the numerical non-Schwarzschild solution coincides with the Schwarzschild one.The fitting of numerical data for various values of p and r_0 shows that ϵ, a_1, and b_1 can be approximated within the maximal error ≲ 0.1% by parabolas as follows: ϵ ≈ (1054 - 1203 p)(3/1271 + p/1529) ,a_1 ≈ (1054 - 1203 p)(7/1746-5 p/2421) ,b_1 ≈ (1054 - 1203 p)(p/1465-2/1585) .These fittings are almost linear in p. Indeed, if one uses k = 1054-1203 p, then the above relations readϵ ≈ 6857795 /2337860877k (1-1271/6857795k)+k^3 , a_1 ≈ 1242869 /565017822k (1+970/1242869 k) +k^3 , b_1 ≈ -370840 /558679215k (1+317/370840k)+k^3 , where one can see that the coefficients of the quadratic form are quite small. Nevertheless they cannot be neglected if one aims at 0.1% accuracy. With the same accuracy we are able to find a_2 and b_2 asa_2 ≈ 6 p^2/17+5 p/6-131/102 ,b_2 ≈ 81 p^2/242-109 p/118-16/89 .a_3 and a_4 diverge atp≈237/223.Therefore we find out that these parameters can be well approximated asa_3 ≈ 9921 p^231-385 p+485729/237-223 p ,a_4 ≈ 9 p^214+3149 p42-280314/237-223 p .In this way, although each of the parameters a_3 and a_4 diverges at p ≈ 237/223, its ratio is finite and thereby has finite contribution into the continued fraction (see fig. <ref>). Finally, we observe that b_3 and b_4 are well approximated by the straight lines asb_3 ≈ -2 p/57+29/56 ,b_4 ≈ 13 p/95-121/98 . Within the chosen accuracy of a fraction of 0.1% for the metric functions f(r) and h(r) (see fig. <ref>) we can set a_5=b_5=0 in (<ref>) and, substituting the found above coefficients ϵ, a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4 into (<ref>), obtain the final analytic expressions for the metric functions as the forth order continued fraction expansion (see appendix <ref>).If one is limited by a rougher approximation of the second order, all the parameters ϵ, a_1, b_1, a_2, b_2 can also be well approximated by linear (in p) polynomials instead of the quadratic ones. Such an approximation would have much simpler analytical form (which will be discussed in subsection C), leading to the larger maximal error of about a few percents. §.§ Expansion of the forth order approximation near the Schwarzschild solutionWhen one is interested in relatively small deviations from the Schwarzschild geometry, a more concise expressions can be obtained by using the expansion in terms of k. In order to have a positive asymptotic mass, the values of k can vary from 0 until approximately -321.727. For example, the final formula for h(r) expanded up to the first order in k then readsh(r) =(1-r_0/r)(1 - kr_0/rh_1(r)/h_2(r)+k^2),where h_1(r)=7094296364854698294656777815 r^3+ 2700140790021572890363934045 r^2 r_0 +32852984866789222219083981378 r r_0^2-4194480693404458083513273360 r_0^3,h_2(r)=61001803863561 r (39646131244569649 r^2-24556525364789942 r r_0+156809140779977329 r_0^2).This formula is considerably shorter than the full formula for h(r), what might be useful when one numerically models various process around the black hole. The ratio of both metric functions isf(r)/h(r) = 1+37793605455056 k r_0^2 (50038777 r+84360383 r_0)/278643 r (6266529735540821295 r^2-3742896005107026923 rr_0+11207698915983181988 r_0^2)+ k^2. When |k| ⪅ 100, first order expansions in k for h(r) and f(r) stay within a few tenths of one percent from the full analytical metric, keeping thereby the same order of the general error. The k-expanded metric is also included into the Mathematica® notebook we share.§.§ Second order approximation: more compact, but robust analytical metricIn case one is interested in a much more robust, but compact expression for the metric, one can be limited by the second order in the expansion (<ref>), i. e. take a_3=b_3=0. Then, it is sufficient to consider a linear fit for ϵ, a_1, a_2, b_1, b_2 as followsϵ ≈ 1054 - 1203 p/326 ,a_1 ≈ 1054 - 1203 p/556 ,a_2 ≈ -18 - 17 p/11 ,b_1 ≈ -1054 - 1203 p/1881 ,b_2 ≈ -2+p/4 . Thus, we obtain even simpler (than in the two previous subsections) form for the metric functions A(r) and B(r) in (<ref>),A(r) = 1-(1054 - 1203 p)r_0^2/2r^2× ×(r+r_0/163r_0+11r_0/278(7r-18r_0-17p(r-r_0))) ,B(r) = 1-4(1054 - 1203 p)r_0^3/1881r^2(2(r+r_0)-p(r-r_0)) .Yet, the obtained metric is considerably less accurate: for p< 0.97 the relative error stays within a fraction of one percent, but for near extremal values of p it may reach a few percents. The accuracy of the approximation for a given value of p of the second and forth order approximation can be learnt from the Mathematica notebook we share with readers. § TESTING THE ACCURACY OF THE APPROXIMATIONThe metric is not a gauge-invariant characteristic and, strictly speaking, comparing the metric functions has no direct physical interpretation. Therefore, the best way to test the accuracy of the analytical metric obtained in the previous section is to calculate basic observable quantities for the analytical metric and compare them with the accurate ones found for the numerical metric. Here we shall consider two kinds of such observable characteristics: the frequency of a massive particle on the innermost stable circular orbit (ISCO) and frequencies of the quasinormal modes in the eikonal (short wavelength) regime.§.§ Innermost stable circular orbitFirst, we shall compute the radius of the smallest circular orbit of a massive test particle rotating around the black hole. The circular movement of a massive particle is described by the following potentialV_m(r)=E^2/h(r)-L^2/r^2-1,where E and L are, respectively, the energy and momentum per unit mass.The innermost stable circular orbit corresponds toV_m(r)=V_m'(r)=V_m”(r)=0,which is reduced to the following equation for the radial coordinate of the orbitr h(r) h”(r)-2 r h'(r)^2+3 h(r) h'(r)=0.We solve the above equation numerically with h(r) given in the Appendix A in the analytical from and in <cit.> numerically.The corresponding orbital frequencies are givenΩ=√(h'(r)/2r).From Fig. <ref> we see thatthe frequency of ISCO decreases quite a few times, as the parameter p grows. This means that the ISCO moves outward the black hole at a great extent. The relative error stays within a few percents for the parametric region under consideration, being much less for the near Schwarzschild and near extremal (almost massless) cases. At the same time, for the intermediate values of p, when the error of the analytical approximation is maximal (see Fig. <ref>), the effect of the deviation from the Schwarzschild metric on the orbital frequency is already much larger than the error. This means that the forth order approximation developed here is adequate. §.§ Analytical formulas for the eikonal quasinormal frequencies Here we shall consider the proper oscillations frequencies, called quasinormal modes <cit.>, of a test field in the background of the black hole in the high frequency (eikonal or high multipole number) regime. The boundary conditions for the quasinormal modes are purely incoming wave on the event horizon and purely outgoing wave at infinity. In the geometrical optic (eikonal) regime perturbations of test fields of any spin are dominated by the same centrifugal-like part of the effective potential. Therefore, it is sufficient to consider here the derivations only for the test massless scalar field, while the resultant eikonal formulas for the test fields of other spin will be the same. Perturbations of a test scalar field obey the general relativistic Klein-Gordon equation1/√(-g)∂ _μ( √(-g)g^μν∂ _νΦ) = 0,Implying thatΦ(t,r,θ,ϕ)= e^-ω tY_ℓ(θ,ϕ)Ψ(r)/r,where Y_ℓ(θ,ϕ) are spherical harmonics, the Klein-Gordon equation can be reduced to the following form: d^2Ψ/dr_*^2+(ω^2-V(r_*))Ψ=0.Here, ω is the frequency; the tortoise coordinate r_* is defined as followsdr_*=dr/√(f(r)h(r)),and the effective potential for the large value of the multipole number ℓ takes the formV(r)=(ℓ+1/2)^2(h(r)/r^2+1/ℓ^2) . From (<ref>) we find that the effective potential (<ref>) has the maximum atr_m=3r_0/2(1+0.000393k)+k^2,ℓ^-2.As k is negative, then the peak of the effective potential is closer to the black hole horizon for non-Schwarzschild solution than for the Schwarzschild one.At high ℓ, once the effective potential has the form of the potential barrier, falling off at the event horizon and spacial infinity, the WKB formula found in <cit.> (for improvements and extensions of this formula, see <cit.>) can be applied for finding quasinormal modes:ω^2=V(r_m)-(n+1/2)√(-2d^2V(r_m)/dr_*^2),which depends on the value and the second derivative of the potential in its maximum r_m. Using the above eikonal formula for the QNMs we find thatω = 2/3√(3)r_0[(ℓ+1/2)(1-0.001308k)-(n+1/2)(1-0.002743k)]+k^2,ℓ^-1,implying higher real (photon circular orbit) frequency and faster decay of the oscillations for the non-Schwarzschild branch. When k=0 the above formula is reduced to its Schwarzschild analogue.We compare the above formulas with the precise values for the real and imaginary parts of the frequencies, obtained by substituting the numerical solutions into the eikonal WKB formula. From Fig. <ref> we see that the first-order in k formula (<ref>) provides quite an accurate result for the real (fractions of a percent) and imaginary (a few percents) parts of the eikonal frequencies. Notice, that the obtained analytical form in terms of the deviation from the Schwarzschild solution k, allowed us to find easily the concise analytical non-Schwarzschild generalization of the well-known eikonal formulas for quasinormal modes of the Schwarzschild black hole. Usually the obtained here eikonal quasinormal modes of a test scalar field determine the parameters of the circular null geodesic: the real and imaginary parts of the quasinormal mode are multiples of the frequency and instability timescale of the circular null geodesics respectively. However, quasinormal modes of non-test, e. g., gravitational, fields may not obey this rule <cit.>. § DISCUSSIONIn the present paper we have obtained the approximate analytic expression for the black-hole solution of the non-Schwarzschild metric in the most general Einstein gravity with quadratic in curvature corrections. The obtained analytical metric represents asymptotically flat black hole which has the same post-Newtonian behavior as in General Relativity, but is essentially different in the strong field regime. The metric is expressed in terms of event horizon radius r_0 and the dimensionless parameter p = r_0/√(2 α), where α is the coupling constant. The minimal value of p≈0.876 corresponds to the merger of the Schwarzschild and non-Schwarzschild solutions, while at p≈1.14 the black-hole mass approaches zero (ϵ=-1).As the metric is written in terms of the black-hole parameter and coupling constant and is accurate everywhere outside the event horizon, it can be used in the study of the basic properties of the black hole and the description of various phenomena in its vicinity in the same way as the exact analytical solution. Our main future aim is to generalize the obtained analytical metric to the case of rotating black holes <cit.>. At the same time a number of other appealing problems associated with the obtained metric could be solved:* Perturbations and analysis of stability of the non-Schwarzschild black hole;* Quasinormal modes of gravitational and test fields in its vicinity (this was partially done for the numerical solution in <cit.> and comparison between analytical and numerical metrics would be appealing). As higher curvature corrections frequently lead to a new branch of non-perturbative (in coupling constant) modes <cit.>, it is interesting to check whether this phenomena takes place for the considered here quadratic gravity.* Analysis of massless and massive particles' motion, binding energy, innermost stable circular orbits, stability of orbits;* Analysis of the accretion disks and the corresponding radiation in the electromagnetic spectra;* Consideration of tidal and external magnetic fields in the vicinity of a black hole, etc.* Hawking radiation in the semiclassical and beyond semiclassical regimes;* Detailed study of the black-hole thermodynamics. The obtained here analytical approximation for the metric (given in Appendix A) has two evident advantages over the numerical solution. First, it allows one to solve all the above enumerated problems in a much more economic and elegant way. Second, the analytical metric allows applications of a greater variety of methods for its analysis. For example, in order to get the full knowledge of the characteristic quasinormal spectrum of a black hole, one has to use the Leaver method <cit.> which simply cannot be applied to the numerical interpolation function and requires the analytically written metric. Applications of other methods, for example (used here for illustration) WKB method <cit.> or the time-domain integration <cit.>, are considerably constrained and give information only about the lowest modes.In addition, we found the eikonal quasinormal frequencies of test fields and the frequency and positions of ISCO for the non-Schwarzschild black hole solution in the higher derivative gravity. Comparison of the data obtained for the analytical and numerical metrics allowed us to test the accuracy of our approximation. It is shown that the non-Schwarzschild black hole is characterized by a much further position of ISCO and much slower rotational frequency of a massive particle. The eikonal quasinormal modes of the non-Schwarzschild black hole have smaller real oscillation frequencies and damping rates.Here we used the expansion up to the forth order and achieved accuracy in the metric functions with the maximal error of fractions of a percent. Once it is necessary, expansion to higher orders will produce much more accurate representation of the metric.R. K. would like to thank H. Lü for sharing his Mathematica® code which produces the numerical solution considered here. R. K. was supported by “Project for fostering collaboration in science, research and education” funded by the Moravian-Silesian Region, Czech Republic and by the Research Centre for Theoretical Physics and Astrophysics, Faculty of Philosophy and Science of Silesian University at Opava. A. Z. thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for support and Theoretical Astrophysics of Eberhard Karls University of Tübingen for hospitality. § ANALYTICAL FORM OF THE METRIC FUNCTIONS Here we provide the obtained analytical form of the metric. In the attachment to this article we share with readers the Mathematica notebook, where the metric functions are explicitly written down.The black-hole metric can be written in the form: ds^2=-(1-r_0/r)A(r)dt^2+B(r)^2dr^2/(1-r_0r)A(r)+r^2(dθ^2+sin^2θ dϕ^2) ,Here, we present the metric functions A(r) and B(r) in terms of the parameters b and r_0. Notice, that one can get the black hole radius r_0 =1 (which leads to the re-definition of the black-hole mass) and express everything in terms of r_0.A(r) = [152124199161 (873828 p^4-199143783 p^3+806771764 p^2-1202612078 p+604749333) r^4+78279(1336094371764p^6-300842119184823 p^5+393815823540843 p^4+2680050514097926 p^3..-9501392159249689 p^2+10978748485369369 p-4249747766121792)r^3 r_0-70372821 (1486200636 p^6+180905642811 p^5+417682197141 p^4-1208134566031 p^3..-324990706209 p^2+3382539200269p-2557857695019) r^2 r_0^2-(104588131327314156 p^6.-23549620247668759617 p^5-435688050031083222417p^4+2389090517292988952355 p^3.-3731827099716921879958 p^2+2186684376605688462974 p-389142952738481370396) r r_0^3+31(3373810687977876 p^6+410672271594465801 p^5-14105000476530678231 p^4+51431640078486304191 p^3. .-71532183052581307042p^2+43250367615320791700 p-9476049523901501640) r_0^4]/[152124199161 r^2((873828 p^4-199143783 p^3+806771764 p^2-1202612078 p+604749333) r^2-2 (873828 p^4-47583171 p^3+386036980 p^2-678598463 p+341153481) r r_0+899(972 p^4+115659 p^3-38596 p^2-1127284 p+1101579) r_0^2)] , B(r) = [464405 (3251230164 p^3-14548777134 p^2+20865434326 p+23094914865) r^3-464405 (6502460328 p^3-52856543928p^2+100077612184 p-32132674695) r^2 r_0-(1244571650887908 p^3+17950319416564777 p^2-53210739821255918p+5097428297648940) r r_0^2 +635371 (4335198168 p^3-42352710803 p^2+90235778452 p-49464019740)r_0^3]/[464405 r ((3251230164 p^3-14548777134 p^2+20865434326 p+23094914865) r^2-(6502460328p^3-52856543928 p^2+100077612184 p-32132674695) r r_0+6(541871694 p^3-6384627799 p^2+13202029643p+2626009760) r_0^2)] . 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"authors": [
"K. Kokkotas",
"R. A. Konoplya",
"A. Zhidenko"
],
"categories": [
"gr-qc",
"astro-ph.HE",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170527225609",
"title": "Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: analytical approximation"
} |
Shell et al.: Bare Demo of IEEEtran.cls for JournalsObservation of Spatio-temporal Instability of Femtosecond Pulses in Normal Dispersion Multimode Graded-Index Fiber Uğur Teğin and Bülend Ortaç U. Teğin and B. Ortaç are with National Nanotechnology Research Center (UNAM) and Institute of Materials Science and Nanotechnology, Bilkent University, 06800 Bilkent, Ankara, Turkey (e-mails: [email protected] and [email protected]) ================================================================================================================================================================================================================================================================================================== We study the spatio-temporal instability generated by a universal unstable attractor in normal dispersion graded-index multimode fiber (GRIN MMF) for femtosecond pulses. Our results present the generation of geometric parametric instability (GPI) sidebands with ultrashort input pulse for the first time. Observed GPI sidebands are 91 THz detuned from the pump wavelength, 800 nm. Detailed analysis carried out numerically by employing coupled-mode pulse propagation model including optical shock and Raman nonlinearity terms. A simplified theoretical model and numerically calculated spectra are well-aligned with experimental results. For input pulses of 200-fs duration, formation and evolution of GPI are shown in both spatial and temporal domains. The spatial intensity distribution of the total field and GPI sidebands are calculated. Numerically and experimentally obtained beam shapes of first GPI features a Gaussian-like beam profile. Our numerical results verify the unique feature of GPI and generated sidebands preserve their inherited spatial intensity profile from the input pulse for different propagation distances particularly for focused and spread the total field inside the GRIN MMF.Ultra-short pulses, Graded-index multimode fibers, Nonlinear fiber optics, Spatio-temporal pulse propagation. § INTRODUCTIONMultimode optical fibers are commonly used in real world applications such as telecommunications, imaging, beam delivery due to the large core size and low nonlinearity. Multimode fibers are generally considered as an unpredictable and random light propagation environments. With recent experimental and theoretical studies, this widely acclaimed attitude is questioned. Plöschner et al. demonstrated that multimode fibers have predictable behaviors and the transferred light stays deterministic <cit.>. Poletti et al. exploited the nonlinear interactions taking place inside of multimode fibers and study supercontinuum generation numerically <cit.>. However, with these astonishing features, multimode fibers require intricate treatments to understand complex nonlinear dynamics. Nowadays, GRIN MMFs, special cases for standard MMFs, are attracting great interest. With the parabolic profile of their refractive index, GRIN MMFs provide novel features. In the last few years, researchers exploited these features and reported new nonlinear dynamics to explore such as GPI <cit.>, supercontinuum generation <cit.>, self-beam cleaning <cit.>, multimode solitons <cit.> and their dispersive waves <cit.>. Among these spatio-temporal effects, GPI, called also as spatio-temporal modulation instability in the literature, excels as new wavelength generation technique since generated GPI sidebands have remarkable frequency shift and inherit the spatial beam shape of pump pulse <cit.>.In 2003, Longhi'stheoretical work predicted GPI effect in multimode fibers <cit.>. Because of the periodic refocusing of the beam, while propagating in GRIN MMF, quasi-phase matching (QPM) (between the pump, signal and idler) resulted as GPI sidebands and discrete peaks appear in the spectrum. In contrast to intermodal four-wave mixing (FWM) which is capable of generating spectral peaks with the same amount of frequency shifts, GPI peaks inherit the spatial mode profile of the pump source <cit.>. Krupa et al. reported the first experimental observation of GPI sidebands by using 900 picosecond pulses with 50 kW peak power inside of GRIN MMF and observed GPI sidebands are detuned more than 120 THz from the pump frequency <cit.>. This study later verified by Lopez-Galmiche et al. <cit.> while demonstrating of supercontinuum generation in GRIN MMF. Their results indicated for a relatively long GRIN MMF cascaded generation of GPI sidebands evolve through the formation of a supercontinuum with contributions of higher-order nonlinear effects such as self-phase modulation (SPM), stimulated Raman scattering (SRS) and FWM generations. Very recently, Wright et al. <cit.> studied the complex background of the self-beam cleaning and GPI sideband generation in GRIN MMF. Their detailed study contains the contributions of higher-order modes to GPI sideband generation and effect of disorder in such a nonlinear system.Aforementioned studies about on spatio-temporal instability focused on quasi-continuous pulse (hundred ps to few ns) evolution in GRIN MMF at normal dispersion regime due to the analogy between GPI and modulation instability presented by Longhi's theoretical work <cit.>. Thus GPI dynamics for femtosecond pulses remained unknown. Here, we present first experimental investigations of the spatio-temporal instability of ultrashort pulses in GRIN MMF at normal dispersion. Femtosecond, linearly polarized pulses at 800 nm start to experience spatio-temporal evolution inside 2.6 m GRIN MMF with 50 μm core diameter and their evolution resulted as the generation of first GPI stokes and anti-Stokes pair in the experiment. Observed GPI sidebands appear in the spectrum as 91 THz detuned with respect to launched pump pulse's central frequency. Formation and the broadening tendency of GPI sidebands with increasing launch pulse energy are reported. Spatial beam shape of first GPI Stokes is measured and features Gaussian-like near-field beam profile. Theoretical calculations and numerical simulations confirm the experimental observations on the GPI sidebands. Simulation results provide detailed information on the generation and the formation behaviors of GPI sidebands. The positions of GPI sidebands in the frequency domain are within the reach of experimental observations. Furthermore, numerical studies provide information on spatial distribution of total field and GPI sidebands inside the GRIN MMF.§ THEORETICAL AND NUMERICAL STUDYLonghi's theoretical model can be simplified and used for estimating frequency offset of GPI sidebands with the pump pulse for given GRIN MMF parameters <cit.>. During sufficient amount of higher-order mode excitation in GRIN MMF, the propagating pulses experience spatial oscillations in longitudinal direction with period ε = π r / √(2Δ), where Δ is the relative index difference and defined as Δ = (n_co^2-n_cl^2)/2n_co^2, r is the core radius of GRIN MMF and n_co (n_cl) is maximum refractive index of fiber core (clad). The required QPM condition for pump, Stokes and anti-Stokes can be expressed as 2k_P-k_S-k_A =-2π h/ε where h = 1, 2, 3, .... Frequency separation between GPI sidebands and pump frequency (f_h) is assumed as :(2π f_h)^2 = 2π h/εβ _2 - 2n_2Îω_0/cβ _2where n_2 is the nonlinear Kerr coefficient, Î is path averaged intensity of beam, ω_0 is the angular frequency of the pump and β _2 is the fiber dispersion parameter. The nonlinear part of the Eq.(<ref>) has weak effect on f_h thus one can approximate f_h = ±√(h)f_n where 2π f_n = √(2π/(εβ _2)). According to this simplified model, a standard GRIN MMF with 50 μm core diameter, 0.01 relative index difference and 36.16 fs^2/mm group velocity dispersion at 800 nm wavelength creates spatial oscillations with ∼ 555 μm period for propagating light beam. For these parameters, theoretical calculations suggest that frequency offset between first GPI sideband pair and the pump pulse is about 89 THz. Regarding the pump wavelength 800 nm, expected first GPI Stokes and anti-Stokes could be located around 1049 nm and 646 nm, respectively.Theoretical calculations are derived independently from the pump pulse duration. To study the detailed evolution of femtosecond pulses in GRIN MMF numerical simulations are needed. Pulse propagation in GRIN MMF can be simulated using the generalize multimode nonlinear Schödinger equation <cit.>. According to this model, the complex electric field can be expanded into a sum for modes p = 0,1,2,…and each mode can be represented with a transverse fiber mode profile. Evolution of temporal envelope of pth mode can be written as:∂ A_p/∂ z = iδβ _0^(p)A_p-δβ _1^(p)∂ A_p/∂ t -iβ_2^(p)/2∂^2 A_p/∂^2 t+iγ/3(1+i/ω_0∂/∂ t) ∑_l,m,nη_plmn[(1-f_R)A_lA_mA_n^*+f_RA_l∫ h_RA_m(z,t-τ)A_n^*(z,t-τ)dτ]where η_plmn is nonlinear coupling coefficient, f_R≈ 0.18 is the fractional contribution of the Raman effect,h_R is the delayed Raman response function and δβ _0^(p) (δβ _1^(p)) is difference between first (second) Taylor expansion coefficient of propagation constant for corresponding and the fundamental mode. Propagation constants for each mode (p) can be written as: β_p(ω)=ω n_0(ω)/c√(1-2c/ω n_0(ω)√(2Δ)/r(2p+1))where n_0 is the refractive index of fiber core, ω is frequency and c is the speed of light in vacuum. Dispersion curve and group velocity dispersion (GVD) for each mode can be derived from Eq.(<ref>). To solve Eq.(<ref>) numerically, we use symmetrized split-step Fourier method <cit.> and include Raman process and shock terms in our simulations. A GRIN MMF with 50 μm core diameter supports approximately 415 modes at 800 nm and simulating all of them will require complex and time-consuming calculations. Thus to achieve manageable computation times, we consider only first six zero-angular-momentum modes in our simulation. We launch pulses with 200 fs pulse duration, 350 nJ pulse energy at 800 nm which has a peak power considerably below the critical power for Kerr-induced self-focusing (∼ 2.44 MW) and this initials pulse energy is distributed among these six modes (50% in p=0, 18% in p=1, 13% in p=2, 10% in p=3, 6% in p=4 and 3% in p=5). Propagation constants, dispersion and nonlinearity parameters are calculated as in <cit.>. We set n_0 as 1.4676, n_2 as 2.7x10^-20 m^2/W, relative index difference as 0.01, integration step as 10 μm, time window width as 15 ps with 2 fs resolution.Spectral and temporal evolution of the pump pulse is first studied numerically for 30 cm GRIN MMF(Fig. <ref>). Femtosecond pulse starts to broaden in frequency domain while propagating in the very first part of the GRIN MMF. The observed relatively large spectral broadening is a unique feature of GRIN MMF and caused by high pump pulse energy <cit.>. Numerical results indicate the generation of GPI sidebands requires approximately 100 oscillations inside the GRIN MMF for our launch conditions and parameters (pulse duration, peak power and fiber core size etc.). Broadened pulse covers the emergence of GPI sidebands and after a certain amount of propagation, discrete peaks become obviously visible. In this aspect, generation of GPI sidebands from ultrashort pump pulses and SPM induced modulation-instability possess similar characteristics in frequency domain <cit.>. Formation of GPI peaks in femtosecond regime is different than the formation in nanosecond regime, in which the GPI sidebands emerge directly from noise level without assisted SPM broadened pump pulses <cit.>. In numerical results, first pairs of GPI sidebands appear at frequencies detuned approximately 90 THz from the launched pump pulse frequency. As shown in Fig.<ref>, first GPI Stokes and anti-Stokes are centered around 1055 nm and 640 nm, respectively. Simulation results indicate that after the formation is established, the intensity of GPI sidebands starts to increase due to constant frequency generation with spatio-temporal propagation. Along the fiber, the positions of GPI sidebands at frequency domains remains stable.Obtaining the spatial evolution of the pulse inside the fiber is important in order to understand spatio-temporal changes. Thus we calculate the spatial evolution numerically at various positions inside the fiber and presented in Fig. <ref> according to the simulation model <cit.>. Our results verify the spatial evolution of the beam experiences periodic refocusing along the GRIN MMF. This periodic behavior preserves the Gaussian-like spatial distribution for all focused points. We show the beam profile for three different focused points (Fig. <ref>(a).a, Fig. <ref>(a).c and Fig. <ref>(a).e). During the GPI sideband generation (∼ 5 cm), we observed non-Gaussian intensity distribution for total field Fig. <ref>(a).b. After the GPI sideband generation occurs, spatial intensity distribution of total field approaches to Gaussian beam shape for spread points Fig. <ref>(a).d and Fig. <ref>(a).f as well. From numerical calculations, we also extract spatial intensity distribution of first GPI Stokes and it indicates that inherited spatial intensity distribution is preserved at focused and spread points Fig. <ref>(a).c-f. This observation is a unique feature of the spatio-temporal evolution of the GPI in GRIN MMF. In addition, we also studied spatial evolution of first GPI anti-Stokes in different positions (focused and spread) along the GRIN MMF. As presented in Fig. <ref>(b), Gaussian-like spatial distributions are preserved for first GPI anti-Stokes as well.We study the effect of different launch conditions on GPI sideband generation in the Fig.<ref>(a). We compare above mentioned result (solid line) with the initial energy distribution between the modes as 30% in p=0, 25% in p=1, 15% in p=2, 5% in p=3, 3% in p=4 and 2% in p=5 (dashed line). Decreasing the energy of fundamental mode (p=0) results in less spectral broadening and slight frequency shift to first anti-Stokes sideband. Intensity difference between first GPI sidebands between the different launch energy distribution is observed as 4 dB. Next, we check the impact of considered number of modes in numerical studies. For longer fiber lengths one may need to decrease number of simulated modes to further reduce calculation times. For this reason, we run simulations with only first three zero-angular-momentum modes with different initial energy distributions. In the literature, it is shown that simulations with three lowest order modes reflect sufficiently similar results with experiments <cit.> thus this approach is an acceptable simplification. Obtained spectra for 30 cm GRIN MMF are presented in the Fig.<ref>(b). First we distribute launch energy as 50% in p=0, 30% in p=1, 20% in p=2 (solid line) then change as 35% in p=0, 35% in p=1, 30% in p=2 (dashed line). Obtained spectra for these energy distributions present nearly identical features.§ EXPERIMENTAL RESULTS AND DISCUSSIONS In the experiments, we use amplified Ti:Sapphire laser (Spitfire by Spectra-Physics) capable to generate linearly polarized, single-mode, 200 femtosecond ultrashort pulses at 800 nm with 1 kHz repetition rate for pump source. The fiber used in the experiment is a commercially available GRIN MMF (Thorlabs-GIF50C) with 50 μm (125 μm) core (clad) diameter and 0.2 numerical aperture. We couple pump pulses into 2.6 m GRIN MMF with plano-convex lens and three-axis translation stage configuration. We test various lenses with different focal lengths and obtain GPI sideband generation in the measured spectrum with beam waists on fiber facet. For small beam waists, observed GPI sidebands are unstable due to an environmental issue such as vibrations and degradation of optical alignment tools. On the other hand, selected 60 mm focal length lens provides ∼ 20 μm waist size, thus excitation of higher-order modes is obtained easily and measured GPI sidebands remain stable for several hours. In general, free space coupling efficiency greater than 80% could be achieved. The generation and formation of experimentally observed GPI sidebands are reported in detail Fig. <ref> for different launched pulse energy conditions. We launch 200 fs pump pulses at 800 nm with ∼ 10 nm FWHM into a 2.6 m GRIN MMF. Pump pulse experiences asymmetric spectral broadening for relatively low pulse energies (for example 270 nJ). The observed asymmetric spectral broadening could be the result of stimulated Raman scattering (SRS). At high launched pulse energy (345 nJ), we observe further spectral broadening on pump region but discrete SRS peaks formation is not detected. With the increasing launched pulse energy for constant fiber length, first GPI Stokes sideband emerges at 295 nJ launched pulse energy. Theory and simulation results indicate that both GPI sidebands should appear at the same time. Thus for launched pulse energy of 295 nJ, first GPI anti-Stokes should lie under the noise level of the optical spectrum analyzer. As launched pulse energy increases (at 320 nJ), amplification and spectral broadening for first GPI Stokes is recorded and in addition, first GPI anti-Stokes also emerges. Similar spectral evolution (amplification and broadening) is also obtained for first anti-Stokes.We successfully generate first GPI peak pair (first Stokes and anti-Stokes) with launched 345 nJ femtosecond pulses into GRIN MMF. First GPI peak pair is observed with ∼ 91 THz separation with respect to pump frequency (f_0) (see Fig. <ref>(a)). First Stokes and anti-Stokes are centered around 1055 nm and 645 nm, respectively. The corresponding optical spectrum bandwidth of first Stokes and anti-Stokes are ∼ 12 nm and ∼ 5 nm, respectively. Even though bandwidths of GPI sidebands seem different in wavelengths, in frequency domain they have approximately close bandwidths of 3.2 THz. We measure the near-field beam profile of the first GPI Stokes sideband for launched pump pulses of 345 nJ pulse energy ( Fig.<ref>-inset). To separate first GPI Stokes from the pump pulse and the GPI anti-Stokes, we use a longpass filter with 1000 nm cutoff wavelength. As expected from a GPI sideband, a clean (speckle free), Gaussian-like near-field beam profile is observed which is similar to the pump beam shape. This feature of GPI sidebands is the signature of GPI in GRIN MMF and significant difference between GPI and intermodal FWM. <cit.>. To compare obtained experimental result with our numerical model, we perform numerical simulations for 2.6 m GRIN MMF (see Fig. <ref>). In order to decrease simulation time to manageable durations we simulate first three zero-angular-momentum modes with included Raman process and shock terms. First, we distribute 345 nJ pulse energy of the launched pulse to modes such as 50% in p=0, 30% in p=1 and 20% in p=2 (solid-line). To check the effect of energy distribution between the modes we run simulations with 35% in p=0, 35% in p=1 and 30% in p=2 distribution as well (dashed-line). For both distributions, positions and bandwidths of GPI sidebands on frequency domain are similar to experimentally obtained results. The low-intensity level of GPI sidebands in simulations may arise from neglecting the contribution of higher-order modes. § CONCLUSIONIn conclusion, we study the spatio-temporal instability of ultrashort pulses in normal dispersion GRIN MMF. Our experimental results present GPI formation with femtosecond pump pulses first time in the literature and the reported GPI sidebands are well-aligned with theoretical predictions and numerical calculations. Detailed numerical studies revealed the generation and propagation behaviors of GPI sidebands inside of GRIN MMF. Observed results provide inside of the GPI formation with ultrashort pulses to complete spatio-temporal pulse evolution studies and indicate that the known attractor effect observed in GRIN MMF for quasi-continuous pump pulses also exists for ultrashort pump pulses <cit.>. This attractor causes first self-beam cleaning then propagating beam experience GPI which manifest itself with sidebands in the frequency domain. For femtosecond pulses, self-beam cleaning is presented by Liu et al. <cit.> and our result complete the information gap for recently emerging research field. We report that, from certain aspects, the generation mechanism of GPI sidebands is analogous to SPM assisted FWM in standard single mode fiber. Experimental observations show that first GPI Stokes features Gaussian-like spatial intensity profile. Numerically calculated spatial intensity profiles verify experimental measurements and provides information on spatial evolution of the beam inside the GRIN MMF. With intrinsic large frequency shift, GPI sidebands can be employed to generate new wavelengths for various application purposes. In future direction, the presented spatio-temporal platform provides potential directions to investigate its complex, nonlinear dynamics with ultrashort pulses such as self-beam cleaning and supercontinuum generation. § ACKNOWLEDGMENTThe authors thank Ç. Şenel for discussions and The Turkish Academy of Sciences — Outstanding Young Scientists Award Program (TUBA-GEBIP); Bilim Akademisi — The Science Academy, Turkey under the BAGEP program; METU Prof. Dr. Mustafa Parlar Foundation; FABED for their supports.IEEEtran[ < g r a p h i c s > ]Uğur Teğinwas born in Bursa, Turkey in 1992. He received the B.S. degree in physics from the Bilkent University, Turkey in 2015. His research interests include mode-locked lasers, fiber amplifiers, nonlinear fiber optics andspatio-temporal nonlinear dynamics. He is currently pursuing the master’s degree in Materials Science and Nanotechnology at Bilkent University, Turkey. He is a member of the Optical Society of America and SPIE. [ < g r a p h i c s > ]Bülend Ortaçreceived the B.S. degree in Physics from the Karadeniz Technical University, Trabzon, Turkey, in 1997, M.S. degree in Teaching and Diffusion of Sciences and Technology from ENS Cachan University, Paris, France, in 2000, and Ph. D. degree in Optoelectronics from Rouen University, Rouen, France, in 2004 respectively. In Mach 2005, he joined the Institute of Applied Physics, Friedrich-Schiller University, Jena, Germany, as a Post-Doctoral Associate. Since November 2009, he has been working as a research assistant professor at Institute of Materials Science and Nanotechnology, Bilkent University. He is the founder and principle investigator of the Laser Research Laboratory. His current research interests include the development of powerful fiber lasers in the continuous-wave regime to pulsed regime (ns, ps and fs) and the demonstration of laser systems for real world applications. He has published more than 100 research articles in major peer-reviewed scientific journals (over 60) and conferences (over 150) in the field of laser physics. | http://arxiv.org/abs/1705.09157v1 | {
"authors": [
"Uğur Teğin",
"Bülend Ortaç"
],
"categories": [
"physics.optics",
"physics.app-ph"
],
"primary_category": "physics.optics",
"published": "20170525131322",
"title": "Observation of Spatio-temporal Instability of Femtosecond Pulses in Normal Dispersion Multimode Graded-Index Fiber"
} |
Yao Qin University of California, San [email protected] Feng Dalian University of [email protected] Lu Dalian University of [email protected] W. Cottrell University of California, San [email protected] * Equal Contribution Hierarchical Cellular Automata for Visual SaliencyYao Qin* Mengyang Feng* Huchuan LuGarrison W. CottrellReceived: date / Accepted: date =================================================================== Saliency detection, finding the most important parts of an image, has become increasingly popular in computer vision. In this paper, we introduce Hierarchical Cellular Automata (HCA) – a temporally evolving model to intelligently detect salient objects. HCA consists of two main components: Single-layer Cellular Automata (SCA) and Cuboid Cellular Automata (CCA). As an unsupervised propagation mechanism, Single-layer Cellular Automata can exploit the intrinsic relevance of similar regions through interactions with neighbors. Low-level image features as well as high-level semantic information extracted from deep neural networks are incorporated into the SCA to measure the correlation between different image patches. With these hierarchical deep features, an impact factor matrix and a coherence matrix are constructed to balance the influences on each cell's next state. The saliency values of all cells are iteratively updated according to a well-defined update rule. Furthermore, we propose CCA to integrate multiple saliency maps generated by SCA at different scales in a Bayesian framework. Therefore, single-layer propagation and multi-layer integration are jointly modeled in our unified HCA. Surprisingly, we find that the SCA can improve all existing methods that we applied it to, resulting in a similar precision level regardless of the original results. The CCA can act as an efficient pixel-wise aggregation algorithm that can integrate state-of-the-art methods, resulting in even better results. Extensive experiments on four challenging datasets demonstrate that the proposed algorithm outperforms state-of-the-art conventional methods and is competitive with deep learning based approaches.§ INTRODUCTION Humans excel in identifying visually significant regions in a scene corresponding to salient objects. Given an image, people can quickly tell what attracts them most. In the field of computer vision, however, performing the same task is very challenging, despite dramatic progress in recent years. To mimic the human attention system, many researchers focus on developing computational models that locate regions of interest in the image. Since accurate saliency maps can assign relative importance to the visual contents in an image, saliency detection can be used as a pre-processing procedure to narrow the scope of visual processing and reduce the cost of computing resources. As a result, saliency detection has raised a great amount of attention <cit.> and has been incorporated into various computer vision tasks, such as visual tracking <cit.>, object retargeting <cit.> and image categorization <cit.>. Results in perceptual research show that contrast is one of the decisive factors in the human visual attention system <cit.>, suggesting that salient objects are most likely in the region of the image that significantly differs from its surroundings. Many conventional saliency detection methods focus on exploiting local and global contrast based on various handcrafted image features, e.g., color features <cit.>, focusness <cit.>, textual distinctiveness <cit.>, and structure descriptors <cit.>. Although these methods perform well on simple benchmarks, they may fail in some complex situations where the handcrafted low-level features do not help salient objects stand out from the background. For example, in Figure. <ref>, the prairie dog is surrounded by low-contrast rocks and bushes. It is challenging to detect the prairie dog as a salient object with only low-level saliency cues. However, humans can easily recognize the prairie dog based on its category as it is semantically salient in high-level cognition and understanding.In addition to the limitation of low-level features, the large variations in object scales also restrict the accuracy of saliency detection. An appropriate scale is of great importance in extracting the salient object from the background. One of the most popular ways to detect salient objects of different sizes is to construct multi-scale saliency maps and then aggregate them with pre-defined functions, such as averaging or a weighted summation. In most existing methods <cit.>, however, these constructed saliency maps are usually integrated in a simple and heuristic way, which may directly limit the precision of saliency aggregation. To address these two obvious problems, we propose a novel method named Hierarchical Cellular Automata (HCA) to extract the salient objects from the background efficiently. A Hierarchical Cellular Automata consists of two main components: Single-layer Cellular Automata (SCA) and Cuboid Cellular Automata (CCA). First, to improve the features, we use fully convolutional networks <cit.> to extract deep features due to their successful application to semantic segmentation. It has been demonstrated that deep features are highly versatile and have stronger representational power than traditional handcrafted features <cit.>. Low-level image features and high-level saliency cues extracted from deep neural networks are used by an SCA to measure the similarity of neighbors. With these hierarchical deep features, the SCA iteratively updates the saliency map through interactions with similar neighbors. Then the salient object will naturally emerge from the background with high consistency among similar image patches. Secondly, to detect multi-scale salient objects, we apply the SCA at different scales and integrate them with the CCA based on Bayesian inference. Through interactions with neighbors in a cuboid zone, the integrated saliency map can highlight the foreground and suppress the background. An overview of our proposed HCA is shown in Figure. <ref>.Furthermore, the Hierarchical Cellular Automata is capable of optimizing other saliency detection methods. If a saliency map generated by one of the existing methods is used as the prior map and fed into HCA, it can be improved to the state-of-the-art precision level. Meanwhile, if multiple saliency maps generated by different existing methods are used as initial inputs, HCA can naturally fuse these saliency maps and achieve a result that outperforms each method.In summary, the main contributions of our work include:(1) We propose a novel Hierarchical Cellular Automata to adaptively detect salient objects of different scales based on hierarchical deep features. The model effectively improves all of the methods we have applied it to to state-of-the-art precision levels and is relatively insensitive to the original maps.(2) Single-layer Cellular Automata serve as a propagation mechanism that exploits the intrinsic relevance of similar regions via interactions with neighbors. (3) Cuboid Cellular Automata can integrate multiple saliency maps into a more favorable result under the Bayesian framework.§ RELATED WORK §.§ Salient Object DetectionMethods of saliency detection can be divided into two categories: top-down (task-driven) methods and bottom-up (data-driven) methods. Approaches like <cit.> are typical top-down visual attention methods that require supervised learning with manually labeled ground truth. To better distinguish salient objects from the background, high-level category-specific information and supervised methods are incorporated to improve the accuracy of saliency maps. In contrast, bottom-up methods usually concentrate on low-level cues such as color, intensity, texture and orientation to construct saliency maps <cit.>. Some global bottom-up approaches tend to build saliency maps by calculating the holistic statistics on uniqueness of each element over the whole image <cit.>.As saliency is defined as a particular part of an image that visually stands out compared to their neighboring regions or the rest of image, one of the most used principles, contrast prior, measures the saliency of a region according to the color contrast or geodesic distance against its surroundings <cit.>. Recently, the boundary prior has been introduced in several methods based on the assumption that regions along the image boundaries are more likely to be the background <cit.>, although this takes advantage of photographer's bias and is less likely to be true for active robots. Considering the connectivity of regions in the background, <cit.> define the saliency value for each region as the shortest-path distance towards the boundary. <cit.> use manifold ranking to infer the saliency score of image regions according to their relevance to boundary superpixels. Furthermore, in <cit.>, the contrast against the image border is used as a new regional feature vector to characterize the background.However, one of the fundamental problems with all these conventional saliency detection methods is that the features used are not representative enough to capture the contrast between foreground and background, and this limits the precision of saliency detection. For one thing, low-level features cannot help salient objects stand out from a low-contrast background with similar visual appearance. Also, the extracted global features are weak in capturing semantic information and have much poorer generalization compared to the deep features used in this paper. §.§ Deep Neural NetworksDeep convolutional neural networks have recently achie- ved a great success in various computer vision tasks, including image classification <cit.>, object detection <cit.> and semantic segmentation <cit.>. With the rapid development of deep neural networks, researchers have begun to construct effective neural networks for saliency detection <cit.>. In <cit.>, Zhao et al. propose a unified multi-context deep neural network taking both global and local context into consideration. Li et al. <cit.> and Zou et al. <cit.> explore high-quality visual features extracted from DNNs to improve the accuracy of saliency detection. DeepSaliency in <cit.> is a multi-task deep neural network using a collaborative feature learning scheme between two correlated tasks, saliency detection and semantic segmentation, to learn better feature representation. One leading factor for thesuccess of deep neural networks is the powerful expressibility and strong capacity of deep architectures that facilitate learning high-level features with semantic information <cit.>.In <cit.>, Donahue et al. point out that features extracted from the activation of a deep convolutional network can be repurposed to other generic tasks. Inspired by this idea, we use the hierarchical deep features extracted from fully convolutional networks <cit.> to represent smaller image regions. The extracted deep features incorporate low-level features as well as high-level semantic information of the image and can be fed into our Hierarchical Cellular Automata to measure the similarity of different image patches.§.§ Cellular AutomataCellular Automata are a model of computation first proposed by <cit.>. They can be described as a temporally evolving system with simple construction but complex self-organizing behavior. A Cellular Automaton consists of a lattice of cells with discrete states, which evolve in discrete time steps according to specific rules. Each cell's next state is determined by its current state as well as its nearest neighbors' states. Cellular Automata have been applied to simulate the evolution of many complicated dynamical systems <cit.>. Considering that salient objects are spatially coherent, we introduce Cellular Automata into this field and propose Single-layer Cellular Automata as an unsupervised propagation mechanism to exploit the intrinsic relationships of neighboring elements of the saliency map and eliminate gaps between similar regions.In addition, we propose a method to combine multiple saliency maps generated by different algorithms, or combine saliency maps at different scales through what we call Cuboid Cellular Automata (CCA). In CCA, states of the automaton are determined by a cuboid neighborhood corresponding to automata at the same location as well as their adjacent neighbors in different saliency maps. An illustration of the idea is in Figure <ref>(b). In this setting, the saliency maps are iteratively updatedthrough interactions among neighbors in the cuboid zone. The state updates are determined through Bayesian evidence combination rules. Variants of this type of approach have been used before <cit.>. <cit.> use the low-level visual cues derived from a convex hull to compute the observation likelihood. <cit.> construct saliency maps through dense and sparse reconstruction and propose a Bayesian algorithm to combine them. Using Bayesian updates to combine saliency maps puts the algorithm for Cuboid Cellular Automata on a firm theoretical foundation.§ PROPOSED ALGORITHMIn this paper, we propose an unsupervised Hierarchical Cellular Automata (HCA) for saliency detection, composed of two sub-units, a Single-layer Cellular Automata (SCA), and a Cuboid Cellular Automata (CCA), as described below. First, we construct prior maps of different scales with superpixels on the image boundary chosen as the background seeds. Then, hierarchical deep features are extracted from fully convolutional networks <cit.> to measure the similarity of different superpixels. Next, we use SCA to iteratively update the prior maps at different scales based on the hierarchical deep features. Finally, a CCA is used to integrate the multi-scale saliency maps using Bayesian evidence combination. Figure. <ref> shows an overview of our proposed method.§.§ Background Priors Recently, there have been various mathematical models proposed to generate a coarse saliency map to help locate potential salient objects in an image <cit.>. Even though prior maps are effective in improving detection precision, they still have several drawbacks. For example, a poor prior map may greatly limit the accuracy of the final saliency map if it incorrectly estimates the location of the objects or classifies the foreground as the background. Also, the computational time to construct a prior map can be excessive. Therefore, in this paper, we build a quite simple and time-efficient prior map that only provides the propagation seeds for HCA, which is quite insensitive to the prior map and is able to refine this coarse prior map into an improved saliency map.First, we use the efficient Simple Linear Iterative Clustering (SLIC) algorithm <cit.> to segment the image into smaller superpixels in order to capture the essential structural information of the image. Let s_i∈ ℝ denote the saliency value of the superpixel i in the image. Based on the assumption that superpixels on the image boundary tend to have a higher probability of being the background, we assign a close-to-zero saliency value to the boundary superpixels. For others, we assign a uniform value as their initial saliency values,s_i = {[ 0.001 i ∈boundary; 0.5 i∉boundary.; ].Considering the great variation in the scales of salient objects, we segment the image into superpixels at M different scales, which are displayed in Figure. <ref> (Prior Maps).§.§ Deep Features from FCN As is well-known, the features in the last layers of CNNs encode semantic abstractions of objects, and are robust to appearance variations, while the early layers contain low-level image features, such as color, edge, and texture. Although high-level features can effectively discriminate the objects from various backgrounds, they cannot precisely capture the fine-grained low-level information due to their low spatial resolution. Therefore, a combination of these deep features is preferred compared to any individual feature map.In this paper, we use the feature maps extracted from the fully-convolutional network (FCN-32s <cit.>) to encode object appearance. The input image to FCN-32s is resized to 500 × 500, and a 100-pixel padding is added to the four boundaries. Due to subsampling and pooling operations in the CNN, the outputs of each convolutional layer in the FCN framework are not at the same resolution. Since we only care about the features corresponding to the original image, we need to 1) crop the feature maps to get rid of the padding; 2) resize each feature map to the input image size via the nearest neighbor interpolation. Then each feature map can be aggregated using a simple linear combination as: g(r_i,r_j)=∑_l=1^L ρ_l·df^l_i-df^l_j_2,where df_i^l denotes the deep features of superpixel i on the l-th layer and ρ_l is a weighting of the importance of the l-th feature map, which we set by cross-validation. The weights are constrained to sum to 1:∑^L_l=1ρ_l = 1. Each superpixel is represented by the mean of the deep features of all contained pixels. The computed g(r_i,r_j) is used to measure the similarity between superpixels.§.§ Hierarchical Cellular AutomataHierarchical Cellular Automata (HCA) is a unified framework composed of single-layer propagation (Single-layer Cellular Automata) and multi-layer aggregation (Cuboid Cellular Automata). It can generate saliency maps at different scales and integrate them to get a fine-grained saliency map. We will discuss SCA and CCA respectively in Sections <ref> and <ref>. §.§.§ Single-layer Cellular AutomataIn Single-layer Cellular Automata (SCA), each cell denotes a superpixel generated by the SLIC algorithm. SLIC takes the number of desired superpixels as a parameter, so by using different numbers of superpixels with SCA, we can obtain maps at different scales. In this section, we assume one scale, denoted m. We denote the number of superpixels in scale m by n_m, but we omit the subscript m in most notation in this section for clarity, e.g., F for F_m, C for C_m and s for s_m.We make three major modifications to the previous cellular automata models <cit.> for saliency detection. First, the states of cells in most existing Cellular Automata models are discrete <cit.>. However, in this paper, we use the saliency value of each superpixel as its state, which is continuous between 0 and 1. Second, we give a broader definition of the neighborhood that is similar to the concept of z-layer neighborhood (here z=2) in graph theory. The z-layer neighborhood of a cell includes adjacentcells as well as those sharing common boundaries with its adjacent cells. Also, we assume that superpixels on the image boundaries are all connected to each other because all of them serve as background seeds. The connections between the neighbors are clearly illustrated in Figure. <ref> (a). Finally, instead of uniform influence of the neighbors , the influence is based on the similarity between the neighbor to the cell in feature space, as explained next. Impact Factor Matrix: Intuitively, neighbors with more similar features have a greater influence on the cell's next state. The similarity of any pair of superpixels is measured by a pre-defined distance in feature space. For the m-th saliency map, which has n_m superpixels in total, we construct an impact factor matrix F∈ℝ^n_m × n_m. Each elementf_ij in F defines the impact factor of superpixel i to j as:f_ij = {[ exp ( - g(r_i,r_j)/σ_f ^2) j ∈NB(i);0j = i or otherwise,;].where g(r_i,r_j) is a function that computes the distance between the superpixel i and j in feature space with r_i as the feature descriptor of superpixel i. In this paper, we use the weighted distance of hierarchical deep features computed by Eqn. (<ref>) to measure the similarity between neighbors. σ_f is a parameter that controls the strength of similarity and NB(i) is the set of the neighbors of the cell i. In order to normalize the impact factors, a degree matrix D=diag{d_1,d_2,⋯,d_n_m} is constructed, where d_i=∑_if_ij. Finally, a row-normalized impact factor matrix can be calculated as F^∗ = D^-1·F.Coherence Matrix:Given that each cell's next state is determined by its current state as well as its neighbors, we need to balance the importance of these two factors. On the one hand, if a superpixel is quite different from all its neighbors in the feature space, its next state will be primarily based on itself. On the other hand, if a cell is similar to its neighbors, it should be assimilated by the local environment. To this end, we build a coherence matrix C = diag{c_1,c_2,⋯,c_n_m} to promote the evolution among all cells. Each cell's coherence towards its current state is initially computed as: c_i = 1/max (f_ij), so it is inversely proportional to its maximum similarity to its neighbors. We normalize this to be in a range c_i∈ [ b , a+b ], where a and b are parameters, via:c_i^ *= a ·c_i - min( c_j)/max( c_j) - min( c_j) + b,where the min and max are computed over j = 1, 2, ..., n_m. Based on preliminary experiments, we set the constants a and b in Eq. (<ref>) to 0.6 and 0.2. If a is fixed to 0.6, our results are insensitive to the value of b in the interval [ 0.1 ,0.3 ]. The final, normalized coherence matrix is then:C^∗ = diag {c_1^∗, c_2^∗, ⋯, c_n_m^∗}.Synchronous Update Rule:In Cellular Automata, all cells will simultaneously update their states according to the update rule, which is a key point in Cellular Automata, as it controls whether the ultimate evolving state is chaotic or stable <cit.>. Here, we define the synchronous update rule based on the impact factor matrix F^ * ∈ℝ^n_m× n_m and coherence matrix C^ * ∈ℝ^n_m× n_m:s^(t + 1) = C^ * s^(t) + ( I - C^ * ) F^ * s^(t),where I is the identity matrix of dimension n_m× n_m and s^(t)∈ℝ^n_m denotes the saliency map at time t. When t = 0, s^(0) is the prior map generated by the method introduced in Section. <ref>. After T_S time steps (a time step is defined as one update of all cells), the saliency map can be represented as s^(T_S). It should be noted that the update rule is invariant over time; only the cells' states s^(t) change over iterations. Our synchronous update rule is based on the generalized intrinsic characteristics of most images.First, superpixels belonging to the foreground usually share similar feature representations. By exploiting the correlation between neighbors, the SCA can enhance saliency consistency among similar regions and develop a steady local environment.Second, it can be observed that there is a high contrast between the object and its surrounding background in feature space. Therefore, a clear boundary will naturally emerge between the object and the background, as the cell's saliency value is greatly influenced by its similar neighbors. With boundary-based prior maps, salient objects can be naturally highlighted after the evolution of the system due to the connectivity and compactness of the object, as exemplified in Figure. <ref>. Specifically, even though part of the salient object is incorrectly selected as the background seed, the SCA can adaptively increase their saliency values under the influence of the local environment. The last three columns in Figure. <ref> show that when the object touches the image boundary, the results achieved by the SCA are still satisfying.§.§.§ Cuboid Cellular Automata To better capture the salient objects of different scales, we propose a novel method named Cuboid Cellular Automata (CCA) to incorporate M different saliency maps generated by SCA under M scales, each of which serves as a layer of the Cuboid Cellular Automata. In CCA, each cell corresponds to a pixel, and the saliency values of all pixels constitute the set of cells' states. The number of all pixels in an image is denoted as H. Unlike the definition of a neighborhood in Section. <ref> and Multi-layer Cellular Automata in <cit.>, here pixels with the same coordinates in different saliency maps as well as their 4-connected pixels are all regarded as neighbors. That is, for any cell in a saliency map, it should have 5M-1 neighbors, constructing a cuboid interaction zone. The hierarchical graph is presented in Figure. <ref> (b) to illustrate the connections between neighbors.The saliency value of pixel i in the m-th saliency map at time t is its probability of being the foreground F, represented as s_m,i^(t) = P(i ∈_m^(t) F), while 1 - s_m,i^(t) is its probability of being the background B, denoted as 1 - s_m,i^(t) = P(i ∈_m^(t) B). We binarize each map with an adaptive threshold usingOtsu's method <cit.>, which is computed from the initial saliency map and does not change over time. The threshold of the m-th saliency map is denoted by γ_m. If pixel i in the m-th binary map is classified as foreground at time t (s_m,i^(t)≥γ_m), then it will be denoted as η_m,i^(t)=+1. Correspondingly, η_m,i^(t)=-1 means that pixel i is binarized as background (s_m,i^(t) < γ_m).If pixel i belongs to the foreground, the probability that one of its neighboring pixels j in the m-th binary map is classified as foreground at time t is denoted as P( η_m,j^(t) = +1|i ∈^(t)_m F ). In the same way, the probability P( η_m,j^(t) = -1|i ∈_m^(t) B ) represents that the pixel j is binarized as B conditioned on that pixel i belongs to the background at time t. We make the assumption that P( η_m,j^(t) = +1|i ∈^(t)_m F ) is the same for all the pixels in any saliency map and it does not change over time. Additionally, it is reasonable to assume that P( η_m,j^(t) = +1|i ∈^(t)_m F ) = P( η_m,j^(t) = -1|i ∈_m^(t) B ). Therefore, we use a constant λ to denote these two probablities:P( η_m,j^(t) = +1|i ∈^(t)_m F ) = P( η_m,j^(t) = -1|i ∈_m^(t) B ) = λ.Then the posterior probability P(i ∈_m^(t) F|η _m,j^(t) = +1) can be calculated as follows: P( i ∈^(t)_m F| η _m,j^(t) =+ 1.) ∝ P( i ∈^(t)_m F)P( η _m,j^(t) =+ 1| i ∈^(t)_m F.)= s_m,i^(t)·λ In order to get rid of the normalizing constant in Eqn. (<ref>), we define the prior ratio Ω(i ∈^(t)_m F) as:Ω( i ∈^(t)_m F) = P( i ∈^(t)_m F)/P( i ∈^(t)_m B) = s_m,i^(t)/1 - s_m,i^(t).Combining Eqn. (<ref>) and Eqn. (<ref>), the posterior ratio Ω(i ∈^(t)_m F | η_m,j^(t)=+1) turns into:Ω( i ∈^(t)_m F| η _m,j^(t) =+ 1.)= P( i ∈^(t)_m F| η _m,j^(t) =+ 1.)/P( i ∈^(t)_m B| η _m,j^(t) =+ 1.)= s_m,i^(t)/1 - s_m,i^(t)·λ/1 - λ. As the posterior probability P(i ∈^(t)_m F|η _m,j^(t) = +1)represents the probability of pixel i of being the foreground F conditioned on that its neighboring pixel j in the m-th saliency map is binarized as foreground at time t, P(i ∈^(t)_m F|η _m,j^(t) = +1) can also be used to represent the probability of pixel i of being the foreground F at time t+1. Then,s_m,i^(t+1) = P(i ∈^(t)_m F|η _m,j^(t) = +1).According to Eqn. (<ref>) and Eqn. (<ref>), we can get:s_m,i^(t + 1)/1 - s_m,i^(t + 1) =P(i ∈^(t)_m F|η _m,j^(t) = +1)/1-P(i ∈^(t)_m F|η _m,j^(t) = +1)= P(i ∈^(t)_m F|η _m,j^(t) = +1)/P(i ∈^(t)_m B|η _m,j^(t) = +1) = s_m,i^(t)/1 - s_m,i^(t)·λ/1 - λ.It is much easier to deal with the logarithm of this quantitybecause the changes in logodds will be additive. So Eqn. (<ref>) turns into:l( s_m,i^(t + 1)) = l( s_m,i^(t )) +Λ,where l( s_m,i^(t + 1))= ln(s_m,i^(t + 1)/1 - s_m,i^(t + 1))and Λ=ln(λ/1-λ) is a constant. The intuitive explanation for Eqn. (<ref>) is that: if a pixel observes that one of its neighbors is binarized as foreground, it ought to increase its saliency value; otherwise, it should decrease its saliency value. Therefore, Eqn. (<ref>) requires Λ > 0. In this paper, we empirically set Λ = 0.05.As each pixel has 5M-1 neighbors in total, the pixel will decide its action (increase or decrease it saliency value) based on all its neighbors' current states. Assuming the contribution of each neighbor is conditionally independent, we derive the synchronous update rule from Eqn. (<ref>) as:l( s_m^(t + 1)) = l( s_m^(t)) +Σ_m^(t)·Λ,where s^(t)_m ∈ℝ^H is the m-th saliency map at time t and H is the number of pixels in the image. Σ_m^(t)∈ℝ^H can be computed by:Σ_m^(t)=∑_j=1^5∑_k=1^Mδ(k=m,j>1) ·sign( s_j,k^(t) -γ _k ·1),where M is the number of different saliency maps, s_j,k^(t)∈ℝ^H is a vector containing the saliency values of the j-th neighbor for all the pixels in the m-th saliency map at time t and 1 = [1, 1, ⋯, 1]^⊤∈ℝ^H. We use δ(k=m,j>1) to represent the occasion that the cell only has 4 neighbors instead of 5 in the m-th saliency map when it is in the m-th saliency map. After T_C iterations, the final integrated saliency map s^(T_C) is calculated bys^(T_C) = 1/M∑_m = 1^M s_m^(T_C).In this paper, we use CCA to integrate saliency maps generated by SCA at M=3 scales. This combination is denoted as HCA, and the visual saliency maps generated by HCA can be seen in Figure. <ref>. Here we use the notation SCAn to denote SCA applied with n superpixels. We can see that the detected objects in the integrated saliency maps are uniformly highlighted and much closer to the ground truth. §.§ Consistent Optimization§.§.§ Single-layer PropagationDue to the connectivity and compactness of the object, the salient part of an image will naturally emerge with the Single-layer Cellular Automaton, which serves as a propagation mechanism. Therefore, we use the saliency maps generated by several well-known methods as the prior maps and refresh them according to the synchronous update rule. The saliency maps achieved by CAS <cit.>, LR <cit.> and RC <cit.> are taken as s^(0) in Eqn. (<ref>). The optimized results via SCA are shown in Figure. <ref>. We can see that the foreground is uniformly highlighted and a clear object contour naturally emerges with the automatic single-layer propagation mechanism. Even though the original saliency maps are not particularly good, all of them are significantly improved to a similar accuracy level after evolution. That means our method is independent of prior maps and can make a consistent and efficient optimization towards state-of-the-art methods. §.§.§ Pixel-wise Integration A variety of methods have been developed for visual saliency detection, and each of them has its advantages and limitations. As shown in Figure. <ref>, the performance of a saliency detection method varies with individual images. Each method can work well for some images or some parts of the images but none of them can perfectly handle all the images. Furthermore, different methods may complement each other. To take advantage of the superiority of each saliency map, we use Cuboid Cellular Automata to aggregate two groups of saliency maps, which are generated by three conventional algorithms: BL <cit.>, HS <cit.> and MR <cit.> and three deep learning methods: MDF <cit.> and DS <cit.>and MCDL <cit.>. Each of them serves as a layer of Cellular Automata s_m^(0) in Eqn. (<ref>). Figure. <ref> shows that our proposed pixel-wise aggregation method, Cuboid Cellular Automata, can appropriately integrate multiple saliency maps and outperforms each one. The saliency objects on the aggregated saliency map are consistently highlighted and much closer to the ground truth.§.§.§ SCA + CCA = HCAHere we show that when CCA is applied to some (poor) prior maps, it does not perform as well as when the prior map is post-processed by SCA. This motivates their combination into HCA. As is shown in Figure. <ref>, when the candidate saliency maps are not well constructed, both CCA and MCA <cit.> fail to detect the salient object. Unlike CCA and MCA, HCA overcomes this limitation through incorporating single-layer propagation (SCA) together with pixel-wise integration (CCA) into a unified framework. The salient objects can be intelligently detected by HCA regardless of the original performance of the candidate methods. When we use HCA to integrate existing methods, the optimized results will be denoted as HCA*. § EXPERIMENTSIn order to demonstrate the effectiveness of our proposed algorithms, we compare the results on four challenging datasets: ECSSD <cit.>, MSRA5000 <cit.>, PASCAL-S <cit.> and HKU-IS <cit.>. The Extended Complex Scene Saliency Dataset (ECSSD) contains 1000 images with multiple objects of different sizes. Some of the images come from the challenging Berkeley-300 dataset. MSRA- 5000 contains more comprehensive images with complex background. The PASCAL-S dataset derives from the validation set of PASCAL VOC2010 <cit.> segmentation challenge and contains 850 natural images. The last dataset, HKU-IS, contains 4447 challenging images and their pixel-wise saliency annotation. In this paper, we use ECSSD as the validation dataset to help choose the feature maps in FCN <cit.>.We compare our algorithm with 20 classic or state-of-the-art methods including ITTI <cit.>, FT <cit.>, CAS <cit.>, LR <cit.>,XL13 <cit.>, DSR <cit.>, HS <cit.>, UFO <cit.>, MR <cit.>, DRFI <cit.>, wCO <cit.>, RC <cit.>, HDCT <cit.>, BL <cit.>, BSCA <cit.>, LEGS <cit.>, MCDL <cit.>, MDF <cit.>, DS <cit.>, and SSD-HS <cit.>, where the last 5 methods are deep learning-based methods. The results of different methods are either provided by authors or achieved by running available code or binaries. The code and results of HCA will be publicly available at our project site [ <https://github.com/ArcherFMY/HCA_saliency_codes>].§.§ Parameter SetupFor the Single-layer Cellular Automaton, we set the number of iterations T_S = 20. σ_f^2 in Eq. (<ref>) is set to 0.1 as in <cit.>. For the Cuboid Cellular Automata, we set the number of iterations T_C = 3. We determined empirically that SCA and CCA converge by 20 and 3 iterations, respectively. We choose M=3 and run SCA with n_1 = 120, n_2 = 160, n_3 = 200 superpixels to generate multi-scale saliency maps for CCA.§.§ Evaluation MetricsWe evaluate all methods by standard Precision-Recall (PR) curves via binarizing the saliency map with a threshold sliding from 0 to 255 and then comparing the binary maps with the ground truth. Specifically,precision = |SF ∩ GF|/|SF|, recall = |SF ∩ GF|/|GF|,where SF is the set of the pixels segmented as the foreground, GF denotes the set of the pixels belonging to the foreground in the ground truth, and |·| refers to the number of elements in a set. In many cases, high precision and recall are both required. These are combined in the F-measure to obtain a single figure of merit, parameterized by β:F_β = ( 1 + β ^2) ·precision·recall/β ^2·precision + recallwhere β^2 is set to 0.3 as suggested in <cit.> to emphasize the precision. To complement these two measures, we also use mean absolute error (MAE) to quantitatively measure the average difference between the saliency map 𝐬∈ℝ^H and the ground truth g∈ℝ^H in pixel level:MAE = 1/H∑_i = 1^H | s_i - g_i|.MAE indicates how similar a saliency map is compared to the ground truth, and is of great importance for different applications, such as image segmentation and cropping <cit.>.§.§ Validation of the Proposed Algorithm§.§.§ Feature AnalysisIn order to construct the Impact Factor matrix, we need to choose the features that will enter into Eqn.( <ref>). Here we analyze the efficacy of the features in different layers of a deep network in order to choose these feature layers. In deep neural networks, earlier convolutional layers capture fine-grained low-level information, e.g., colors, edges and texture, while later layers capture high-level semantic features. In order to select the best feature layers in the FCN <cit.>, we use ECSSD as a validation dataset to measure the performance of deep features extracted from different layers. The function g(r_i, r_j) in Eqn. (<ref>) can be computed asg(r_i, r_j) = df_i^l - df_j^l_2,where df_i^l denotes the deep features of superpixel i on the l-th layer. The outputs of convolutional layers, relu layers and pooling layers are all regarded as a feature map. Therefore, we consider in total 31 layers of fully convolutional networks. We do not take the last two convolutional layers into consideration as their spatial resolutions are too low.We use the F-measure (the higher, the better) and mean absolute error (MAE) (the lower, the better) to evaluate the performance of different layers on the ECSSD dataset.The results are shown in Figure. <ref> (a) and (b). The F-measure score is obtained by thresholding the saliency maps at twice the mean saliency value.We use this convention for all of the subsequent F-measure results. The x-index in Figure. <ref> (a) and (b) refers to convolutional, ReLu, and pooling layers as implemented in the FCN. We can see that deep features extracted from the pooling layer inandcan achieve the best two F-measure scores, and also perform well on MAE. The saliency maps in Figure. <ref> correspond to the bars in Figure <ref>. Here it is visually apparent that salient objects are better detected with the final pooling layers ofand. Therefore, in this paper, we combine the feature maps fromandwith a simple linear combination. Eqn. (<ref>) then turns into: g(r_i,r_j)=ρ_1·df^5_i-df^5_j_2 + ρ _2·df^31_i-df^31_j_2,where ρ_1 and ρ_2 balance the weight ofand . In this paper, we empirically set ρ_1=0.375 and ρ_2=0.625 and apply them to all other datasets.To test the effectiveness of the integrated deep features, we show the Precision-Recall curves of Single-layer Cellular Automata with each layer of deep features as well as the integrated deep features on two datasets. The Precision-Recall curves in Figure. <ref> (c) demonstrate that hierarchical deep features outperform single-layer features, as they contain both category-level semantics and fine-grained details. §.§.§ Component EffectivenessTo demonstrate the effectiveness of our proposed algorithm, we test the results on the standard ECSSD and PASCAL-S datasets. We generate saliency maps at three scales: n_1 = 120, n_2 = 160, n_3 = 200 and use CCA to integrate them. FT curves in Figure. <ref> indicate that the results of the Single-layer Automata are already quite satisfying. In addition, CCA can improve the overall performance of SCA with a wider range of high F-measure scores than SCA alone.Similar results are also achieved on other datasets but are not presented here to be succinct. §.§.§ Performance ComparisonAs is shown in Figure <ref>, our proposed Hierarchical Cellular Automata performs favorably against state-of-the-art conventional algorithms with higher precision and recall values on four challenging datasets. HCA is competitive with deep learning based approaches and has higher precision at low levels of recall. Furthermore, the fairly low MAE scores, displayed in Figure. <ref>, indicate that our saliency maps are very close to the ground truth. As MCDL <cit.> trained the network on the MSRA dataset, we do not report its result on this dataset in Figure. <ref>. In addition, LEGS <cit.> used part of the images in the MSRA and PASCAL-S datasets as the training set. As a result, we only test LEGS with the test images on these two datasets. Saliency maps are shown in Figure. <ref> for visual comparison of our method with other models. §.§ Optimization of state-of-the-art methodsIn the previous sections, we showed qualitatively that our model creates better saliency maps by improving initial saliency maps with SCA, or by combining the results of multiple algorithms with CCA, or by applying SCA and CCA.Here we compare our methods to other methods quantitatively. When the initial maps are imperfect, we apply SCA to improve them and then apply CCA. When the initial maps are already very good, we show that we can combine state-of-the-art methods to perform even better by simply using CCA. §.§.§ Consistent ImprovementIn Section <ref>, we concluded that results generated by different methods can be effectively optimized via Single-layer Cellular Automata. Figure <ref> shows the precision-recall curves and mean absolute error bars of various saliency methods and their optimized results on two datasets. These results demonstrate that SCA can greatly improve existing results to a similar precision level. Even though the original saliency maps are not well constructed, the optimized results are comparable to the state-of-the-art methods. It should be noted that SCA can even optimize deep learning-based methods to a better precision level, e.g., MCDL <cit.>, MDF <cit.>, LEGS <cit.>, and SSD-HS <cit.>. In addition, for one existing method, we can use SCA to optimize it at different scales and then use CCA to integrate the multi-scale saliency maps. The ultimate optimized result is denoted as HCA*. The lowest MAEs of saliency maps optimized by HCA in Figure <ref> (c) show that HCA's use of CCA improves performance over SCA alone.§.§.§ Effective IntegrationIn Section. <ref>, we used Cuboid Cellular Automata as a pixel-wise aggregation method to integrate two groups of state-of-the-art methods. One group includes three of the latest conventional methods while the other contains three deep learning-based methods. We test the various methods on the ECSSD dataset, and the integrated result is denoted as CCA. PR curves in Figure. <ref> demonstrate the effectiveness of CCA over all the individual methods. FT curves of CCA in Figure. <ref> are fixed at high values that are insensitive to the thresholds. In addition, we binarize the saliency map with two times mean saliency value. From Figure. <ref> we can see that the integrated result has higher precision, recall and F-measure scores compared to each method that is integrated. Also, the mean absolute errors of CCA are always the lowest. The fairly low mean absolute errors indicate that the integrated results are quite similar to the ground truth. Although Cuboid Cellular Automata have exhibited great strength in integrating multiple saliency maps, they have a major drawback in that the integrated result highly relies on the precision of the saliency detection methods used as input. If saliency maps fed into Cuboid Cellular Automata are not well constructed, it cannot naturally detect the salient objects via interactions between these candidate saliency maps. HCA, however, can easily address this problem by incorporating single-layer propagation and multi-layer integration into a unified framework. Unlike MCA <cit.> and CCA, HCA can achieve better integrated saliency map regardless of their original detection performance through the application of SCA to clean up the initial maps. PR curves, FT curves and MAE scores in Figure. <ref> show that 1) CCA has a better performance than MCA, as it considers the influence of adjacent cells on different layers. 2) HCA can greatly improve the aggregation results compared to MCA and CCA because it is independent of the initial saliency maps.§.§ Run TimeThe run time to process one image in the MSRA5000 dataset via Matlab R2014b-64bit with a PC equipped with an i7-4790k 3.60 GHz CPU and 32GB RAM is shown in Table <ref>. The Table displays the average run time of each component in our algorithm, not including the time for extracting deep features. We can see that the Single-layer Cellular Automata and Cuboid Cellular Automata are very fast at processing one image, on average 0.06s. Their combination HCA takes only 0.2421s to process one image without superpixel segmentation and 1.0240s with SLIC.We also compare the run time of our method with other state-of-the-art methods in Table <ref>. Here we compute the run time including superpixel segmentation and feature extraction for all models. Our algorithm has the least run time compared to other deep learning based methods and is the fifth fastest overall.§ CONCLUSIONIn this paper, we propose an unsupervised Hierarchical Cellular Automata, a temporally evolving system for saliency detection. It incorporates two components,Single-layer Cellular Automata (SCA), which can clean up noisy saliency maps, and Cuboid Cellular Automata (CCA), that can integrate multiple saliency maps. SCA is designed to exploit the intrinsic connectivity of saliency objects through interactions with neighbors. Low-level image features and high-level semantic information are both extracted from deep neural networks and incorporated into SCA to measure the similarity between neighbors. With superpixels on the image boundary chosen as the background seeds, SCA iteratively updates the saliency maps according to well-defined update rules, and salient objects naturally emerge under the influence oftheir neighbors. This context-based propagation mechanism can improve thesaliency maps generated by existing methods to a high performance level. We used this in two ways: First, given a single saliency map, SCA can be applied to superpixels generated from the saliency map at multiple scales, and CCA can then integrate these into an improved saliency map. Second, we can take saliency maps generated by multiple methods, apply SCA (if necessary) to improve them, and then apply CCA to integrate them into better saliency maps. 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In: Proceedings of the IEEE International Conference on Computer Vision, pp 406–414 | http://arxiv.org/abs/1705.09425v1 | {
"authors": [
"Yao Qin",
"Mengyang Feng",
"Huchuan Lu",
"Garrison W. Cottrell"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170526034316",
"title": "Hierarchical Cellular Automata for Visual Saliency"
} |
Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA Department of Quantum Physics and Astrophysics and Institute of Cosmos Science , Universitat de Barcelona, Martí Franquès 1, E08028-Spain Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA Department of Physics, The City College of New York, New York, NY 10031, USAGraduate School and University Center, The City University of New York, New York, NY 10016, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USANPLQCD CollaborationINT-PUB-17-016 NT@UW-17-10 MIT-CTP-4909 In this comment, we address a number of erroneous discussions and conclusions presented in a recent preprint by the HALQCD collaboration, arXiv:1703.07210 <cit.>.In particular, we show that lattice QCD determinations of bound states atquark masses corresponding to a pion mass of m_π=806 MeV are robust, and that the extracted phases shifts for these systems pass all of the “sanity checks” introduced in arXiv:1703.07210 <cit.>. Comment on “Are two nucleons bound in lattice QCD for heavy quark masses? - Sanity check with Lüscher's finite volume formula -” Frank Winter December 30, 2023 ================================================================================================================================ In the last decade, significant progress has been made inthe study of multi-hadron systems using lattice QCD, with the first calculations of multi-baryon bound states and their electroweak properties and decays having been performed <cit.>. It is imperative that the methods used in these calculations be robust;investigations such as those of the HALQCD collaboration in Ref. <cit.> are vital provided they are carried out correctly. However, as we show in detail, many of the conclusions reached in Ref. <cit.> (henceforth referred to as ), that cast doubt on the validity of multi-baryon calculations, are incorrect. Since we have recently refined one of the analyses that is criticized in , we focus our attention on the conclusions drawn regarding this case in particular, see Ref. <cit.>. The central point addressed byis whether there exist bound states in theandtwo-nucleon channels at heavy quark masses. Three independent groups have analysed lattice QCD calculations at quark masses corresponding to a heavy pion mass of ∼800 MeV (one set of calculations used quenched QCD) and found that there are bound states in these channels. Each of these groups has concluded this by extracting energies from two-point correlation functions (with the quantum numbers of interest) at two or more lattice volumes and demonstrating, through extrapolations based on the finite-volume formalism of Lüscher <cit.>, that these energies correspond to an infinite-volume state that is below the two-particle threshold and is hence a bound state. Each group has used different technical approaches, and all are in reasonable agreement given the uncertainties that are reported. The HALQCD collaboration has also investigated these two-particle channels using a method (also based on the work of Lüscher <cit.>) that involves constructing Bethe-Salpeter wavefunctions, but do not find evidence for bound states in these channels <cit.>.[In the ΛΛ channel, the HALQCD approach does indicate a bound state, but the binding energy is found to be significantly different from that determined by extrapolating finite-volume energy levels <cit.>.] We note, however, that the HALQCD method introduces unquantified systematic effects as discussed in, e.g., Refs. <cit.> and the nuclear physics overview talks in recent proceedings of the International Symposium on Lattice Field Theory <cit.>). Here, we focus our criticisms ofon several specific points. §.§.§ Misinterpretation of energies and source independence Figure 2 ofcontains a compilation of results for the ground states of theandtwo-nucleon channels. Unfortunately the figure includes a second statefrom Ref. <cit.> that the authors of Ref. <cit.> explicitly indicate is not the ground state, and reporting it as such is a significant error on which many of the invalid arguments of HAL are based.[Whether the quoted value for the second energy in Ref. <cit.> is a true estimate of an excited-state energy is a question for future discussion. However for the ground states, all results unambiguously agree.] There is a small scatter in the remaining results that is due to statistical fluctuations, discretisation artifacts and exponentially-small residual finite-volume effects, but, taken as a whole, there is no inconsistency in these results. In addition, a further recent study of axial-current matrix elements using a different set of interpolators <cit.> (denoted in Fig. <ref> by NPLQCD17) also finds a consistent negatively-shifted energy on the 32^3×48 ensemble used in this comparison.Figure 2 inalso fails to include the energies extracted in Ref. <cit.> on the largest volume, which dominate the extraction of the binding energy. Without the results from this large volume, the confidence in the binding energy in Ref. <cit.> would be significantly diminished.It is therefore vital that this information be included in any discussion of these results. Figure <ref> below shows a (corrected) summary of the energy levels extracted for the ground states of theandtwo-nucleon systems in different volumes that are published in the literature at this particular quark mass.No significant interpolator dependence is observed, as is indicated by simple fits to the reported results for each volume,with all these fits having acceptable values of χ^2 per degree offreedom.Figure 13 ofis also erroneously described as indicating that scattering state results are not source independent. The results show three energy levels where different interpolating operators are consistent within one standard deviation, and one energy level that differs at two standard deviations. This indicates broad agreement within the reported uncertainties and, contrary to statements in HAL, does not provide a sound statistical basis for a claim of inconsistency. In summary, comparison of results from the different interpolators in Refs. <cit.> shows that both bound and scattering-state energy levels are source-independent within reported uncertainties. This is contrary to the claims in .§.§.§ Volume scaling of energiesThe authors of HAL claim that the single-exponential behaviour found in our work, Refs. <cit.>, and in that of Ref. <cit.>, is a “mirage” arising from the cancellation of two or more scattering eigenstates[The scattering states are loosely used here to denote states in a finite volume that correspond to the continuum states of infinite volume.] contributing to the correlation functions with opposite signs (see Ref. <cit.> for elaborations on possible “mirage” plateaus). This interpretation of the negatively-shifted states in these works is exceedingly unlikely, however, as such cancellation would need to occur in an almost identical way for multiple different volumes. For each of the different analyses of the 806 MeV ensembles in Fig. <ref> (NPLQCD2013 d=0, NPLQCD2013 d=2 and Berkowitz2016 d=0), identical sources and sinks were used in each of three volumes (two volumes in the case of Berkowitz2016). Scattering-state eigenenergiesnecessarily change significantly with volume, having power-law dependence as dictated by the Lüscher quantisation condition. While it is possible that, in a given volume, a correlator for a particular source-sink interpolator combination could exhibit a cancellation between contributions of two scattering states that produces an energy level below threshold, it is very unlikely that the cancellation would persist in different volumes as the scattering-state eigenenergies change significantly with volume. As shown in Fig. <ref>, for example, the volume-independent interpolators used in Ref. <cit.> produce energy levels in the three different volumes that are statistically indistinguishable, and even the approach to single-exponential behaviour does not depend on volume. The figure shows the effective masses of the smeared-point correlation functions, but the same features are seen in all other source-sink interpolator combinations that are studied. This rules out the possibility that the negatively-shifted signals are caused by cancellations between scattering states. The largest volume used in our works <cit.> makes this an extremely robust statement as the spatial volumes from which we draw these conclusions vary by a factor of eight. §.§.§ Consistency of Effective Range Expansion (“Sanity Check (i)”) If the effective range expansion (ERE) is a valid parametrization of the scattering amplitude at low energies, the analyticity of the amplitude as a function of the centre-of-mass energy implies that theERE obtained from states with positively-shifted energies (k^*^2>0, where k^* is the centre-of-mass interaction momentum) must be consistent with that obtained from states with negatively-shifted energies (k^*^2<0). Althoughfinds that the NPLQCD results pass this test, wedemonstrate how robust the results in Refs. <cit.> are in this regard through the plots presented in Fig. <ref>. This figure shows fits to the ERE using both ground states (n=1) and first excited states (n=2) (color-shaded bands). These are overlaid on ERE fits using only the ground states (hashed bands). The two sets of bands are fully consistent with each other, proving that this check is unambiguously passed. The same feature is seen for three-parameter ERE fits, with significantly larger uncertainty bands (see also Ref. <cit.>).[Our analysis of two-nucleon correlation functions generated from these ensembles of gauge-field configurations has been recently refined in a comprehensive re-analysis <cit.>,including results at additional kinematic points. This new analysis has been used in obtaining the results shown in Figs. <ref> and <ref>.All of the energies extracted from the three lattice volumes, and the binding energies and ERE parameters subsequently obtained, are in agreement with our previous results; i.e., the differences in the mean values of the results from the previous and the new analyses are within one standard deviation as defined by the (statistical and systematic) uncertainties of the results combined in quadrature <cit.>.] The difference in the size of uncertainties in the phase shift between the fits with and without the n=2 data shows that conclusions about the behaviour and/or validity of the ERE for datasets only near the bound-state pole are likely subject to significant uncertainties. We note that scattering parameters extracted in the region near k^*^2=0 from a linear ERE will in general differ from those determined in the vicinity of a bound-state pole due to higher order terms in the ERE. Indeed, it is known that in nature, the ERE of thephase shift around k^*^2=0 and around the deuteron pole are different (albeit slightly) <cit.>.§.§.§ Residue of the S-matrix at the bound-state pole (“ Sanity check (iii)”) The sign of the residue of the S-matrix at the bound-state pole is fixed. This requirement leads to the following condition onk^*δ : . d/dk^*^2(k^*δ+√(-k^*^2))|_k^*^2=-κ^(∞)^2 < 0, where κ^(∞) is the infinite-volume binding momentum.As is seen from Fig. <ref>, which displays the results of the 2017 refined analysis <cit.> of the correlation functions analyzed in Refs. <cit.>,the slope of the two-parameter ERE fit to the k^*δ function (colored regions) is less than the slope of -√(-k^*^2) (grey regions)at the corresponding bound-state pole in both channels. In thechannel the difference is at the 1σ level, while the difference is more than 3σ in the coupled - channels. The uncertainty in the tangent line to the -√(-k^*^2) function at k^*^2=-κ^(∞)^2 arises from the uncertainty inthe values of κ^(∞) (see also Ref. <cit.>). A similar conclusion can be drawn from three-parameter ERE fits.For the sake of clarity, the two-parameter ERE fits to the results of only the 2013 analysis of the same correlation functionsare shown in Fig. <ref>, and are seen to beconsistent with the criterion in Eq. (<ref>) as well. For comparison,in Fig. <ref> we show the results of two-parameter ERE fits obtained in the 2013 analysis <cit.> and in the2017 refined analysis of the same correlation functions <cit.>. Both analyses of these channels yield results that are consistent with each other and with the criterion in Eq. (<ref>) within the uncertainties of the calculations, thus passing check (iii).§.§.§ DiscussionGiven the discussion above,the NPLQCD results presented in the “NPL2013” row of Table IV of the published version of <cit.>, reproduced below,where we have taken the liberty of changing the notation (in their published version)used to indicate passing a “sanity check” infrom a “ * ” entry to “Passed”.We are currently revisiting the other NPLQCD analyses discussed in . Ref. <cit.> refutes thecriticisms ofsource-dependence leveled at the works of the PACS-CS collaboration <cit.>. Ref. <cit.> provides a summary of the evidence for the validity of ground-state identifications in two-nucleon systems. With the robust conclusion of the existence of bound states reached by independent groups, and argued in this Comment, the systematic uncertainties of the potential method used by the HALQCD collaboration requires further investigation to better understand the origin of its failure to identify two-nucleon bound states. SRB was partially supported by NSF continuing grant number PHY1206498 and by the U.S. Department of Energy through grant number DE-SC001347. EC was supported in part by the USQCD SciDAC project, the U.S. Department of Energy throughgrant number DE-SC00-10337,and by U.S. Department of Energy grant number DE-FG02-00ER41132. ZD, WD and PES were partly supported byU.S. Department of Energy Early Career Research Award DE-SC0010495 and grant number DE-SC0011090. KO was partially supported by the U.S. Department of Energy through grant number DE- FG02-04ER41302 and through contract number DE-AC05-06OR23177 under which JSA operates the Thomas Jefferson National Accelerator Facility. A.P. is partially supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia ’María de Maeztu’), and, with additional European FEDER funds, under the contract FIS2014-54762-P, by the Generalitat de Catalunya contract 2014SGR-401, and by the Spanish Excellence Networkon Hadronic Physics FIS2014-57026-REDT.MJS was supportedby DOE grant number DE-FG02-00ER41132, andin part by the USQCD SciDAC project,the U.S. Department of Energy through grant number DE-SC00-10337. BCT was supported in part by the U.S. National Science Foundation, under grant number PHY15-15738.MLW was supportedin part by DOE grant number DE-FG02-00ER41132. FW was partially supported through the USQCD Scientific Discovery through Advanced Computing (SciDAC) projectfunded by U.S. Department of Energy, Office of Science, Offices of Advanced Scientific Computing Research,Nuclear Physics and High Energy Physics and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177. | http://arxiv.org/abs/1705.09239v3 | {
"authors": [
"Silas R. Beane",
"Emmanuel Chang",
"Zohreh Davoudi",
"William Detmold",
"Kostas Orginos",
"Assumpta Parreño",
"Martin J. Savage",
"Brian C. Tiburzi",
"Phiala E. Shanahan",
"Michael L. Wagman",
"Frank Winter"
],
"categories": [
"hep-lat",
"nucl-th"
],
"primary_category": "hep-lat",
"published": "20170525155521",
"title": "Comment on \"Are two nucleons bound in lattice QCD for heavy quark masses? - Sanity check with Lüscher's finite volume formula -\""
} |
*mps* | http://arxiv.org/abs/1705.09670v2 | {
"authors": [
"Patrick Tunney",
"Jose Miguel No",
"Malcolm Fairbairn"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526180116",
"title": "Probing the Pseudoscalar portal to Dark Matter via $\\bar{b} b Z( \\rightarrow \\ell\\ell) + E_{T} \\hspace{-8mm}/\\hspace{4mm}$: From the LHC to the Galactic Centre Excess"
} |
=1 π̃b →s μ^+ μ^-K^*B_s →μ^+ μ^-B_s →μ^+ μ^- γB^+ → K^+ μ^+ μ^-B →μ^+ μ^-B → X_s μ^+ μ^-[Δ C_9^e][Δ C_9'^e][Δ C_10^e][Δ C_10'^e][Δ C_S^e][Δ C_S'^e][Δ C_P^e][Δ C_P'^e][Δ C_T^e][Δ C_T5^e] [Δ C_9^μ][Δ C_9'^μ][Δ C_10^μ][Δ C_10'^μ][Δ C_S^μ][Δ C_S'^μ][Δ C_P^μ][Δ C_P'^μ][Δ C_T^μ][Δ C_T5^μ][Δ C_9^ℓ][Δ C_9'^ℓ][Δ C_10^ℓ][Δ C_10'^ℓ][Δ C_S^ℓ][Δ C_S'^ℓ][Δ C_P^ℓ][Δ C_P'^ℓ][Δ C_T^ℓ][Δ C_T5^ℓ] [email protected] of Theoretical Physics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, [email protected] of Theoretical Physics, Indian Association for the Cultivation of Science, 2A & 2B, Raja S.C. Mullick Road,Jadavpur, Kolkata 700 032, India [email protected] of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel The recent LHCb measurement of R_K^* in two q^2 bins, when combined with the earlier measurement of R_K, strongly suggestslepton flavour non-universal new physics in semi-leptonic B meson decays. Motivated by these intriguing hints of new physics, severalauthors have considered vector, axial vector, scalar and pseudo scalar operators as possible explanations of these measurements. However,tensor operators have widely been neglected in this context. In this paper, we consider the effect of tensor operators in R_K and R_K^*.We find that, unlike other local operators, tensor operators can comfortably produce both of R_K^* ^ low and R_K^* ^ centralclose to their experimental central values. However, a simultaneous explanation of R_K is not possible with only Tensor operators, and othervector or axial vector operators are needed. In fact, we find that combination of vector and tensor operators can provide simultaneous explanationsof all the anomalies comfortably at the 1 σ level, a scenario which is hard to achieve with only vector or axial vector operators.We also comment on the compatibility of the various new physics solutions with the measurements of the inclusive decay B_d → X_s ℓ^+ ℓ^-.Role of Tensor operators in R_K and R_K^* Diptimoy Ghosh========================================= §.§Introduction The LHCb collaboration has recently announced measurements ofR_K^*≡ B(B̅_d →K̅^* μ^+ μ^-)/ B(B̅_d →K̅^* e^+ e^-) in two q^2 (≡ (p_ℓ^+ + p_ℓ^-)^2) bins,[0.045,1.1] and [1.1,6] GeV^2 (referred to as low and central bins respectively) <cit.>. In both the bins, they observe deviation from the Standard Model (SM), atthe 2.1-2.3 σlevel in the low bin and at the 2.4-2.5 σ level in the central bin <cit.>. Interestingly, in the summer of2014, a similar LHCb measurement of the ratioR_K≡ B(B^+ → K^+ μ^+ μ^-)/ B(B^+ → K^+ e^+ e^-) for q^2 ∈ [1,6] GeV^2 also showed a 2.6 σdeviation from the SM <cit.>.The experimental measurements as well as the latest SM predictions for these ratios are summarised inthe first 3 rows of Table-<ref>.As the theoretical predictions of R_K and R_K^* in the SM are rather reliable <cit.>, these measurements highly suggestfor lepton non-universal new physics (NP). This has spurred a lot of activities in the recent past, both in the language of model independent higherdimensional operators and specific models beyond the SM <cit.>.In the context of dimension-6 NP operators, it has been pointed out that short distance NP operatorsof certain types can provide an overall good fit to the data. However, a discussion of the tensor operators was missing.In this paper, we fill this gap with a detailed analysis of the role of tensor operators in R_K and R_K^*[In the context ofB̅_d →K̅^* ℓ^+ ℓ^- decay, the tensor operators with m_ℓ≠ 0 was first considered by one of the authors in<cit.> and later in <cit.>.].Note that, it is not possible to generate tensor operators at the dimension-6 level if the Standard Model gauge symmetry isimposed <cit.>. However, tensor operators can be generated at the dimension-8 level, see the end ofsection <ref> for more details. Besides R_K and R_K^*, we also consider the branching ratios of B_s →ℓ^+ ℓ^- (ℓ = μ, e) as they are reliably predicted in theSM. Furthermore, we also show the compatibility with measurements of the branching ratios of the inclusive decay B_d → X_s ℓ^+ ℓ^-.The experimental measurements of these observables are summarised in Table-<ref>. In the table and the subsequent text, we use thefollowing short-hand notations (q^2 is given in GeV^2)B_Kℓℓ^ cen≡ B(B^+ → K^+ ℓ^+ ℓ^-),q^2 ∈ [1,6]B_K^*ℓℓ^ low(cen)≡ B(B̅_d →K̅^* ℓ^+ ℓ^-),q^2 ∈ [0.045,1.1]( [1.1,6] )ℬ_ℓℓ≡ B(B̅_s →ℓ^+ ℓ^-) B_X_sℓℓ^ low (high)≡ B(B̅_d → X_s ℓ^+ ℓ^-),q^2 ∈ [1,6] ( [14.2,25] )We will not consider any angular observables (P_5^', for example) in this analysis because their SM predictions are debatable<cit.>[Note however that, large deviations from the SM expectations in two q^2-bins of P_5^' μ have beenclaimed in the literature <cit.>. Interestingly, the Belle collaboration has provided the first measurementof P_5^' in the electron mode <cit.>, and indeed, the central value for P_5^' μ deviatesmore than that of P_5^'e. However, at this point the statistics is low, and the jury is still out on this.].In our calculations of B(B̅_d →K̅^(*)ℓ^+ ℓ^-) we only include the factorizable part described by the form-factors,and no non-factorizable corrections are included. However, this is good enough for the theoretically clean observables R_K and R_K^*.As for the form-factors, we use <cit.> for B → K matrix elements and <cit.> for the B → K^* matrix elements. §.§Effective operatorsThe SU(3) × U(1) invariant effective Lagrangian at the dimension-6 level for b → s transition is given byL_eff = -4G_F/√(2) α_ em/4 πV_tb V_ts^* H_eff^(t)+h.c.where, H_eff^(t) = C_1 O_1^c + C_2 O_2^c + ∑_i=3^6 C_i O_i +∑_i=7^10 C_i O_iIn models beyond the SM, new operators can be generated. The complete basis of dimension-6 operatorsincludes new operators given by H_eff^(t), New =∑_i=7, 9, 10 C_i^'O_i^' +∑_i=S, P, S', P', T, T5 C_i O_iwhere, the various operators above are defined by𝒪_7(7') =1/em_b [sσ_μν P_R(L) b] F^μν 𝒪_9(9') =[sγ_μ P_L(R) b][lγ^μ l],𝒪_10(10')=[sγ_μ P_L(R) b][lγ^μγ_5 l]𝒪_S(S') =[s P_R(L) b] [l l],𝒪_P(P')=[s P_R(L) b][lγ_5 l]𝒪_T =[sσ_μν b][lσ^μν l],𝒪_T5=[sσ_μν b][lσ^μνγ_5 l] Note that, the Wilson coefficients for the photonic dipole operators 𝒪_7 and 𝒪_7^' are lepton universalby definition, and lead to lepton flavour non-universality only through lepton mass effects, which is not enough to provide explanationof the R_K^* anomalies once bound from B_d → X_s γ is taken into account <cit.>. So we neglectNP effect in these operators. For all the other operators, we write their Wilson coefficients as C_i = C_i^ SM + Δ C_i whereΔ C_i corresponds to the shift in the Wilson coefficient from its SM value due to short distance NP.§.§Tensor operatorsIn this section, we study the effect of the two tensor operators, O_T, O_T5, on R_K and R_K^*.In Eq. <ref> - <ref> below we show numerical formulae for the various branching ratios (normalised to their SMpredictions) as functions of Δ C_T5^μ and Δ C_T5^e:B_Kℓℓ^ cen/ B_Kℓℓ^ cen|_ SM≈1+0.02 [Δ C_T5^ℓ]^2B_K^*ee(μμ)^ low/ B_K^*ee(μμ)^ low|_ SM≈ 1 - 0.00 (0.24)[Δ C_T5^e(μ)]+ 0.30 [Δ C_T5^e(μ)]^2 B_K^*ee(μμ)^ cen/ B_K^*ee(μμ)^ cen|_ SM≈1 + 0.00(0.06) [Δ C_T5^e(μ)] + 0.53 [Δ C_T5^e(μ)]^2 The full set of numerical formulae valid in the presence of all the operators are presented in Appendix <ref>. These formulae can be usedto perform very quick analysis of models as the only required inputs in these formulae are the short distance Wilson coefficients.In Fig. <ref> we show how R_K ^ cen, R_K^* ^ low and R_K^*^ cen vary withΔ C_T5^e.It can be seen from the left panel of Fig. <ref> that Δ C_T5^e ∼± 1 not only explains R_K^*^ cen andR_K^*^ low simultaneously but also brings them close to the experimental central values. As pointed out by one of theauthors in <cit.>, this is not possible naturally by any other local operator at the dimension-6 level, and in this sense, the tensoroperators are unique. However, as can be seen from the right panel of Fig. <ref>, Δ C_T5^e ∼± 1 can not reduce R_K^ cen muchfrom its SM value of unity[That the tensor operators alone can not explain R_K was also pointed out in <cit.>.], and hence a simultaneous explanation of R_K^ cen, R_K^*^ cen and R_K^*^ lowis not possible. All statements made here for Δ C_T5^e applies equally for the other tensor Wilson coefficientΔ C_T^e. Note that, any non-zero value for Δ C_T^μ and Δ C_T5^μ leads to values for R_K and R_K^* greater than theirSM values and thus, tensor operators in the muon sector are ruled out as possible explanation of these anomalies. In the following section, we will investigate whethera simultaneous solution is possible when other additional operators are also considered. While we consider only unprimed operators in the main text, the effect of the primed operators in conjunction with the tensor operators can be found in Appendix <ref>.§.§Combination of Vector and Tensor operators In Fig. <ref>, we show the regions in Δ C_9^e - Δ C_T5^e plane allowed by the experimental measurementsof the various observables listed in Table-<ref>. In the left panel, the blue, red and yellow shaded regions correspond tothe 1σ experimental ranges of R_K^*^ low, R_K^*^ cen and R_K^ cen respectively. The black shaded regionsare the overlap of the three. It should be noticed that the black shaded region is outside the Δ C_T5^e = 0 line, and hence nosimultaneous solutions are possible with onlyΔ C_9^e. In the right panel, we also show the regions allowed byℬ^low_X_see(in blue) and ℬ^high_X_see (in red). The black shaded region from the left panelis also superimposed there. It can be seen that there is a small overlap of the black, blue and red regions in the right panel where all theconstraints including those from the inclusive decay are satisfied. In Fig. <ref>, we show the allowed regions in the Δ C_9^μ - Δ C_T5^e plane. The various shadedregions in the left panel have the same meaning as in Fig. <ref>. The grey vertical (horizontal) band corresponds tothe experimental 1σ allowed region of ℬ^low_X_sμμ (ℬ^low_X_see).Similar to the previous case, here also a simultaneous solution is not possible with only Δ C_9^μ, and non-zero tensor contributionis required. However, as can be seen from the right panel of Fig. <ref>, this scenario is in tension with the measurementsofℬ^high_X_sℓℓ.We now consider the two cases Δ C_9^e = -Δ C_10^e vs. Δ C_T5^e andΔ C_9^μ = -Δ C_10^μ vs. Δ C_T5^e.In Fig. <ref> we show our results. It can be seen from the upper panel thatΔ C_9^μ = -Δ C_10^μ alone (i.e., with Δ C_T5^e = 0) can not explain R_K^*^ low,R_K^*^ cen and R_K^ cen simultaneously within their experimental 1σ regions. However, a simultaneoussolutions is possible if a non-zero Δ C_T5^e ∼± 0.6is considered.Note that, the Wilson coefficient C_10^μ also modifies ℬ_μμ which gives a bound0 ≲Δ C_10^μ≲ 0.7 at the 1σ level <cit.>. Hence, the black overlap region in the upper left panelis allowed by ℬ_μμ. However, as in Fig. <ref>, this scenario also is in tension with the measurements of ℬ^high_X_sℓℓ. The situation is better for Δ C_9^e = -Δ C_10^e vs. Δ C_T5^e as shown in the bottom panel ofFig. <ref>. Here, a simultaneous solutions to not only R_K^*^ low,R_K^*^ cen and R_K^ cen, but also the inclusive decay B̅_d → X_s e^+ e^- is possible. Thiscorresponds to the small overlap of the black, red and blue shaded regions in the lower right panel ofFig. <ref>.Before closing this section, we would like to mention that the tensor operators do not get generated at the dimension-6 level ifSU(2) × U(1)_Y gauge invariance is imposed, which was also pointed out in <cit.>. However, it can be generated at the dimension-8 level. For example,one can write down the operator (1/Λ^4)(s_R L_1 H̃)(μ_R Q_3 H̃) which, after electroweaksymmetry breaking, generates the operator (v^2/2Λ^4) (s̅ P_L μ)(μ̅P_L b). This operator can be Fierz transformedinto (v^2/2Λ^4) (s̅ P_L b)(μ̅P_L μ) and the tensor operator (v^2/8Λ^4) (s̅σ_μν P_L b) (μ̅σ^μν P_L μ).For more details, see Appendix <ref>. §.§Summary Motivated by the recent measurements of R_K^* in two q^2 bins by the LHCb collaboration, we have performed a detailed analysis of the roleof tensor operators in R_K and R_K^*, for the first time in the literature. We show that, unlike the vector, axial vector, scalar or pseudo scalar operators, tensoroperators can comfortably explain R_K^*^ cen and R_K^*^ low simultaneously. Hence, if the experimental measurement ofR_K^* in the low q^2 bin stays in the future, either a very light vector boson (as shown by one of the authors in <cit.>) orthe existence of tensor operators would be unavoidable. However, we find that a simultaneous explanation of R_Kalso would require the existence of other Wilson coefficients (of vector and/or axial vector operators, for example) in conjunction with the tensor operators.We study the interplay of the vector and axial vector operators with the tensor structures, and obtain the regions allowed by the 1σ experimentalvalues of R_K and R_K^*.We further show that the measured branching ratios for the inclusive B_d → X_s ℓ^+ ℓ^- decay provide very important constraints on the various solutions. We also present completely general numerical formulae which can be used to effortlessly compute R_K^ cen, R_K^*^ cen, R_K^*^ low and the inclusive branching fractions just knowing the short distance Wilson coefficients at the m_b scale.——————————————————————————– § COMPLETE EXPRESSIONS FOR THE BRANCHING RATIOS B_Kee^ cen B_Kee^ cen|_ SM=1 +0.2429 +0.0274 ^2+0.2429 +0.0549 +0.0274 ^2- 0.225 [-2mm]+0.0274 ^2-0.225 +0.0549 +0.0274^2+0.0092 ^2+0.0184 +0.0092 ^2+0.0092 ^2 +0.0184 +0.0092 ^2 +0.0002 +0.0171 ^2+0.0171 ^2 [2mm] B_Kμμ^ cen B_Kμμ^cen|_ SM = 1 +0.2427 +0.0274 ^2+0.2427+0.0548 +0.0274 ^2-0.2253[-2mm] +0.0275 ^2-0.225 +0.055 +0.0275^2+ 0.009 ^2 +0.018+0.009 ^2-0.0187+ 0.0046 (+)(+ ) +0.0091 ^2-0.0187 +0.0182 +0.0091^2+0.0168 ^2 +0.0457 +0.0103(+) +0.0174 ^2 [2mm] B_K^*ee^ low B_K^*ee^ low|_ SM= 1+0.0764 +0.0136 ^2-0.1048 -0.0257 +0.0136 ^2-0.1118 [-2mm] +0.0136 ^2+0.1054 -0.0257 +0.0136^2+0.0006(-)^2 +0.0006 (-)^2 -0.0015 +0.2901 ^2-0.0013+0.2901 ^2 B_K^*μμ^ low B_K^*μμ^ low|_SM = 1+ 0.0806 +0.0144 ^2-0.1103 -0.027 +0.0144 ^2-0.1167 [-2mm]+0.0142 ^2+0.1106 -0.027 +0.0142^2+0.0006 ^2 -0.0012 +0.0006 ^2-0.0078 +0.0019 -0.0019 + 0.0006 ^2+0.0078 -0.0019 +0.0019 -0.0013+0.0006 ^2-0.2362 +0.0165 -0.0165+0.3057 ^2 -0.2676 +0.0088 +0.0088 +0.305 ^2 B_K^*ee^ cen B_K^*ee^ cen|_SM = 1 + 0.2187 +0.032 ^2-0.1998 -0.0474 +0.032 ^2-0.2629 [-2mm] +0.032 ^2+0.1945 -0.0474 +0.032^2+0.0067 ^2 -0.0134 +0.0067 ^2+0.0067 ^2-0.0134 +0.0067 ^2 +0.5349 ^2+0.0003 +0.0001 -0.0001 +0.5349 ^2 B_K^*μμ^ cen B_K^*μμ^cen|_ SM =1 + 0.2194 +0.0321 ^2-0.2004-0.0476 +0.0321 ^2-0.2622[-2mm] +0.032 ^2+0.1949 -0.0475 +0.032^2+0.0066 ^2 -0.0132 +0.0066 ^2-0.0138 +0.0034 -0.0034+0.0067 ^2+0.0138 -0.0034 +0.0034 -0.0134+0.0067 ^2+0.0638 +0.0295 -0.0295+0.5373 ^2 +0.0039 +0.0154 +0.0154 +0.5359 ^2 10^6B_X_sℓℓ^ low =10^6B_X_sℓℓ^low|_ SM +0.4156 +0.0647 ( ^2+ ^2+ ^2+ ^2) -0.5308+0.0108 ( ^2+ ^2+ ^2+ ^2) +0.8615( ^2+^2 ) [2mm] 10^6B_X_sℓℓ^ high =10^6B_X_sℓℓ^high|_ SM + 0.1187 +0.0143( ^2+ ^2 + ^2) -0.1171 + 0.0143^2 +0.0063 ( ^2+ ^2+ ^2+^2) + 0.1272 ( ^2+ ^2 ) §PRIMED OPERATORS Earlier we considered only the unprimed vector and axial vectoroperators namely, C_9^μ, e and C_10^μ, e, and neglectedtheir primed counterparts C_9^'^μ, e and C_10^'^μ, e.It has been shown (see for example, <cit.>) thatthe primed operators alone are unable to produce the experimental measurementsof R_K and R_K^* simultaneously. In this section, wewill investigate whether the situation can improve in the presence of tensoroperators. Fig. <ref> shows the allowed regions in Δ C_9'^μ -Δ C_T5^e plane. It can be seen that in order to satisfy R_K^ cen, R_K^∗^ lowand R_K^∗^ cen simultaneously in thepresence of Δ C_9'^μ, large value of Δ C_T5^e ≈±1.3 is also needed. However, this solution is in tension with ℬ^low_X_see ascan be seen from the grey region in the left panelof Fig. <ref>. Note that, in the right panel ofFig. <ref> the blue region covers the whole plane, and hencethis solution is consistent with ℬ^high_X_sℓℓ. Similar statements can be made also for Δ C_10'^μ, as can be seenfrom Fig. <ref>. Fig. <ref>and Fig. <ref> show the allowed regions inΔ C_9'^evs. Δ C_T5^eandΔ C_10'^e vs. Δ C_T5^e planes respectively. In these casesalso, the primed operators can be allowed if a largetensor contribution exists at the same time. §SU(2) × U(1)_ Y GAUGE INVARIANCEAs mentioned in the main text, the tensor operators do not get generated at thedimension-6 level if SU(2) × U(1)_ Y gauge invarianceis imposed[Tensor operators have also been considered in the context ofthe charged current anomalies R_D and R_D^*, seefor example <cit.>. In that case, however,tensor operator can be generated already at the dimension6 level <cit.>.]. However, they can be generated at thedimension-8 level. Here we show a few examples,1. C_Y_d Y_ℓ/Λ^4 [s_Rσ^μν Q_3 H̃] [e_ℓ Rσ_μν L_ℓH̃ ]→1/2 C_Y_d Y_ℓv^2/Λ^4[s_Rσ^μν b_L ] [e_ℓ Rσ_μνe_ℓ L ] = 1/4 C_Y_d Y_ℓv^2/Λ^4( O_T-O_T5)2. C_sLeQ/Λ^4[s_R L_ℓH̃][e_ℓ R Q_3 H̃ ] →C_sLeQ/Λ^4(1/2[s_R Q_3H̃] [e_ℓ R L_ℓH̃ ] + 1/8 [s_Rσ^μν Q_3H̃][e_ℓ Rσ_μν L_ℓH̃ ])= 1/8 C_sLeQv^2/Λ^4(O_S' - O_P'+ 1/4 O_T - 1/4 O_T5) It is hard to generate only the tensor operators in a complete field theorymodel. The second operator above is much easier to generate(it can be generated even at the tree level). In this case, however, both scalarand tensor operators are generated with the following relationsamong the Wilson coefficients, Δ C_S'^e = - Δ C_P'^e = 4 Δ C_T5^e = - 4 Δ C_T^e . Note that, gauge invariance at the dimension 6 levelalways leads to the relation Δ C_S'^e = + Δ C_P'^e<cit.>, which is nowbroken by the dimension 8 operators. In Fig. <ref>, we show the various allowed regions in the ΔC_S'^e (= - Δ C_P'^e) vs.Δ C_T5^e (= - Δ C_T^e)plane. It is interesting that the black overlap regions in the left panelsatisfy Eq. (<ref>) approximately. In fact, there is tiny regionin the right panel which satisfies the inclusive measurements too.Note that, the value of Δ C_S'^e = - Δ C_P'^e ≈ 3correspondsto a NP scale Λ∼ (C_sLeQ)^1/41.5 TeV. While the scale israther low, it is still intriguing that one local operator inEq. (<ref>) can explain all the anomalies (including R_K^*^low) simultaneously. Unfortunately, for such large valueΔ C_S'^e = - Δ C_P'^e ≈ 3, ℬ_ee exceeds theexperimental upper bound, and some cancellation, eitherfrom other dimension-8 operators or from dimension-6 operators would benecessary for this operator to be viable. More detailedexploration of such dynamics is left for future work. ——————————————————————————–10Aaij:2017vbb LHCb, R. Aaij et al.,(2017), 1705.05802. Aaij:2014ora LHCb collaboration, R. Aaij et al.,(2014), 1406.6482. Hiller:2003js G. Hiller and F. Kruger,Phys. Rev. D69 (2004) 074020, hep-ph/0310219. Bordone:2016gaq M. Bordone, G. Isidori and A. Pattori,Eur. Phys. J. C76 (2016) 440, 1605.07633. Altmannshofer:2008dz W. Altmannshofer et al.,JHEP 0901 (2009) 019, 0811.1214. Alok:2009tz A.K. Alok et al.,JHEP 1002 (2010) 053, 0912.1382. Alok:2010zd A.K. Alok et al.,JHEP 1111 (2011) 121, 1008.2367. Alok:2011gv A.K. Alok et al.,JHEP 1111 (2011) 122, 1103.5344. 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"authors": [
"Debjyoti Bardhan",
"Pritibhajan Byakti",
"Diptimoy Ghosh"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170525180034",
"title": "Role of Tensor operators in $R_K$ and $R_{K^*}$"
} |
Device-independent point estimation from finite dataand its application to device-independent property estimation Yeong-Cherng Liang Accepted .... Received ...; in original form 2017 May 17 =================================================================================================================== Polarisation of radio pulsar profiles involves a number of poorly understood, intriguingphenomena, such as the existence of comparable amounts of orthogonal polarisation modes (OPMs), strongdistortions of polarisation angle (PA) curves into shapes inconsistent with the rotating vector model (RVM),and the strong circular polarisation V whichcan be maximum (instead of zero) at the OPM jumps. It is shown thatthe existence of comparable OPMs and of the large V results from a coherent addition of phase-delayed waves in natural propagation modes,which are produced when a linearly polarisedemitted signal propagates through an intervening medium on its way to reach the observer. The longitude-dependent flux ratio of two OPMscan be understood as the result of backlighting the intervening polarisation basis by the emitted radiation. The coherent mode summation implies opposite polarisationproperties to those known from the incoherent case,in particular, the OPM jumps occur at peaks of V, whereas V changes signat a maximum of the linear polarisation fraction L/I.These features are indispensable to interpret variousobserved polarisation effects. It is shown that statistical properties of the emission mechanism and ofpropagation effects can be efficiently parametrised in a simplemodel of coherent mode addition,which is successfully applied to complex polarisation phenomena, such as the stepwise PA curve of PSR B1913+16 and the strong distortions of the PAcurve within core components of pulsars B1933+16 and B1237+25.The inclusion of coherent mode addition opens the possibility fora number of new polarisation effects,such as inversion of relative modal strength, twin minima in L/Icoincident with peaks in V, 45^∘ PA jumpsin weakly polarised emission, and loop-shaped core PA distortions. The empirical treatment of the coherencyof mode additionmakes it possibleto advance the understanding of pulsar polarisation beyond the RVM model.pulsars: general – pulsars: individual: PSR J1913+16 – pulsars: individual: PSR B1237+25 – pulsars: individual: PSR B1919+21 – pulsars: individual: PSR B1933+16 – radiation mechanisms: non-thermal.1.5pt 9pt<1.5pt 9pt∼ 0.02mm R_W R_W R_W W_in W_out ϕ_m ϕ_m, i θ_m Δϕ_resW_in W_out ρ_in ρ_out ϕ_in ϕ_out x_in x_out θ_min^m θ_max^m § INTRODUCTIONThe origin of orthogonal polarisation modes (OPMs) in pulsars has been unknown,except for the general notion that they are probably propagation effects inbirefringent pulsar plasma. The observed properties of the modesmake up for a real panopticon of peculiarities, which is well documentedin pulsar literature.[For example in Xilouris et al. (1998), Stairs et al. (1999), Weltevrede & Johnston (2008), Gould & Lyne (1998), Manchester & Han (2004), Tiburzi et al. (2013), Karastergiou & Johnston (2006),Johnston & Weisberg (2006), Keith et al. (2009). xkj98, stc99, wj08, gl98, mh04, tjb13, kj06, jw06 ]It is possible to find polarisation angle (PA) curves which follow the rotating vector model (RVM), e.g. B0301+19, B0525+21 (Hankins & Rankin 2010, hereafter HR10). hr10Other PA curves follow RVM, but with frequent transitions (jumps)between the modes. Other cases seeminglydo not obey any simple model (erratic, e.g. B1946+35, Mitra & Rankin 2017).mr2017 The PA distributions of single samples (recorded in single pulse observations) reveal that the polarisation modes can be very well definedby narrow peaks, or oppositely, be very wide, almost merging with each other, or filling the whole 180^∘ PA range (Stinebring et al. 1984, Mitra et al. 2015, hereafter MAR15).scr84, mar2015 In the latter case the associated linear polarisationfraction L/I is very low (eg. B2110+27, Fig. 19 in MAR15),but in other cases L/I may be very high.This complex picture is spiced up with cases which generally follow RVM, but locally, especially at the central component,exhibit extremely complicated non-RVM distortions (e.g. B1933+16, Mitra et al. 2016, hereafter MRA16; B1237+25, Smith et al. 2013, hereafter SRM13). mra2016, srm13 Several objects, including those with complex corepolarisation, reveal curious symmetry of their single-pulse PA tracks:the strongest (primary) polarisation mode is acompanied by short patches of the secondary mode in the profile peripheries.The scant attempts to understand the complex polarisation distortionsinvolved both coherent and incoherent summationof radiation in different modes.The coherent interaction of modes has long been considered a possible source of observed circular polarisation (Cheng & Ruderman 1979;Melrose 2003; Lyubarskii & Petrova 1998),cr79, lp98, m03 since the latter appears when the modal waves combine with a non-zero phase lag (which is not a multiple of π). However, propagation processes were also considered unlikelyto induce appropriately small phase delays (of the order of the wavelength)to avoid complete cancelling of V (Michel 1991, p. 30).m91 Moreover, the appreciable amount of V can also be directlyproduced by the emission mechanism (e.g. Sokolov & Ternov 1968; Gangadhara 2010). st68, g10The influence of magnetised plasma on propagating wavesis known (e.g. Barnard & Arons, 1986; Wang et al. 2010;Beskin and Philippov 2012),ba86, bp12, wlh10 so the properties of the outgoing pulsar radiation can in principle be estimated.However, the result is sensitive to the poorly known and likelycomplex properties of the intervening medium(propagation direction and the spatial distribution of plasma density)which makes the detailed modellingof propagation physics ambiguous. The complexity of such calculations(e.g. Wang et al. 2010), and the vague correspondence between their resultswlh10 and data offers limited practical help in interpretingthe observed polarisation.The alternative empirical approach attempts to reproduce the observed non-RVM effects through some specific combination of radiationin each polarisation mode. This is typically a data-guided guess, also suffering from the problem of ambiguity.The central profile region of PSR B0329+54, for example,exhibits the elliptically polarised radiationwith the PA which deviates from the RVM. Edwards & Stappers (2004) interpret this es04 through the coherent combination of radiation in two natural propagationmodes, with some relative phase delay.Melrose et al. (2006), however, attribute this effect to incoherent mmk2006 combination of specific distributions of radiation in both modes.Both these so different models focus on the single pulse PA behaviourat a fixed pulse longitude, and achieve reasonable consistency with data.Contrary to that approach, in this paper I interpret variationsof the non-RVM effects across the pulse window, i.e. the goal is tounderstand the distorted shape of complex PA curves. The statistical single pulseproperties need to be taken into accountin such analysis, but fortunately this can be done through their simpleparametrisation. Sect. <ref> describes observations that can be considered a clear manifestation of the coherent addition of modes,and which inspired the analysis of this paper.After introducing the needed mathematics (Sect. <ref>),the coherent mode addition is applied to a few peculiar polarisation effects (Sect. <ref>).§ OBSERVATIONAL EVIDENCE FOR THE COHERENT ADDITION OF MODES A strong signature of the coherent mode addition (CMA),far more convincing than the mereexistence of V, is presented by thoseorthogonal PA transitions, that are accompanied by high levels of |V|. Especially thosewhich occur at the peak of |V|.This type of behaviour suggests CMA, because it is expected in the case oforthogonal modes added with a phase lag which changes with pulse longitude.A clear example of such phenomenon is provided by the profile ofthe relativistic binary pulsar B1913+16 (Fig. <ref>, based on Fig. 7 in Weisberg & Taylor 2002). wt02 The PA of this pulsar exhibits a continuous increase across a larger than π interval of195^∘. On the inner side of two maxima that flank/surround the profile, the PA makes two steep transitions between three approximately orthogonal values. Both these PA transitions are accompanied by a high level of circular polarisation (|V|/I ∼ 0.3)which peaks at pulse longitudeswhere the PA assumes a midpoint value, located half way betweenthe nearly constant values of apparently pure orthogonal modes.Several properties of these PA transitions differ strikingly fromthe well known properties of transitions that are produced bythe incoherently combined modes. The latter appear when the amount of one mode starts to exceed another one, and are expected to occur at longitudeswhere L/I drops to zero and V crosses zero on its way to change sign. In B1913+16 the linear polarisation fraction (top thick grey line)does not even reach the proximity of zero. Neither does the circularpolarisation change sign in coincidence with the PA transition. Another example of similar effect is provided by PSR B1933+16. As can be seen in Fig. 1 of MRA16mra2016 at the longitude of-5^∘ the PA curve is almost transiting to another polarisation mode. Linear polarisation degree is fairly low at the jump, however, |V|again reaches a maximum on the way between the orthogonal modes(instead of crossing zero).The average PA does not quite reachthe other mode, and immediately retreats (at longitude of -4^∘),to forma narrow V-shaped distortion of the PA curve. The feature makes impression of a failed mode jump, or some superficially similar jump-likebehaviour.Even more striking view of this jump, however, is provided by the dottydistribution of PA recorded in individual samples. The PAdistortion is possibly bidirectional,with the PA of some samples increasing, while most of them have the PA decreasing in accordance with the average PA curve. As a result, the V-shaped distortion of the average PAforms only a lower half of a loop-like structure,created by the PA of individual samples. Just four degrees ahead of this loop-shaped distortion in B1933+16, the profile presents a well behaving, regular orthogonal mode jump, with little V or L, i.e. consistent with the “incoherent" mode summation(coherent sum of orthogonally-polarised waves of equal amplitudegenerally does not make L vanishing). Note that `incoherent' means coherentaddition of waves with a large mixture of different phase lags, sampled from a wide phase lag distribution (wider that 2π). If the loop-like (or V-shaped) distortion is interpreted as acoherent sum of modal contributions, the observed signal would have toquickly change (within 4^∘ of pulse longitude) from an incoherent sum of modesto a coherent sum ofmodes. This suggests it may be worth to attempt to interpet the wholepulsar emission in terms of CMA, albeit with longitude-dependent width ofthe phase lag distribution. Further examples of similar phenomena include the infamous PA distortions observed within core components of M-type pulsars: B1237+25 (SRM13)and B1857-25 (Mitra & Rankin 2008).srm13, mr2008 With its loop-like shape,the distortion in B1237+25 is superficially similarto that in B1933+16. However, there are two key differences:1) In B1237+25 V has an antisymmetric sinusoid shape, with a sign change near the middle of the core component (in B1933+16 V is negativethroughout the loop). 2) In B1237+25 the PA loop is createdas a split of the primary mode PA track, however, the loop convergesback to the secondary mode, which is visible in the peripheries of the profilein the form of short patches of an orthogonal PA. In B1933+16the PA loop opens and closes at theprimary mode. Remarkably, in these complicated-core objects, V changes sign not atthe minima of L. The minima are located on both sides of the disappearing V, and again seem to be roughly coincident with peaks of |V|. § THE MODEL§.§ Simple model of coherent mode addition I assume that radiation observed at any pulse longituderesults from coherentcombination of linearly polarised natural mode waves. These are represented as two monochromatic sinusoids:E_x = E_1cos(ω t),E_y = E_2cos(ω t - Δϕ),where Δϕ is the phase delay between the modes.The Stokes parameters for such wavesare calculated from:I= E_1^2 + E_2^2Q = E_1^2 - E_2^2U = 2 E_1 E_2 cos(Δϕ)V = -2 E_1 E_2 sin(Δϕ)(Rybicki & Lightman 1979). Therl79polarisation fractions and angle are calculated in the usual way:Π_ tot=I_ pol/I = (Q^2 + U^2 + V^2)^1/2/I Π=L/I=(Q^2 + U^2)^1/2/I ψ= 0.5arctan(U/Q).It is assumed thatthe observed polarisation results from an intrinsic signal(likely produced directly by the emission mechanism) which has propagated through some intervening magnetospheric plasma,possibly located at the polarisation limiting radius(PLR, e.g. Petrova & Lyubarskii 2000). pl00 The intervening matter is represented by a polarisation basis (x⃗,y⃗) and the signal enters the basis at a polarisation anglegiven by:tan=E_2/E_1=R,where R is the mode amplitude ratio,E_1=Ecos,E_2=Esin,and E is the incident wave amplitude. The PA observed in single pulses typically exhibits wide distributions at a fixed longitude, which are assumed to be caused by thestochastic nature of the input signal. I assume that typically a wide distribution of , denoted ,which at some longitudes may possibly be even quasi-isotropic,enters the polarisation basis (differentcontributeat a given pulse longitude in different single pulses). The PA at whichis maximum is denoted by , and the width ofby. The incident signalis split into components E_x and E_y parallelto the polarisation direction of the proper propagation modesin the intervening matter, which induces some phase lag Δϕbetween E_x and E_y.The intervening matter is assumed to be highly modulated andirregular, hence the lag is also expected to be highly random, and represented by a wide distributionwith the maximum atand the width .The whole propagation physics is parametrised with these two quantities.It is assumed that the observed variations of polarisation as a function of pulse longitude[The capital letter Φ refersto the pulse longitudein the profile, and should not be mistaken with the small ϕ which denotes oscillationphase in the wave given by eq. (<ref>).Hereafter I always use the word `longitude'to refer to Φ,whereas `phase' is used for ϕ.] Φ, mostly result from the longitude dependenceof the mode ratio and phase lag, i.e.:E_2/E_1 = tan((Φ)) = (Φ)= (Φ)= (Φ).The function (Φ) is determined by the relativeorientation of a distribution of incident PAs and the intervening basis,as described in Section <ref>. Since the emission and propagationphysics is neglected, the other functions need to be found by a guess,hopefully guided by the qualitative properties of available data.Equations (<ref>)-(<ref>) make the Stokes parameters dependent on the pulse longitude.Hereafter I will be interested in polarisation fractions and angle, since they depend only on the ratio R=E_2/E_1.This makes it possible to ignore the absolute profiles ofI, L and V, i.e. the incident wave amplitude E=1 in eqs. (<ref>).[Unfortunately, in observationalworksusually only the observed I, L and V are published. This makes it very difficult toassess L/I and V/I whenever I is changing steeply or is very low. This plotting tradition seriously hampers any attempts to interpret the observed pulsar polarisation.] The polarisation angle ψ,as given by eq. (<ref>), so far only includes the effects of mode addition, withoutthe likely variations caused by the changes of projected magnetic fieldwith Φ. If these regular RVM variations of PA are denoted as ψ_ RVM(Φ), the observed PA is:ψ_ obs = ψ + ψ_ RVM,i.e. ψ_ RVM serves as a reference value forthe non-RVM effects. §.§ The origin of orthogonal polarisation modesIt is then assumed that the input wave becomes split into components parallel to the natural polarisation directions,then the components acquire some phase lag , and finally combine.Fig. <ref> presents the polarisation characteristics forvalues within one full wave oscillation period, and for a setofmode amplitude ratios that correspond to the input PA values separated by 5^∘. The bottom panel presents the PA as a function of thephase lag, withfixed on each curve of ψ().For each point in the bottom panel, can be established by following the closest curve of ψ(Δϕ)towards the left vertical axis, i.e. ψ_in= ψ(Δϕ=0).The PA of the natural modes of the intervening polarisationbasis is shown with the dot-dashed horizontal linesat ψ=±90^∘ and 0^∘ (the latter is overlapping with ψ(=0)). The straight lines that correspondto equal mode amplitudes(incident PA of ±45^∘, hereafter called intermodes)consist of separate sections that jump discontinouslyto orthogonal values at =90^∘ and 270^∘ (as marked with ticks and thick line sections). This corresponds to the transformation of the polarisation ellipseinto a circle, and further into an orthogonally oriented ellipse(as shown near the top of the figure).Changes of L/I and V/I for this equal amplitude case (=45^∘)are shown in the top panel with thick lines, which clearly demonstratethe anticorrelation of L/I and |V|/I inherent for the CMA.The orthogonal polarisation modes of radio pulsar emissionclearly stand out at the locationswhere several different curves of ψ(Δϕ)cross at the same point. In single pulse observations,at a fixed pulse longitude, different points of this figure are quasi randomly sampled, with some spreadin the input PA (, shown left of the plotting box)and some spread in the phase lag (, shown below the plot).The origin of these OMPs, emerging as the nodesin the lag-amplitude ratio diagram, is simple: the input signal which ispolarised at an arbitrary ψ_in, produces polarisation ellipsesparallel to either natural polarisation mode, whenever the lagΔϕ=π/2. That mode will be observed, which corresponds to the larger componentof the split input vector. The phenomenon is presented in Fig. <ref>a, in which components ofa dotted electric field vector, tilted at an arbitrary input angle of ψ_in=25^∘combine at Δϕ=90^∘into the dotted polarisation ellipse,whereas another input vector at ψ_in=65^∘ (solid), similarly produces the solid ellipse. This way the two orthogonal modes,always polarised at 0^∘ or 90^∘ are produced.[The PA, L, and V of the elliptically polarised radiationare determined assumingdecomposition intothe fully circular part, and the phase coherent linear part in the direction of the major axis of the ellipse.]Ifis slightly different from 90^∘, the polarisation ellipses are no longer strictly parallel to the x⃗ or y⃗vector of the basis, however, for most R the misalignment is small(this can be seenby making a vertical cut through Fig. <ref>, e.g. at =80^∘).If the input vectors appear on both sides of the dashed diagonal inFig. <ref>a,the two orthogonal modes will be present in the observed signal whenever the medium imposes phase delaysin the vicinity of π/2 on the propagatingsignal. If Δϕπ/2, variety of PA values will be observed,however, their fixed-longitude distributions will reveal maxima at the modes. This can be understood by considering the local line density along some vertical cut of Fig. <ref>(any vertical slice which is not at a multiple of Δϕ=π). Thus, the modes will emerge statistically in the observations, even for phase lags much different from 90^∘,although in such case they will form a distribution with wide peaksconnected with bridges. Therefore, if the lag andare sampled from a wide or random distribution, the modes stand out clearly in statistical sense, i.e. appear frequently.When both the distributions( and ) are uniform,the bottom panel ofFig. <ref> will not be uniformly covered by samples. Instead, the PA observed in different samples will bunch at the orthogonal modes, following the densityof the lines in the figure. In spite of the pure randomnessof input polarisation parameters,the orthogonal modes will anywaystand out in single pulse observations with equal strength, but the average profile will be fully depolarised.The apparent OPMs generated through the coherent mode combinationare then an extremelyrobust feature, easily surviving most of sampling styles that can be imagined for Fig. <ref>. Only the zero lag, or a multiple of π, does not produce the OPMs. In other words, the density of lines in Fig. <ref> presents the probability of measuring a given PA for a totally randomorientation of the input polarisation and for a random(uniform) distribution of phase lags (by `measuring' I meanthe measuring of PA at short time samples always taken at a fixed pulse longitude).Equations psiang, qst, and ust giveK≡1/R-R/2=cos/tan2ψ.A square equation R^2 + 2KR - 1 = 0 gives the solution for(K),which provides the input PA for any point in Fig. <ref>:(, ψ). The above-described probability distribution(for the uniformand ) is then calculated as the rate at which the lines of fixedare crossed vertically: P(ψ,)_rndm = |d/dψ()|and is presented in Fig. <ref>. If in Fig. <ref> is narrow, whereas is very wide (or isotropic) then clearly defined OPMs will be observed when peaks near =90^∘ or 270^∘.If the peak ofis at =0^∘, 180^∘, or a multiple of thereof, the quasi-uniform distribution of the input PA will be observed.Intermediate cases produce wide observed PA distributions, with broad tracks of quasi-orthogonal modes,and many PA values between them. Inclusion of a narrow PA distribution (shown on the left in Fig. <ref>)further modifies the picture, as will be described below. For a symmetrical orientation of input vectorswith respect to ψ_in=45^∘, equal amounts of same-handednesscircularly polarised signal are produced (Fig. <ref>a). It is important to note that this case is different from the casewhen the input signal is split into two natural orthogonal modes which are elliptically polarised and reach the observer withoutbeing combined on the way. In the last case the handedness of V is opposite (Fig. <ref>c) and the resulting V=0.A symmetrical distribution of the input PA,centered at ψ_in=45^∘ (marked with the dashed line in Fig. <ref>a) will produce equalamounts of the two OMPs plus strong V. The same distributioncentered at ψ_in=90^∘ (or 0^∘) will produce one strongpolarisation mode with little V, as illustrated in Fig. <ref>b.This is again different from the elliptical mode case, in which a single mode signal should reveal the elliptical polarisation intrinsic to the naturalpropagation mode.It is a notable feature that for the vectors located on different sidesof ψ_in=45^∘ (Fig. <ref>a) the orthogonal OPMsare produced with the same handedness of circular polarisation. This phenomenon is observed in the single pulse emission,where samples of the same handedness appear on both orthogonal mode PA tracks. This is well illustrated in Figs. 1 and 2 of Mitra et al. (2015)who plot opposite-handedness samplesin different colors. Another very important point is that in the caseof the coherent mode additionV changes sign when the peak of the input PA distributionmoves across one of the natural modes (Fig. <ref>b). Therefore,different samples of the same polarisation mode may haveopposite sign of V, as can also be seen in Figs. 1 and 2 of MAR15.Both the above-described properties are opposite to what is known about thenoninteracting elliptically polarised propagation modes.As shown in Fig. <ref>a the answer to the question ofwhich mode will be detected, depends on the orientation of the input polarisation vector with respect to the separating angle of 45^∘.Accordingly, in the case of symmetricaldistributions that peak at ,the predomination of a given mode depends on the position ofrelative to 45^∘. To learn what Stokes parameters contribute at a given pulse longitude,and , shown on the margins of Fig. <ref>, must beconvolved with the line density distribution on the lag-PA diagram. The convolution ofis simple: thelag distribution selects the whole vertical area in Fig. <ref>, located directly above , and the contribution ofdifferent phase lags to the average signal is weighted by . The PA distribution on the left, however, is convolved indirectly: for the input PA of, say, 40^∘, it is necessary to start at this value on the left axis of Fig. <ref>, and follow the fixedline rightwards,until the vertical region selected by the lag distribution is reached. This implies that whenever the PA distributionextends to both sides of =45^∘,both OPMs are produced and they may appear in single pulse data (at a given phase in different rotation periods,i.e. they may bevisible as the parallel tracks of orthogonal PA).[The visibilityof both OPMs is known to depend on the question of whether the two OPMs, i.e. thedifferent results of the coherent addition,can simultaneously contribute to a single time sample (Stinebring et al. 1984; McKinnon& Stinebring 2000). scr84, ms00]Forwhich does not extend beyondthe interval of [-45^∘,45^∘](or beyond [135^∘, 215^∘]), a single OPM will be observed. An arbitrary example of combined effects ofandis shown in Fig. <ref>, where a moderately widePA distribution peaking at=30^∘, and a lag distribution peaking at=60^∘, generate two approximately orthogonal polarisation modes of unequal magnitude (at Δϕ≈90^∘).The thicknessof ψ() curves in that figure is made proportional to the product of appropriate values in the distributions:()(), where should be understood as =f(ψ, ).The imperfect orthogonality of OPMs, often observed in pulsars, is a natural consequence of the convolution presented in Fig. <ref>.The PA and lag distributions do not necessarilyneed to be as narrow as shown in Fig. <ref>.In particular,can span much more than 2π. The resulting polarisation characteristics are then a residual effectdetermined by the horizontal misalignment ofwith respect to vertical symmetry lines of the lag-PA pattern(e.g. Δϕ=90^∘, or 180^∘). Possible asymmetry of the lag distribution wings could also be in such case decisive for the ensuing PA. § RESULTSThe OPM model of previous section will now be employed tounderstand selected examples of pulsar polarisation zoo.§.§ The origin of polarisation in B1913+16The PA curve of B1913+16 (Weisberg and Taylor 2002, reproduced here in Fig. <ref>), if considered alone, looks quite innocuous:there are three PA flattenings that seem to present the two natural OPMs separated by two gradual transitions between them.However, |V|/I astonishingly peaksat the transitions (instead of crossing zero),and both the transitions are gradual. Equally strange, L/I ishigh everywhere (top grey line in Fig. <ref>).Instead of reaching zero, L/I decreases near the trailing PA jump from 0.65 to 0.45.Since generally |V|/I>0 in the profile, the peakof thedistribution must either be displaced from zero, or the distribution must be asymmetric, because otherwise V would vanish (e.g. see Fig. <ref>b,and note that a change of the lag from π/2 to -π/2 is equivalentto a change of sign in one componentof the incident vector). Two examples of such distributions (denoted )are shown on the left margin of Fig. <ref>. The asymmetricwith only positive is expected for the PLR scenario (Cheng & Ruderman 1979; Lyubarskii & Petrova 1998; ES04)cr79, lp98, es04 although both the distributions oftenproduce similar results if the symmetric one is not too wide and is displaced to some>0.[The observation of slightlyelliptical modes by ES04 in B0329+54 is not necessarily inconsistent with the PLR effects on the purely linear natural modes,becausethe observedOPMs arethe net resultof the coherent combination of the natural propagation modes with some and . A slight asymmetryof(and ) around zerois sufficient forthe observed mode to acquire some ellipticity even when the natural modesare perfectly linear (Fig. <ref>b).Still, the naturalelliptical modes are supported by the observedOPM jumps at which V does change the sign.For simplicity I ignore this ellipticity in this paper. ] If the small positive V/I on the leading edge of the profile is ignored(Φ≈-20^∘ in Fig. <ref>), a sinusoid-like V/I is observed in the rest of the pulse window, i.e. V/Idecreases from zero to a negative minimum at Φ=-7^∘, crosses zeroat 7^∘, reaches maximum at Φ=15^∘ and finally drops to zero at the trailing edge. Since V should change sign whenevercoincides with one of the naturalmodes, the observed V profile suggests that the input vector must rotate by slightly more than 180^∘, say betweenthe values of -100^∘ and 100^∘ in Fig. <ref>. This interval ofis hereafter assumed to be cast onto the observed pulse window of B1913+16. The initial position of thedistributionis shown at the bottom of the figure, andit is assumed to roughly correspond to the leading edge of the observed profile. While the pulsar rotates, moves rightwards. The top panel of Fig. <ref> shows the probability distribution that needs to be convolved withandtodetermine the polarisation characteristics at a given longitude. In a general case of arbitrary distributions,the procedure is as follows: for each pulse longitude(i.e. for a given position ofat ) the Stokes parameters, as determined byand (,ψ), need to be integrated over to obtain the observed distribution of PA (i.e. ψ) at a fixed Φ (or distributions of L/I and V/I at that Φ).To obtain a single value of PA in an average profile, the integration needs to be done over bothand .For B1913+16 I employ moderately narrow distributions, which makes itpossible to read out the result directly from Fig. <ref>. The modal peaks, located at =0 and ±90^∘ in the P distribution (top panel) are very strong features. Therefore, when(i.e. )coincideswith oneof the modal PA values at some pulse longitude Φ,that OPM value statisticallydominates in single pulse samples observed at that Φ. In other words,the peak of the convolved net probability distributionwill be located close to the modal peaks.However, ifis located at an intermode (e.g. =45^∘), then the modal points are located in the low-level wings of bothand .The strongest (most likely) contribution then comes from the location in the(ψ, ) diagram which corresponds to the peaks ofand . For the symmetricdistribution shown on the left margin ofFig. <ref>, this corresponds roughly to (ψ,)=(45^∘, 40^∘).Therefore,the steady motion ofalong the horizontalaxis of the diagram (as caused by the pulsar rotation) translates into the wavy thick arrow marked in the bottom part of all panels in Fig. <ref>. The observed polarisation will be dominatedby the Stokes parameters recorded along this thick wavy line. Accordingly, Fig. <ref> provides L/I (middle panel) and V/I(bottom panel) with the same wavy track as in the top panel.In agreement with the observed properties of B1913+16,at both OPM transitions (=±45^∘)the wavy line omits the region of low L/I (bright in the middle panel),staying much larger than zero for all the time.|V|/I is maximum at the OPM transitions (=±45^∘),and V changes signin the middle (=0), in the region dominated by one polarisation mode. A numerical code which convolves Gaussian distributions ofandgives the result shown in Fig. <ref>.[For each longitude Φ, i.e. for a givenand , the code simply runs over theandloops and calculates the Stokes parameters scaled by . A calculation of ψ then makes it possibleto gather the Stokesin separate arrays indexed by ψ and Φ.] It is obtained for =40^∘, σ_=70^∘ and σ_ψ,in=30^∘(parameters selected to obtain a rough by eye fit). The value ofchanges linearly as defined by the horizontal axis.This is likely unrealistic, but appears sufficient to reproduce the observed polarisation approximately (the full length of theaxis must be cast onto the pulse window of B1913+16). Modelled single pulse PA distributions(grey patches in Fig. <ref>)have the familiar form of OPM bands observed in other pulsars.In the intermode region(near =±45^∘)these bands overlap in Φ, because one wing ofdistributionextends across the intermode value.Similar result is obtained for the distribution shown with the thin line on the left marginof Fig. <ref>. This is because in both cases one wing ofreaches the modal points at =90^∘, and is close to zero.Even if this last distribution is made symmetricaround =0^∘, the PA would not change much, but V would vanish.The direction of the step-wise PA variationsin B1913+16 (Fig. <ref>) is determined by the steady motion of towards largerwith the increasing Φ. The directionof OPM transition is not accidental in the model: within the modaltransition the derivative dψ/dΦ has the same sign as d/dΦ.This tells us thatincreases monotonically within the full pulse window of B1913+16, as marked in Fig. <ref>. Although no RVM effect is included in the model, the PA curve is fairlysimilar to the observed one. The RVM PA (which may be considered constant at this stage of calculation) can be imaginedas a perfectly horizontal line at ψ=0. It can be seen that neither the average PA nor the grey single-pulse PA track follow the horizontal line.This slope is introduced by the steady motion ofacross the naturalmode at ψ=0.Therefore, in B1913+16 the slope of RVM PA is likely biased by the modecoherency effects,and is unlikely to be determined through the RVM fitting,unless a precise model of the coherent mode combination is included as a part of the RVM fitting procedure.Unlike in Fig. <ref>,the full span of the observed PA(195^∘, as noted by Weisberg & Taylor 2002) exceeds180^∘, which is the maximum possible value for the equatorward viewing geometry.This slight vertical extension is likely caused by the RVM, which only partially contributes to the PA slope observed in the middle of the profile. §.§ Other numerical examples If thedistribution is very wide (π) or centered close to the modal points (≈90^∘) then thin well-defined bands of bothOMPs extend for mostof the profile at a longitude-dependent strength ratio (see Fig. <ref>).This is because the modal pointson the P(ψ,) plane are always within the reach ofand keep to be selected byfor any .In Fig. <ref> V/I is close to zero because=40^∘ is much smaller than σ_=130^∘ (symmetry ofwith respect to =0 suppresses V). L/I hasthe profile of |cos2| withdeep minima at sharp OPM jumps.[L/I has the profile of |cos2|, becausethe input signal of amplitude E, linearly polarised at an anglewith respect toone of the natural modes, produces two waves of amplitudesE_x = Ecos, and E_y = Esin. Ifis very wide, i.e. the waves combine incoherently, thenL/I = |E_x^2 - E_y^2|/(E_x^2+E_y^2)=|cos(2)|, i.e. L/I vanishes four timesper a single revolution of(each time when the projected componentsare equal).]As soon as σ_ψ,in is increased above ∼50^∘,both sharp modes of similaramplitude are present at all Φ and there is a nearly complete depolarisation (L/I=V/I≈ 0). Overall, very wide distributions tend to depolarise for obvious reasons, however, the wide suppresses both L and V(at whatever width of ), whereas the widesuppressesV, but L only at the modal transitions.Moderately widecanproduce broad clouds of PA centered at the modal values,with the average PA slowly traversing from one mode to another.The transition may occur at a small, but non-vanishing L/I and V/I.Effects of this type are often observed (e.g. B0823+26 in Fig. 12 of MAR15,B2110+27 in Fig. 19 therein) and occur whenis crossing the intermodeand the widths of the distributions are moderate (σ_ψ,in∼45^∘, ∼45^∘, σ_∼45^∘). Otherwise, i.e. for a large ,can reach the modal points on the lag-PA plane, and the observed modal tracks become narrow. §.§ Other pulsarsTwo prototypical single pulse PA scatter plotsare schematically presented in Fig. <ref>. The top one, based on B1237+25 (Fig. 1 in SRM13), shows the PA of anM-type profile (with multiple components, Rankin 1983; Backer 1976) ran83, bac76and with a complicated PA distortion at the profile center.Similar distortions,associated withthe core component, are also observed in B1933+16 and B1857-26. Double `conal' profiles (Fig. <ref>b), on the other hand, exhibit the well defined S-shaped PA swing following a single OPMacrossthe full profile, with roughly symmetricpatches (short fragments of a PA track)of the secondary OPM in the profile peripheries. As shown in Fig. <ref>a, similar orthogonal PA patches also appear in the pulsars with the complexcore properties (e.g. B1237+25). Unfortunately, in these objectsthe complexity of the central PA distortion often makes it difficult to consistently identify a single OPM throughout the whole pulse window. In B1933+16 the loop-like PA distortion converges back at the original (strong) mode.In B1237+25, however, the core PA loop spreads from the primary mode, but appears to converge at another, secondarymode, as identified by the short PA patch marked in Fig. <ref>a.Moreover, Mitra & Rankin (2008) note that flat sections of the PA curve observed in B1857-26 imply equatorward viewing geometry for whichthe full span of the RVM PA cannot exceed π, contrary to what seems to beobserved, if the RVM is consistently attributed to the primary (strong) mode.The authors were then tempted to present a PA fit based on`unsavory assumption' that the primary (brighter) mode on the leading side continues into the `patch mode' on the right.In principle, however, such change of mode identity (with the RVMcontinuing from the primary into the secondary mode) occurs naturallyin the CMA model, whenevermoves across 45^∘ and starts to mostly contribute to the other polarisation mode.This is because for a phase lag increasing from zero, the lines of fixed- in the lag-PA diagram (Fig. <ref>)diverge upwards for ψ_in45^∘, whereas they diverge downwards for ψ_in45^∘. Thus, at an intermode the strength of the modes is exchanged and the primary mode becomes the patch mode. Such inversion of themode amplitude ratio is also explained in Fig. <ref>a.This possibility allows one to construct a naive polarisation model of pulsars such as B1237+25 and B1857-26, which resembles the generic case shown Fig. <ref>a.§.§.§ Tentative discussion of polarisation for pulsars withcomplicated core emission At the leading edge of the profile of B1237+25,thedistribution may be thoughtto be located not too far from 45^∘ (as markedon the left margin of Fig. <ref>), since both OPMs are observed(the primary mode and the secondary or patch mode) and L/I is about50%. This is caused by awing ofreaching across =45^∘.The observed zero value of V, generally expected whenis displaced from the modal value (=0 or ±90^∘),might be justified by assuming thatthe distribution is symmetric and centered at =0. With increasing longitude,moves away from 45^∘ sincethe patch mode disappears, whereas theprimary mode nearly totally dominates the observed flux (L/I≈90%).This corresponds to the peak of crossing through =0in Fig. <ref>. Before the core is reached from left(at Φ≈-2^∘ in Fig. 1 of SRM13),L/I starts to quickly decrease, because one wing of(the bottom wing in Fig. <ref>)extends across -45^∘.This time no patch of the secondary mode appears, but this might be explained by arguing thatboth modes are simultaneously present in individual samplesof single pulse emission.At the central loop-like distortion would have to cross -45^∘, which produces the OPM transition and changes the mode illumination, thus replacing the identity of modes (the bright mode now follows another RVM track offset by 90^∘). Thencould continue decreasing (as marked with the M arrow inFig. <ref>) and the patternwould repeat in a reversed order,i.e. L/I would increase, andwould approach ψ_in=-90^∘,since again one mode dominates in the data on the right hand side of the core,where L/I is very high. Finallyapproaches the intermodal=-135^∘, and the patch of the now-secondary mode lights up. In this scenario, within the profile window of `complex core pulsars',the peak ofmoves by nearly 180^∘, a valuethat may be associated with a sightline passing very closeto the magnetic pole.This interpretation may seem to be consistent withthe upper branch of the PA loop in B1237+25, which separates from the primary mode at Φ=-0.8^∘, but converges at the patch mode PA at longitude Φ=1.5^∘(see Fig. 1 in SRM13). Had it been correct, such model would imply that the PA curve of B1237+25 should be fitted withthe RVM which follows the primary mode on the left hand side, butcontinues into the patch mode on the right hand side of the profile. The `unsavory assumption' of Mitra &Rankin (2008), made for B1857-26 (Fig. 2 therein) then may be considered a realistic possibilityfor pulsars such as B1237+25 and B1857-26. With the odd number of intermode crossings within the core,the distinction of the primary mode and the secondary mode (patch mode)is meaninglessif it is supposed to reflect a single RVM track through the entire pulse window. However, this picture ignores some details of the observed core PA distortion, and the geometric origin of themotion, so itwill berectified further below.§.§.§ Tentative polarisation model for classical D-type pulsars Unlike the `complex core' pulsars, pulsars such as B0301+19 and B0525+21exhibit textbook PA variations that seem to stay in a single RVM trackthrough the whole pulse window. Interpretationbased on the motion of(Fig. <ref>)implies thatmoves within much smaller intervalin these objects.On the leading edge of their profiles, one wing ofmust extend across ψ_in=45^∘, sincethe `patch mode" and low L/I are observed there. In the profile center crosses zero, which would explain why L/I is larger in the central region(which is counterintuitive, since two profile components that overlap in pulse longitude, may be expected to depolarise each otherat the center). At the trailing sidewould approach -45^∘, again feedingthe existence of unequal amounts of two OPMs.The range of traversed ψ_in is smaller than 90^∘, which may seem consistent withapassage of sightline at a larger distance from the magnetic pole.The primary and secondary modes do not have their identity replaced in D-type pulsars,so the traditional RVM fitting may be applied. This said, it must be noted that alternative interpretation of D pulsarsis possible at least for L/I,because L/I is symmetric with respect to =0on the lag-PA diagram. Accordingly,may move from the vicinity of 45^∘ towards some value close to zero, and then retreat towards the original position.In this casechanges nonmonotonically andhas a minimum in the middle of the profile. The resulting linearpolarisation isnearly identical to the case with the steadily increasing .Section <ref>, which derives the motion offrom the magnetospheric geometry, implies yet a different variations of.§.§ Core PA distortion in B1933+16 The core PAdistortions look like fast modal transitions whichhave got reversed, or failed to be completed. As can be seen in Fig. <ref>, any manipulations with (displacements ofalong theaxis) do not allow usto leave the single mode space between the consecutive intermodes. On the other hand,a single passageofthrough one ofthe intermodes (at ±45^∘ or ±135^∘)produces both the orthogonal transition of PAand a change of the modal amplitude ratio,thus redefining which mode is primaryand which is secondary.Still, such replacement of the mode illuminationrequires an odd number of intermode crossings, whereasthe core PA loops have the `up-and-down' form, which suggests an even number(on the way to another mode and back). Even in B1237+25 the odd-numberedtransition of the core PA (from the primary to the secondary mode)is followed by one more PA transition back to the primary mode, at a trailing-sidelongitude where the core emission seems to cease and subdue to the peripheric emission. In B1933+16 the core loop opens and closes at the same (say primary) mode, so it will now be interpreted in terms of thereversed mode transition with no mode identity replacement, ascaused by nonmonotonic variationsof . On the left side of the PA loop in B1933+16(see Fig. 1 in MRA16)one polarisation mode dominates which means that ∼0 andthe observed ψ, V/I, and L/I can be representedby the values at the median line of Fig. <ref> (assuming similar distribution for B1933+16 and B1913+16).Let us assume that within the loop of B1933+16 changes with Φ nonmonotonically,in the way which is presented by the backward-bent arrow in Fig. <ref>b. This qualitatively reproduces the majorobserved properties of the loop:while moving leftward L/I decreases, |V|/I increases, and the average PAalmost makes the full OPM transition. Shortly after passing through the minimum in L/I (at =-45^∘), starts to increase, immediately passing again through the L/I minimum, and|V| starts to drop.Numerical simulation of such case is shown in Fig. <ref>,where simple linearchanges ofwere asumed ( in radians is shown in the bottom panel with a dashed line).Despite such linearvariations of are far from those expected at an even-numbered intermode crossing in pulsar magnetosphere(see Section <ref>) they produce polarisation similar to that observed. In particular,the modelled minima in L/I have the characteristic twin-like look as in the data: they are close to each other,and connected with only a slightly increased L/I in between them. The sharp tip at the center of L/I and the slanted PA on both sides of the V-shaped PA feature(Fig. <ref>) are artifacts of the linear. The V-shaped distortion of Fig. <ref> does not extend beyond Δψ = 90^∘ hence it does not have the loop-shaped form.The observed shape may be caused by a more complicated structureofthan the Gaussian form assumed above.However, the observed loop can also be interpreted purely in terms of displacement, with a fixed∼45^∘ within the entire feature. The PA curve distortion is then bidirectional, i.e. the primary PA track bifurcatesinto a loop. §.§.§ Longitude-dependentand the loop of B1933+16Ifis fixed at a value close to 45^∘ (=41^∘ inFig. <ref>) then the loop-like bifurcation of the PA trackcan be interpreted through the motion ofalong theaxis.When the fullinterval (360^∘)is cast onto some Φ interval,then the loop of Fig. <ref> is formed.The loop appears because the PA track nearly follows the disconnected sections of the broken intermodal PA (thick in the bottom panel of Fig. <ref>).Since < 45^∘, with the increasing most of input power (in ) is displaced down,towards the lower intermode, whereas a part of thewing (where> 45^∘) follows the upper branch of the loop.At the proper modes (ψ=0 or 90^∘, =90^∘ or 270^∘) the PA track is strongly enhanced[The blankbreaks at the proper modes of Fig. <ref>result from a limited resolution of the calculation.] and narrow. This results from the fixed orientation of the assumedintervening polarisation basis, which in reality can fluctuate. Thiswould reduce the modal enhancement and make the PA track at the modal pointswider.Fig. <ref> does not reproduce the 1.5 GHz PA loop observedin B1933+16,because V>0 on the leading side (see. Fig. 1 in MRA16).However, a nonmonotonic change of within the loop such as marked with the backward-bent arrowin Fig. <ref>b, produces the result of Fig. <ref>, which is qualitatively consistent with data. In particular, the twin minima in L/I appear in the middle of the intermodal transitions(i.e. coincident with the proper modes) and V/I stays negative throughout the loop. The maxima of |V| coincide with the minima in L/I, as observed on the leading side of the loop in B1933+16.At a higher frequency ν=4.5 GHz the upper branch of the loop disappears(Fig. 1 in MRA16), the amplitude of the V-shaped average-PA distortiondecreases, L/I increases, whereas V/I decreases. All these properties are qualitatively reproducedwhen the misalignment offrom the intermode is increased at the larger ν. This can be seen inFig. <ref> which presents the result for =31^∘.The lag-based intermodal split of Fig. <ref>is then a successful model, capable of reproducing several observedproperties at both frequencies. The nonmonotonic increase and drop of || suggests a passage of a signal through a stream or stripe of intervening matter, whereas the change ofwith νcould be associated with the ν-dependent emission altitudeor PLR altitude. However, the model is more complex thanthe -based model of Fig. <ref>. Namely, if the primary mode observed just left of the loop in B1933+16results from the intervention of some PLR polarisation basis, thenanother intervening basis, offset by about 45^∘,is required within the loop. Therefore, in the following material I rather focus on the motionofand on the subject of intermodecrossing in the central part of the profile.§.§ Magnetospheric picture Previous sections suppose and imply that the input PA distributionrotates with respect to some intervening polarisation basis. A possible interpretation of this is to place the intervening basis at the PLR andorient it along the sky-projectedlocal magnetic field. The maximum of , on the other hand, is attributedto the projected directionof magnetic field line planes in a low-altitude emission region. When the emitted beam is propagating outwards through the rotating magnetosphere,it bends backwards in the reference frame whichcorotates with the star (Fig. <ref>).With the increasing radius r, dipole axismoves out of the beam in the forward direction. This suggests the structure of two displaced B-field line patterns (Fig. <ref>), one of which represents the symmetry at the low altitude of emission (radial distance r_em), whereas the other – at the PLR. Lines in both these patterns rotate at different rate while they are probed by the horizontallypassing line of sight.As marked in the figure, the input polarisation angle ψ_inis determined by the angle at which the lines in these patterns cut each other at a point selected by the line of sight.The material of previous sections implies that the average PA observed at a given Φ usually coincides with the OPMs of the intervening basis, hence the observed RVM curves must be attributed to the PLR. The low-r emission,on the other hand, is only backliting the PLR polarisation basis. The polarisation direction of the low-r emission determinesthe flux ratio of observed modes,e.g. it can produce the orthogonal mode transitions, but otherwisedoes not affect the observed PA value.Fig. <ref> suggests that the PLRline structure should be positioned on the left hand-side (the leading side) of the emitted beam. However, this would move the RVM PA curvestowards the leading side of profiles, in conflict with most observations (Blaskiewicz et al. 1991;Krzeszowski et al. 2009). bcw91, kmg09 Therefore, below I risk the assumption that dynamics of plasma at the PLRis influenced by noninertial effects of corotation,in such a way thatthe effective `B-field line' structure[Actually the structureof electron trajectories in the observer reference frame,see Fig. 2 in Dyks et al. (2010), cf. Blaskiewicz et al. (1991), Dyks (2008), and Kumar & Gangadhara (2012). dwd10, d08, bcw91, kg12]is displaced rightwards with respect to the sky-projected structure of the low-r B-field. To gain a quick insight into the phenomenon, a flat geometry of Fig. <ref> is assumed (instead of the spherical). If the radiobeam center(and the center of an observed pulse profile) is placed at Φ=0, thebeam and PLR polarisation angles are:ψ_bm=arctan(β/Φ) ψ_plr=arctan(β/(Φ+Δ_plr)),where β=ζ-α is the impact angle i.e. the angle of closest approach of sightline to the magnetic pole, and Δ_plr is the angular displacement between the poles at r_em and PLR. The input PA is equal to=ψ_plr-ψ_bmand is shown in Fig. <ref> for a set of β values ranging between0.1Δ_plr and 5Δ_plr. As might be expected, the curves keep close to =0at large |Φ| andare symmetric with respect to the midpoint betweenthe beam center at Φ=0 and the PLR symmetry pointat Φ=Δ_plr=5^∘.For a nearby pole passage,follows the upper curves (hereafter I refer to the curves' position at the midpoint Φ), which extend vertically for a large fraction of π. Some of them cut the intermodes (at 45^∘ or 135^∘)twice or four times. For β = 0.2Δ_plr (second line from top)the top intermode (at =135^∘) is approached from below with a subsequent retreat. For a more distant polarpassage,can approach the 45^∘ intermode from below,or stay close to zero throughout the profile.In the extreme case ofβ≪Δ_plr, the geometry shown in the bottom of Fig. <ref> holds, with≈π-ψ_bm, i.e. increases monotonically.In the last case the intermodes (at =45^∘ and 135^∘)are cut twice at two longitudes located symmetrically on both sides of Φ=0. This resembles the case of B1913+16. If the location of r_em (or PLR)changes withthe observation frequency ν, then Δ_plr can changewhich leads to an increase (or a decrease) ofconsistently at alllongitudes (dotted lines in Fig. <ref>). The black portions of lines in Fig. <ref> mark whichparts of thecurves are detectable, if the radio emission is limited to a dipole-axis-centred cone with a half-opening angle ρ=0.5, 1, and 3Δ_plr (panels a, b, and c, correspondingly).As can be seen in Fig. <ref>, the detectable variations of can either bemonotonic or have a maximum located asymmetrically in the profle. When the PLR displacement is small (Δ_plr≪ρ, Fig. <ref>c), fast symmetric changes ofare located in the center of the profile.Even for the large displacement of the PLR line structure (Δ_plr=5^∘ in Fig. <ref>),the tendency to cross intermodes near the center of the profile is clearly visible. If Δ_plr is five times smaller (and equal to 1^∘)the curves are horizontally compressed as shown in Fig. <ref>,which puts all the intermodecrossings really near the profile center (in the `core' region). The passage of sightline which is sufficiently close to thepole to cross at least the lower intermode is then less likely.A simple calculation shows that (Φ) crosses the lower intermode (at =45^∘) two times for β<0.5tan(3π/8)Δ_plr=1.21Δ_plr.Both intermodes (at 45^∘ and 135^∘) are crossedwhen β<0.5tan(π/8)Δ_plr=0.21Δ_plr (in which casethere may be up to four detectable crossings). Thus, to produce a complicated,loop-like PA feature in the profile, the line of sight must pass at about the same distance from the poleas Δ_plr (or closer). When β1.2Δ_plr, the increasingapproaches the lower intermode and drops to the original low value without the crossing (5th line from bottom inFigs. <ref> and <ref>).Moreover, if the angular radius ρ of the emission coneis much larger than Δ_plr(Figs. <ref>c and <ref>c) then the number of intermodecrossings is even, which implies no replacementof the mode illumination or identification in the profile peripheries. Therefore, the presence of the PA loop under the core component doesnot necessarily mean that the primary mode on one side of the profileshould continue into the secondary (patch) mode on the other side. However, for the bottom black casesin Fig. <ref>a and b,such replacement of the mode identification does occurat the lower intermode crossing near the center of the pulse window. §.§ Distribution displacement vs widening If theprofiles of Fig. <ref> are considered as a valid model for the motion of across the lag-PA diagram of Fig. <ref>, thena problem appearswith the above described interpretation of D-type profiles. Namely, the symmetric PA patches in the outer profile region (see Fig. <ref>b) have been attributed to the proximity oftothe intermodes, whereas for the distant polar passage approaches ∼ 45^∘ in the middle of the profile. A doubletraverse through 45^∘ matches the behaviour of B1913+16 (with its `another' mode at the center), but not B0301+19, or B1133+16.The magnetospheric interpretation (Figs. <ref> and <ref>) suggests thatshould be close to zero in the profile periphery,and closer to 45^∘ in the center. A possible way out is to assume that β≫Δ_plr in D type pulsars, which is consistent with the peripheric viewing geometry.Thenfollows the lowest curves shown in Fig. <ref>, staying close to zero throughout the entire profile.Thedistribution is then pinned at ≈0 and a methodto produce the peripheric secondary mode is needed. A natural solution isto assume thatbecomes much wider in the profile periphery. Both wings ofextend across both intermodes (at ±45^∘)thus producing the peripheric patches of the secondary mode. This interpretationis supported by the single pulse observations of D pulsars, which demonstrate intense clouds of PA samples filling in the space betweenthe peripheric OPM tracks (HR10; Young & Rankin 2012;MAR15). hr10, yr12, mar2015 Most importantly,in the PA distributions observed at a fixed Φ,the modal peaks are bridged with wings that seem to extend at equal strengthtowards both the larger and smaller PA.This is a signature of a broadenningof the incidentdistribution, not a displacement.Similar effects are involved in the peripheric double mode PA tracksobserved in the M type pulsars. The widening of the incidentis revealed by thick PA blobs centeredat the PA track of the primary mode in B1237+25(at Φ=-4.7^∘ and 3.7^∘ in Fig. 1 of SRM13, also shown with bullets in Fig. <ref>a).The incidentdistribution must become wider right at these longitudes. Closer to the profile edgethe primary mode trackbecomes apparently thin again, most likely becausehasbecome even wider and the power of the primary modeis leaking into the secondary mode.[Hence only the thin topmostpart of the primary track leaves a visible trace in the figure.] It should be noted that diversity ofvaluesat a given Φ can spread evena single inputvalue into a wide PA distribution (see Fig. <ref>), however,the phase lags alone usually cannot make the secondary mode to appear.[Unless the intermodes are misidentified as the natural modes, which might happen for thecase shown in Fig. <ref>.A widening ofcan also switch on the secondary mode trackif ∼0 andreaches across ±45^∘, cf. Fig. <ref>.]Therefore, the increase of the observed PA distribution, noted already by SRM13, likely needs that the intrinsic becomes wider.The core PA distortions may then be also associatedwith a fast increase of , as evidenced by the low L/I. Fig. <ref> shows thatis really likely to shoot upin the center of the profile, which directly contributes to the increaseof . The fast motion ofand the large are then positively corellated. It is then found that the widthof the incidentdistributionis an important parameter which can vary strongly across the pulse windowand influence the PA observed in single pulse data. §.§ Polarisation of the core emissionThe tools of previous sections may now be used to interpretthe polarisation of the core component in PSR B1237+25. At Φ=-3^∘ in the profile of B1237 (see Fig. 1 in SRM13)L/I is almostequal to 1 and V/I is negligible which means thatthe observed radiation is totally confined in a single polarisation mode, with ∼ 0 andnot extending beyond ±45^∘.This corresponds to the locus (,)=(0^∘,45^∘) on the lag-PA diagram of Fig. <ref>. While moving towards the core, i.e. rightwards in Fig. <ref>,at Φ≈ -2^∘ (in Fig. 1 of SRM13), the observed L/I decreases and V/I increases in roughly reciprocal relation, becausestarts to deviate from =0, as implied by Fig. <ref>(the (Φ) curves are more clear in Fig. <ref>,so below I refer to the third from top case in Fig. <ref>b). The anticorrelation of L/I and |V|/I is characteristic of the coherently combined modes,since the minima of L/I clearly overlap with the maxima of |V|/I in the whole parameter space (compare the locations of the “oval" contoursin the two bottom panels of Fig. <ref>). At Φ≈ -0.9^∘,approaches the intermode crossing(∼45^∘) andbecomes wider,so L/I nearly vanishes,[In the context of the CMA model,the following statement fromSRM13: `the deep linear minimum just prior to the core shows that the core and the [adjacent]emission represent different OPMs' is understood as astatement about components of a single incident polarisation vector,i.e. only involves a single input polarisation mode.]and the top branch of the PA loop detaches from the primary PA track (or the primary PA track bifurcates, see Sect. <ref>).The large width of thedistribution keeps the L/I at a low level throughout the entire core.At Φ=0, i.e. slightly on the left side of the core maximum,the peak ofpasses throughthe natural mode at =90^∘ which produces the sign change of V,and the slight increase of L/I in between the twin minima.In Fig. <ref>a this corresponds to the passage of the `a' branchof the core loop through the dotted RVM curve of the secondary polarisation mode. Then, while following the third from top curve in Fig. <ref>b,keeps increasing up to a maximum value (larger than 90^∘ but smaller than 135^∘), which is reached at the minimum observed V (i.e. at the maximum amplitude of the negative V).So far we can imagine that we moved horizontally across the ovalcentered at (, )=(45^∘, 90^∘) in Fig. <ref>,and that we stopped near the tip of the wavy arrow. According to the third from top line in Fig. <ref>b, is now decreasing towards the vicinity of the 90^∘ mode,i.e. V/I comes back to zero, and the distortedPA track (marked `b' in Fig. <ref>a) approaches the RVM curveof the secondary (patch) mode. At this longitude the core emission appears to cease, consistentlywith the third from top case in Fig. <ref>b. Therefore,the average observed PA leaves the loop-shaped distortion andjumps to the primary mode of the peripheric (“conal") emission. Thus, the mode identity replacement in B1237+25occurs only within the core component, and does not propagateonto the peripheric emission. The primary mode on both sides of the profilemust then follow the same RVM track. A full derotation ofback to 0^∘ wouldproduce a positive V on the right-hand side of the core. This would create a symmetric feature shown in Fig. <ref>, with two humps of positive V separated by a negative V in the middle.However, in B1237+25 the core beam is narrow in comparison to Δ_plr and the right hand side of the (Φ) profile is missing,as implied by Figs. <ref>b and <ref>b.Such interpretation is supported by the polarisationprofiles of B1541+09 at 430 MHz, and B1839+09 at 1.4 GHz (see Figs. 8 and 9 in HR10) which exhibit the full symmetric V profile,with the negative V surrounded by the positive maxima of V.§.§.§ The abnormal mode branch of the core loop in B1237+25It is not clear ifbifurcates on the left side of the core PA loop in B1237+25,because the bottom branch of the loop (marked `c' in Fig. <ref>a) may represent the primary OPM track.Moreover, the `a-b' and 'c' branches of the loop dominatein different profile modes (normal N, and abnormal Ab, SRM13),i.e. the bifurcation may be considerednonsimultaneous. The enhancement of the `c' branch in the Ab modeis associated with a strong change of the profile shape: the fourth component(counting from left) disappears,and new emission appears betweenthe second component and the core (see Fig. 6 in SRM13).In terms of the CMA model, the downward deflection of the 'c' branchin the Ab mode requires thatmoves in the opposite directionthan in the N mode, i.e. towards the smaller . According to Fig. <ref>, this should reverse the sign of V, which is not observed.Therefore, the change of themotion directionshould be accompanied by a change of the sign of , i.e. shouldbecome displaced to the other side of =0.[It is possibleto interpret this phenomenonin terms of the spiral radio beam geometry (Dyks 2017). dyk17 A change of sign of the electric field in the polar region would change thedirection of the E⃗×B⃗ drift, so the spiral wouldrevolve in the opposite direction, while possibly beinganchored at the same point.This could displace the emission of the 4th component to the spacebetween the component no. 2 and the core. Simultaneously, charges of opposite sign (than in the N mode) could be accelerated and emittingtowards the observer. The change of the spiral geometry, as drivenby the reversal of the electric field, would then produce the profile mode change, or, in general, a change in drifting or fluctuatingsingle pulse properties.] §.§ Separation of profiles into OPMs The coherent nature of the observed non-RVM PA distortions impliesthat the incoherent separation of pulsar emission into OPMs, which is based onRVM fits, may not give meaningfull results in the presence of strong PA distortions. Moreover, since the flux ratioof the OPMs mostly reflects the relative magnitude of components ofthe input polarisation vector (and the width of ),the intensity profiles of separated OPMs tell little on the OPMs themselves. McKinnon & Stinebring (2000) provide other argumentsagainst the modal separation of profiles. §.§ PA distortions caused by a complicated magnetic field The complicated loop-like distortions of the core PAare then a result of fast and multipleintermode crossing that occurs when the sightline is passing near the pole, and the projected B field rotates with respect to thePLR basis (which is also rotating).Similar fast variations ofmay be expected when the magnetic field in the radio emission region has a complicated multipolar structure(e.g. Petri 2017). pet17The CMA model implies that such multipolar distortions of Bdo not leave a direct imprint of this local B in the observed PA curves. As in the core case,the multipolar B barely changes the illumination of the PLR basis, sothe resulting PA distortions appear as transitions between(or departures from) the RVM tracks. These `transitional' distortions are caused by the intermode crossing and reflect the relative orientation of B in the emission and PLR regions, not theabsolute direction of B in the multipolar emission region. The numerousdistortions of a single PA curve observed in millisecond pulsars (e.g. J0437-4715) may be caused by such multipolar B. §.§ The 45^∘ PA jumps in weakly polarised profiles A rare but striking polarisation effect is presented bypulsars B1919+21 (Fig. 18 in MAR15) and B0823+26 (Fig. 7 inEverett & Weisberg 2001),ew2001 in which the PA makes a 45^∘ jump and follows a continuousPA curve associated with a very weakly polarised emission. The low L/I suggests thatis close to 45^∘ or thatis very wide. The first case is shown in Fig. <ref>, in which=40^∘ and σ_ψ, in=30^∘. At the profile edges in B1919+21,must be wide enough toreach the modes at =90^∘, as suggested by the orthogonal PA patches observedat Φ≈-1.5 in Fig. 18 in MAR15.This is presented in the top panel of Fig. <ref>,where σ_=90^∘. Sinceis not perfectly aligned with =45^∘, one of the OPM spots is slightly stronger, which defines which OPM is observed as the averagePA.In the middle of the profile of B1919+21, becomes narrower, as shown in the bottom panel of Fig. <ref>. Therefore, the intrinsic (input) PA distribution (also shown with the line thickness at =0) dominates there. Sincepeaks near 45^∘, the average PA offset by ∼45^∘is observed, and the single pulse samples reveal the variety of incident PAsin the intrinsicdistribution. The `dirty', erratic single pulse PAobserved in the inner parts of B1919+21 profile must then representthe intrinsic (incident) PA, or at least be closer to the intrinsic PA than in the profile edges where the OPMs dominate.Needless to say, the 45^∘ jumps also appear whenstays narrow,but moves from ∼0 to ∼90^∘(as in Fig. <ref>).The transition from the wide to narrowshould in general be associated with a change of L/I at the 45^∘ jump.This is because for the wide(top in Fig. <ref>)the two OPM spots of comparable strength produce nearly total depolarisation,whereas for the narrow(bottom) L/I has the intrinsic value,which can be large ifis narrow. The magnitude of L/Iobserved on both sides of the 45^∘ jump in B0823+26 (Fig. 7 in Everett & Weisberg 2001,Φ=-2^∘) is indeed very different.If, on the other hand, the incidentis wide (and centred close to 45^∘), then L/Iis low on both sides of the 45^∘ jump. This must be the caseof B1919+21 which seems to have low L/I both at the edges andin the middle of profile.This interpretation suggests that thewidening ofin the profile peripheries contributes to the `edge depolarisation' observed in pulsar profiles (Rankin & Ramachandran 2003). rr03§.§ PA jumps of arbitrary magnitude Whenis not close to 45^∘, the fast narrowing of produces PA jumps of arbitrary magnitude. These should be associated with a generally larger L/I (than in the 45^∘ case). §.§ Frequency dependent profile depolarisation Since the refraction coefficient (hence the speed)of a modal wave likely dependson the frequency ν, the phase lag distribution in Fig. <ref> may be expected tomove horizontally (or change width) with varying ν. This may be relatedto the observed ν-dependent profile depolarisation. However, early venturesinto the interpretation of the observed ν-dependent effects,suggest that the distribution ofstrongly influences the lookof profile polarisation at different frequencies. This subject is deferred to future work. § SUMMARY It has been found that the observed pulsar OPMs and the strong circular polarisation, are the statistical result of coherent additionof waves in two natural propagation modes. Precisely, the modes appearbecause the distribution of phase lags of combining waves extendsup to =90^∘. The combiningwaves represent thecomponents of a single incident signal, which at lower altitudesmay well be in a pure linearly polarised state.The pulsar emission mechanism (such as the extraordinary mode curvature radiation, see Dyks 2017) dyk17may then emit a linearly polarised radiation which initiallypropagates in a single mode (Melrose 2003). The production of OPMs and V is indeed a propagation effect. The original (emitted) signal is just illuminating thepolarisation basis of intervening matter at a higher altitude. The observed RVM-like variations of PA mostlyreflect the orientation of magnetic field in this high-altitude interveningregion.The orientation of B-field in the emission region, on the other hand,barely determines the ratioof flux in both observed modes(the emitted radiation is illuminating the PLR B-field structure from below).The polarisation characteristics that result from such mode originare completely different from the properties of incoherently summedelliptically polarised natural modes. In the coherent case, a vector split at the angle of =45^∘ generates strong nonzero V (and in general a nonzero L/I, depending on the modal phase lag). The change of V sign occurs at a mode maximum intensity (in general at large L/I). L and V areanticorellated (unlike in the incoherent case). As shown in previous sections, these characteristics are consistentwith numerous observed properties of pulsar polarisation. However, the CMA does not exclude effects governed bythe standard incoherent summation of elliptically polarised natural modes(the incoherent summation corresponds to the widecasesdiscussed in this paper, but the ellipticity of the proper modes has not yet been included).Despite the suggestive look of the antisymmetric V at core components,(which in the incoherent case would imply an OPM transition),it has been found that the change of V sign is instead associated withthe alignment of incidentwith one of the modes. The minimum in L/Iis indeed observed to coincide with peaks in |V|/I, not with the sign change. The generally low L/I within the entire corehas its origin in wide or multiply peakeddistribution. The antisymmetry of V/Iis consistent with the model provided thatthe symmetry point of the PLR polarisation basis is displacedrightwards with respect to the core. For a negligible displacement, the symmetric V profile is expected,as indeed observed in some objects. The characteristic symmetry of the observed PA pattern, with thepatches of the secondary mode in the profile peripheries,can be mostly attributed to the broadening of the input PA distribution in the profile peripheries.Because of the `PLR basis illumination', the ratio of power in the observed orthogonal modes,as defined by their apparent location at presumed RVM tracks,is inversed whenevercrosses the intermode at ±45^∘. An odd number of crossings makes the impression that the identity ofprimary/secondary modes has been replaced.This happens within the core of B1237+25, but at the vanishingtrailing edge of the core, the mode designation is set back to original, through the orthogonaljump to the dominating primary mode of the “conal" emission. There has been a tendency among pulsarists to interpretvarious kinks in PA curves as a result of adjacent components being emitted from different altitudes, or in terms of longitudinal polar currents. Many of those distortions may in fact result from coherent mode combination.For example, the tiny distortion of the average PA curve under thecomponents C_3 and C_4 in J1024-0719 (Fig. 5 in Craig 2014,longitude interval[-50^∘,-20^∘]), is clearly coincident with an increase in Vcra2014 which suggests the coherent origin.Incoherent separation of intensity profiles into OPMs,as defined by an RVM fit to the average PA data may not produce a correct result in the presence of strong distortions, since the latter have coherent origin. It has been shown that the empirical approach to thecoherent mode addition is the missing ingredient needed to understand numerous polarisation properties of radio pulsars.With the coherency allowed, it is possible tocomprehend polarisation effects that are beyond the reachof the RVM model with the incoherently added modes. § ACKNOWLEDGEMENTSI thank Joel Weisberg for permission to reproduce the figure showing B1913+16. I am grateful to Adam Frankowski fordetailed comments on the manuscript and discussions.I appreciate discussions with Bronek Rudak. This work was supported bythe National Science Centre grant DEC-2011/02/A/ST9/00256. mn2e | http://arxiv.org/abs/1705.09238v1 | {
"authors": [
"J. Dyks"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170525155004",
"title": "The origin of radio pulsar polarisation"
} |
cm12 =2500 plain thmTheorem[section] *thmaTheorem A *thmbTheorem B *thmcTheorem C prop[thm]Proposition *propaProposition A lemma[thm]Lemma claim[thm]Claim cor[thm]Corollary *corcCorollary C fact[thm]Fact subsec[thm] summary[thm]Summarydefinition defn[thm]Definition example[thm]Example examples[thm]Examples notation[thm]Notation assume[thm]Assumptionsremark remark[thm]Remark aside[thm]Asidemyeq[1][] thm #1 mysubsection[2][] #2.#1 mysubsect[2][] #2.#1Dept. of Mathematics, U. Haifa, 3498838 Haifa, [email protected] Dept. of Industrial Engineering, Dept. of Mechanical Engineering Ariel U., 4076113 Ariel, [email protected] 70G40; Secondary 57R45, 70B15 We construct a completed versionof the configuration space of a linkage Γ in [,]3 which takes into account the ways one link can touch another. We also describe a simplified versionwhich is a blow-up of the space of immersions of Γ in [.]3 A number of simple detailed examples are given. Configuration spaces of spatial linkages: Taking Collisions Into AccountNir Shvalb December 30, 2023 ========================================================================= § INTRODUCTION A linkage is a collection of rigid bars, or links, attached to each other at their vertices, with a variety of possible joints (fixed, spherical, rotational, and so on). These play a central role in the field of robotics, in both its mathematical and engineering aspects: see <cit.>.Such a linkage Γ, thought of as a metric graph, can be embedded in an ambient Euclidean space d in various ways, called configurations of Γ. The spaceof all such configurations has a natural topology and differentiable structuresee <cit.> and Section <ref> below.Such configuration spaces have been studied extensively, mostly for simple closed or open chains (cf. <cit.>; but see <cit.>). Inthe plane, the convention is that links freely slide over each other, so that a configuration is determined solely by the locations of the joints. This convention is usually extended to spaces of polygons in 3 (see, e.g., <cit.>), so they no longer provide realistic models of linkages (since in this model the links can pass through each other freely). Alternatively, some authors have studies spaces of embeddings of sets of disjoint lines in [,]3 for which the issue does not arise (cf. <cit.>).The goal of this paper is to address this question for spatial linkages, by constructing a model taking into account how different bars touch each other. We still use a simplified mathematical model, in which the links have no thickness, and so on. Our starting point is the spaceof embeddings of Γ in [.]3 If we allow the links to intersect, we obtain the larger spaceof immersions of Γ in [.]3 However,disregards the fact that in reality two bars touch each other on one side or the other. To take this into account, we construct the completed configuration spacefromby completing itwith respect to a suitable metric. The new points of ∖ are called virtual configurations: they correspond to immersed configurations decorated with an additional (discrete) set of labels. See Section <ref>.Unfortunately, the completed configuration spaceis very difficult to describe in most cases. Therefore, we also construct a simplified version, called the blow-up, denoted by [.] Here the labelling is described explicitly by a finite set of invariants (see Proposition <ref> below), called linking numbers, which determine the mutual position of two infinitely thin tangent cylinders in space (cf. <cit.>). See Section <ref>.One advantage of the blow-up is that its set of singularities can be filtered in various ways, and the simpler types can be described explicitly. See Section <ref>.The relation between these spaces can be described as follows:[] @^(->[rr] @->>[rr] @->>[rr] . The second half of the paper is devoted to the study of a number of examples:*The simplest “linkage” we describe consists of two oriented lines in [.]3 In this case [,]=and the configuration space is described fully in <ref>.A. *More generally, in Section <ref> we consider a collection of noriented lines in space, and show that its completed configuration space ishomotopy equivalent to that of a linkage consisting of n lines touchingat the origin.We give a full cell structure forwhen n=3 in Section <ref>. *Finally, the case of a closed quadrilateral chain is analyzed in <ref>.A and that of an open chain of length three in <ref>.B. § CONFIGURATION SPACES Any embedding of a linkage in a (fixed) ambient Euclidean space d is determined by the positions of its vertices, but not all embeddings of its vertices determine a legal embedding of the linkage. To make this precise, we require the following: An linkage type is a graph [,]=(V,E) determined by a set V of N vertices and a set E⊆ V^2 of k edges (between distinct vertices). We assume there are no isolated vertices.A specific linkageΓ=(,)of typeis determined by a length vector[,]:=(ℓ_1,…,ℓ_k)∈E_+ specifying the length ℓ_i>0 of each edge (u_i,v_i) in E[.]i=1,…,k Thisis required to satisfy the triangle inequality where appropriate. We call an edge with a specified length a link, (or bar) of the linkage Γ, and the vertices of Γ are also known as joints.An embedding ofin the Euclidean space d is an injective map :V→d such that the open intervals ((u_i),(v_i)) and ((u_j),(v_j)) in d are disjoint if the edges (u_i,v_i) and (u_j,v_j) are distinct in E, and the corresponding closed intervals [(u_i),(v_i)] and [(u_j),(v_j)] intersect only at the common vertices. The space of all such embeddings is denoted by [;]() it is an open subset of [.](d)^VWe have a moduli function[,]λ_:(d)V→[0,∞) written [,]↦λ_():E→[0,∞) with (λ_())(u_i,v_i):=(u_i)-(v_i) for [.](u_i,v_i)∈ E We think of Λ:=(λ) as the moduli space for [.]The immersion space of the linkage Γ=(,) is the subspace :=λ_^-1() of [.](3)V A point ∈ is called an immersed configuration of Γ: it is determined by the condition (u_i)-(v_i)=ℓ_ifor each edge (u_i,v_i)in E .Finally, the configuration space of the linkage Γ=(,) is the subspace :=∩() of [.]() A point ∈ is called an (embedded) configuration of Γ. Since () is open in [,](d)V is open in [.]We may also consider linkage typescontaining lines (or half lines) as “generalized edges”[:]e∈ E in this case we add two (or one) new vertices of e to V, in order to ensure that any embedding ofin d is uniquely determined by the corresponding vertex embedding [.]:V→dThe simplest kind of connected linkage is that of k-chain, with k edges (of lengths [),]ℓ_1,…,ℓ_k in which all nodes of degree ≤ 2.If all nodes are of degree 2, it is called a closed chain, and denoted by [;]k otherwise, it is an open chain, denoted by [.]k Isometries acting on configuration spaces The group d of isometries of the Euclidean space d acts on the spaces () and [,] and the action is generally free, but certain configurations (e.g, those contained in a proper subspace W of [)]d may be fixed by certain transformations (e.g., those fixing W) (see <cit.>).Note in particular that we may choose any fixed node u_0 as the base-point of Γ, and the action of the translation subgroup T≅d of d on (u_0)is free.Therefore, the action of T onis also free. We call the quotient space :=()/T the pointed space of embeddings for [,] and :=/T the pointed configuration space for Γ. Both quotient maps have canonical sections, and in fact ()≅×d and [.]≅×d A pointed configuration (i.e., an element [] of [)] is equivalent to an ordinary configurationexpressed in terms of a coordinate frame for d with (u_0)= at the origin.If we also choose a fixed edge (u_0,v_0) instarting at [,]u_0 we obtain a smooth map p:()→ S^d-1 which assigns to a configurationthe direction of the vector from (u_0) to [.](v_0) The fiberof p at _1∈ S^d-1 will be called thereduced space of embeddings of [,] and the fiberof p at 1 will be called the reduced configuration space of Γ. Note that the bundles ()→ Sd-1 and → Sd-1 are locally trivial. § VIRTUAL CONFIGURATIONS The spaceof immersed configurations can be used as a simplified model for the space of all possible configurations of Γ. However, this is not a very good approximation to the behavior of linkages in 3-dimensional space. We now provide a more realistic (though still simplified) approach, as follows: Note that since j:()→(3)V is an embedding into a manifold, it has a path metric: for any two functions ,':V→d we let δ'(,') denote the infimum of the lengths of the rectifiable paths fromto ' in () (and δ'(,'):=∞ if there is no such path). We then let [.](,'):=min{δ'(,'),1} This is clearly a metric, which is topologically equivalent to the Euclidean metric on () inherited from (3)V (cf. <cit.>).The same is true of the metricrestricted to the subspace(compare <cit.>). Since any continuous path γ:[0,1]→() has an ϵ-neighborhood of its image still contained in [,]()⊆(d)V by the Stone-Weierstrass Theorem (cf. <cit.>) we can approximate γ by a smooth (even polynomial) path γ̂. Thus we may assume that all paths between configurations used to defineare in fact smooth. We define the completed space of embeddings ofto be the completion () of () with respect to the metric(cf. <cit.>). The completed configuration space of a linkage Γ is similarly defined to be the completion of the embedding configuration spaceswith respect to [.] The new points in ∖ will be called virtual configurations: they correspond to actual immersions of Γ in d in which (infinitely thin) links are allowed to touch, “remembering” on which side this happens.The spaces we have defined so far fit into a commutative diagram as follows:[] @^(->[r] @^(->[d]@^(->[r] @^(->[d]@^(->[r] @^(->[d] @/_1.3pc/[l]@->>[r] [d]@^(->[d]@^(->[r]@^(->[r]() @^(->[r] @/^1.0pc/[l]() @->>[r] (3)V. The pointed and reduced versions [,][,][,]) and [,] are defined as in <ref>, and fit into a suitable extension of [.]eqrelconfspNote that the moduli function λ_:(V)→E of <ref> extends to [,]λ̂:()→E and in factis just the pre-image λ̂() for the appropriate vector of lengths . Even though the metricis topologically equivalent to the Euclidean metric d_2 on [,]() its completion with respect to the latter is simply [,](d)^V so the corresponding completion ofis the space of immersed configurations of <ref>. In fact, from the properties of the completion we deduce: For any linkage Γ of type [,] there is a continuous map [.]q:→The map q is a quotient map, as long asis dense in [.]This may fail to hold if Γ has rigid non-embeddedimmersed configurations, which are isolated points in [).] In such cases we add the associated “virtual completed configurations” as isolated points in [,] so that q':→extends to a surjection. § BLOW-UP OF SINGULAR CONFIGURATIONS As in the proof of Lemma <ref>, we may think of points in () as Cauchy sequences (x_i)_i=1^∞=(γ(t_i))_i=1^∞ lying on a smooth path γ in [,]() which we can partition intoequivalence classes according to the limiting tangent direction :=limi→∞γ'(ti) of γ.We can use this idea to construct an approximation to [,](n) by blowing up the singular configurations where links or joints of Γ meet (cf. <cit.>). For our purpose the following simplified version will suffice: Given an abstract linkage [,]=(V,E) we have an orientation for each generalized edge (that is, edge, half-line, or line) [,]e∈ E by Remark <ref>. Letdenote the collection of all ordered pairs (e',e”)∈ E2 of distinct edges of Γ which have no vertex in common.For each embedding :V→3 ofin space and each pair [,]ξ:=(e',e”)∈ letdenote the linking vector connecting the closest points ∈(e') and ∈(e”) on the generalized segments (e') and (e”) (in that order). Sinceis an embedding, [.]≠ We define an invariant ϕξ()∈/3={-1,0,1} by:ϕξ() := (·((e')×(e”)))ifis interior to[,](e') to [,](e”)and (e') and (e”) are not coplanar0otherwise. This is just the linking number(ℓ',ℓ”) of the lines ℓ' and ℓ” containing (e') and [,](e”) respectively, although the usual convention is that (ℓ',ℓ”) is undefined when the two lines are coplanar (cf. <cit.>). If we letdenote the product space [,](3)V×(/3) the collection of invariants ϕξ() together define a (not necessarily continuous) function [,]Φ:()→ equipped with a projection [,]π:→(3)V such that π∘Φ is the inclusion j:()(3)V of <ref>.We now define an equivalence relation ∼ ongenerated as follows: consider a Cauchy sequence (i)i=1∞ in X=() with respect to the path metric(cf. <ref>). Since j:()(3)V is an inclusion into a complete metric space, andbounds the Euclidean metric in [,](3)V the sequence j(i)i=1∞ converges to a point [.]∈(3)VIf there are two (distinct) sequences α⃗=(αξ)ξ∈ and β⃗=(βξ)ξ∈ in (/3) and a Cauchy sequence (i)i=1∞ as above such that for each N>0 there are m,n≥ N with ϕξ(m)=αξ and ϕξ(n)=βξ for all [,]ξ∈ then we set (,α⃗)∼(,β⃗) in , where =limi j(i) in [.](3)VFinally, let :=/∼ be the quotient space, with q:→ the quotient map. Note that the projection π:→(3)V induces a well-defined surjection [.]:→(3)V The other projection →(/3) induces the completed linking number invariants ϕξ→/3 for each ξ∈ (where ϕξ() is set equal to 0 if [).]αξ≠βξ By Definition <ref> we have the following:The function Φ:()→ induces a continuous map [.]:()→Given an abstract linkage [,]=(V,E) the image of the map [,]:()→ denoted by [,]() is called the blow-up of the space of embeddings [.]()It contains the blow-up of the configuration spaceas a closed subspace; this is defined to be the closure of the image of [.] The singular set of It is hard to analyze the completed space of embeddings () or the corresponding configuration space [,] since the new points are only describable in terms of Cauchy sequences in [.]() However, for most linkages Γ, the spaceof embedded configurations is dense in the space of immersed configurations [.] We denote its complement by [.]Σ:=∖ A point (graph immersion) ∈Σ must have at least one intersection between edges not at a common vertex.Note that generically, in a dense open subset U of Σ, for any two edges (e',e”)∈ the intersection of (e') and (e”) is at an (isolated) points internal to both edges, and each of the intersections are independent.All fibers ofare finite, the restriction ofto(or [)]() is an embedding, while the restriction ofto -1(U)→ U is a covering map. Observe that the identifications made by the equivalence relation ∼ on(or ) do not occur over points of U, since for any ∈ U and [,]ξ=(e',e”)∈ the intersection of (e') and (e”) is at a single point internal to both (and in particular, (e') and (e”) are not parallel). Therefore, for any Cauchy sequence (i)i=1∞ in () (or [)] converging to , there is a neighborhood N ofin () (or [)] where ϕξ(_i) is constant +1 or constant -1 for all [.]ξ∈ We may summarize our constructions so far in the following two diagrams: [] () @^(->[d] @^(->[rrd] @^(->[rr]^Φ=(3)V×(/3)@->>[rd]q() @->>[rr]<<<<<<<<<<<[d]π() [lld]@^(->[r][llld]^(3)Vand similarly for the various types of configuration spaces:[] @^(->[d] @^(->[rrd] @^(->[rr]^Φ=×(/3)@->>[rd]q@->>[rr]<<<<<<<<<<<[d]π[lld]@^(->[r][llld]^U @^(->[r]Σ@^(->[r] where π is generically a surjection (unlesshas isolated configurations). § LOCAL DESCRIPTION OF THE BLOW-UP As we shall see, the global structure of the blow up () (or [)] can be quite involved, even for the simple linkage Γ2 consisting of two lines. The local structure is also hard to understand, in general, since even the classification of the types of singularities can be arbitrarily complicated. In the complement of the generic singularity set U (cf. <ref>) we have virtual configurations where:* k≥ 3 edges meet at a single point P (k will be called the multiplicity of the intersection at P); *Three or more edges meet pairwise (or with higher multiplicities); *One or more edges meet at a vertex (not belonging to the edges in question); *Two or more vertices (belonging to disjoint sets of edges) meet; *Two or more edges coinciding; *Any combination of the above situations (including the simple meeting of two edges at internal points, as in U above) may result in a higher order singularity if one situation imposes a constraint on another (as when intervening links are aligned andstretched to their maximal length).The goal of this section is to initiate a study of the simpler kinds of singularity as they appear in the blow-up.Double points The simplest non-trivial case of a blow-up occurs for a blown-up configuration ∈() where two edges e' and e” have a single intersection point P interior to both (e') and [.](e”) We assume that restrictingto the submechanism Γ':=∖{e',e”} obtained by omitting these two edges yields an embedding [.]'∈(Γ') In this case we have a neighborhood N of () in (3)V for which -1(N) is a product [,]N'× M where N' is an open set in a Euclidean space k corresponding to a coordinate patch around ' in [,](Γ') while M is diffeomorphic to a suitable open set in the blow-up (Γ2) for the two-line mechanism analyzed in <ref>A. below. Thus M is homeomorphic to the disjoint union of two half-spaces: [.]8×{± 1} Nevertheless, we can list the simpler types of singularity (outside of U).Edge and elbow Now consider the case where an interior point of one edge e1 meets a vertex v common to two other edges e2' and e2” (thus forming an “elbow”[),]Λ2 as in Figure <ref>: We can think of {e1,Λ} as forming a (disconnected) abstract linkage [,]=(V,E) so as in <ref>, for each embedding :V→3 ofin space we have two invariants ϕξ'(),ϕξ”()∈/3={-1,0,1} namely, the linking numbers of (e1) with (e2') and of (e1) with [,](e2”) respectively. Together they yield [.]ϕ⃗()∈(/3)=/3×/3For example, if in the embeddingshown in Figure <ref>(a) we have chosen the orientation for 3 so as to have linking numbers [,]ϕ⃗()=(+1,-1) say, then Figure <ref>(b) will have [.]ϕ⃗()=(-1,+1)On the other hand, for the embedding of Figure <ref>(c) we have [,]ϕ⃗()=(-1,-1) while for Figure <ref>(d) we have [,]ϕ⃗()=(0,0) since the nearest points to (e1) on (e2') or (e2”) are not interior points of the latter.Thus we see that the immersed configuration represented by Figure <ref>(a), in which (e1) passes through the vertex [,](v) but is not coplanar with (e2') and [,](e2”) has two preimages in the blowup [,]()= one of which corresponds to Figure <ref>(a) (with invariants [),](+1,+1)while the other preimage corresponds to both Figures <ref>(c)-(d), under the equivalence relation of <ref>, since we can have Cauchy sequences of either type converging to <ref>(a).On the other hand, the immersed configuration represented by Figure <ref>(b), in which (e1) passes through the vertex [,](v) and all edges are coplanar, is represented by three distinct types of inequivalent Cauchy sequences in [,] corresponding to Figure <ref>(a), Figure <ref>(b), and Figure <ref>(c)-(d), respectively.Thus it has three preimages in the blowup [.]()= This is the reason we used invariants in [,]/3 rather than [.]/2 One further situation we must consider in analyzing the edge-elbow linkage is when two or more edges coincide:*When only (e2') and (e2”) coincidethat is, the elbow is closedwe still have the two cases described in Figure <ref>. *If (e1) coincides with [,](e2') say, with (v) internal to [,](e1) the pre-image in () is a single virtual configuration (since all cases are identified under ∼). This is true whether or not the elbow is closed. Two elbows Next consider two elbows: [,]Λ1 consisting of two edges e1' and e1” with a common vertex v1 and [,]Λ2 where two other edges e2' and e2” are joined at the vertex [,]v2 as in Figure <ref>: Now we have four two-edge configurations, consisting of pairs of edges [,](e1', e2')[,](e1', e2”)(e1”, e2') and (e1”, e2”) respectively, so ϕ⃗ takes value in [.](/3)=(/3)4For example, in the embedding ofFigure <ref>(a) wehave [,]ϕ⃗():=(+1,+1,+1,-1) in Figure <ref>(b) we then have [,]ϕ⃗():=(+1,-1,-1,-1) while in Figure <ref>(c) we have [.]ϕ⃗():=(+1,+1,+1,+1)The virtual configuration we need to consider is one in which the vertices of the two elbows coincide. As in Section <ref>, we must consider a number of mutual positions in space, as in Figure <ref>. Assuming the edges (e2') and (e2”) do not coincide, they span a plane E. If at least one of the edges e1' and e1” does not lie in E, as in Figure <ref>(a)-(b), we generally have three possibilities for the blow-up invariants: namely, the limits of the three cases shown in Figure <ref>, where case (c) (and a number of others) are identified with the case of the two elbows being disjoint. The same holds if all four edges lie in E, as in Figure <ref>(c).On the other hand, if e1' and e1” are on opposite sides of E, we cannot have the mutual positions described in Figure <ref>(b), so only two blow-up invariants can occur.We do not consider here the more complicated cases when one or two of the elbows are closed, so that the edges e2' and [,]e2” say, coincide (and thus we have no plane E)even though similar considerations may be applied there. § PAIRS OF GENERALIZED INTERVALS Even for relatively simple linkage types [,] the global structure of the blow up () (or [)] can be quite complicated.However, the local structure is more accessible to analysis.The simplest non-trivial case of a blow-up for a general linkage typeoccurs when two edges e' and e” have a single intersection point P interior to both (e') and [.](e”) Such a configuration behaves locally like the completed configuration space of two (generalized) intervals in [,]3 which we analyze in this section. <ref>.ATwo lines in 3We begin with a linkage type 1 of two lines ℓ1 and [.]ℓ2 In this case there is no length vector, so 1=1 and [.]1=(1) Note that the convention of <ref> implies that each line has a given orientation.For simplicity, we first consider the case where the first line ℓ_1 is the (positively oriented) x-axis , so we need to understand the choices of the second line ℓ=ℓ_2 in [.]3Inside the spaceof oriented lines ℓ in 3 (that is,[,]=0 where Γ0 consists of a single line) we have thesubspaceof lines intersecting the x-axis. We have [,]=(× S^2)/∼ where (x,)∼(x',') for any x,x'∈ if and only if [,],'∈{±1} with x↦ (x,0,0) sendingto [.]3 Moreover, we have a map φ:1→ from the (unpointed) reduced configuration space 1=1 for the original linkage, which is a homeomorphism onto [,]∖_ since any reduced embedding of 1 in 3 is determined by a choice of an (oriented) line ℓ not intersecting the x-axis.By taking an appropriate -tubular neighborhood of the two lines, we may assume that we have two -cylinders tangent to each other in [,]3 with one of them symmetric about the x-axis (see Figure <ref>). If we denote the space of such tangent cylinders by [,]X we see that 1 is a disjoint union [.]_>0 X_ Moreover, by re-scaling we see that all the spaces X_ are homeomorphic, so in fact [,]1≅ X_1×(0,∞) say. Inside the space X1 we have a singular locusof configurations where the two tangent cylinders are parallel. Thus [,]≅1×{± 1} since such configurations are completely determined by the rotation ϕ of the cylinder T_1(ℓ) around ℓ with respect to the cylinder T_1() around the x-axis, together with a choice of the orientation ± 1 of ℓ (relative to ).The (open dense) complement X_1∖ consists of configurations of a cylinder T_1(ℓ) tangent at a single point Q on the boundary of both cylinders. Such a configuration is determined by:*The projection x of the point Q on ; *The angle θ between the (oriented) parallels to the respective axesand ℓ through Q; and *The rotation ϕ of Q about(i.e., the rotation of the perpendicular x⃗Q⃗ torelative to ).(see Figure <ref>).Thus U:=X_1∖ is diffeomorphic to the open manifold [,]×1×(1∖{0,π}) with global coordinates [,](x,ϕ,θ) identifying U with an open submanifold of the thickened torus [.]1×1×1 The boundary 1×1×{0,π} of U is identified in X_1 withby collapsing 1 to a point.In summary, X_1 is homeomorphic to the “tightened” torus [,](1×1×1)/∼ in which two opposite thickened circles (open annuli) are tightened to ordinary circles (see Figure <ref>). Thus X_1 is homotopy equivalent to a torus [.]1×1 The full space of reduced embeddings 1=1 is still homotopy equivalent to [,]X_1 and thus to [.]1×1 Its completion 1 allows configurations with [,]=0 so it is a quotient of [.]X_1×[0,∞) When [,]θ≠ 0,π we need no further identifications, since the new “virtual” configurations must still specify on which sides of each other the two lines touch. However, when ℓ is parallel to , this has no meaning, so we find: 1 = 1 = ([0,∞)×1×1×1)/∼ ,where [:](_1,x_1,ϕ_1,θ_1)∼(_2,x_2,ϕ_2,θ_2)*for any [,]x_1,x_2∈1 if [,]θ_1=θ_2∈{0,π}[,]ϕ_1=ϕ_2 and [;]_1=_2>0*for any ϕ_1,ϕ_2∈1 and [,]x_1,x_2∈1 if θ_1=θ_2∈{0,π} and [.]_1=_2=0 This is because (,ϕ) serve as polar coordinates for the plane perpendicular tothrough the point of contact Q, and ·ϕ is the length of the path rotating ℓ aboutwhen the two are parallel.We still have a natural map φ̂:1→ to the space of oriented lines in [,]3 taking a completed configuration to the location of [,]ℓ=ℓ_2, but it is two-to-one on all points of(exceptitself), since a pair of intersecting lines corresponds to two different completed configurations. To distinguish between them, we need the following: As in <ref>, given two non-intersecting lines ℓ1 and ℓ2 in [,]3 their linking vector =(ℓ1,ℓ2) is the shortest vector from a point on ℓ1 to a point on [,]ℓ2 perpendicular to their direction vectors 1 and [.]2When the lines are skew,is thus a multiple of 1×2 by a scalar [,]a≠ 0 and the linking number (ℓ1,ℓ2) of the two lines is [,](a)∈{±1}with (ℓ1,ℓ2):=0 when ℓ1ℓ2 are parallel, as above.By definition, pointsin the completed configuration space 1 correspond to (equivalence classes of) Cauchy sequences in the original space [,]1 and thus sequences of pairs of -cylinders tangent at a common point Q. Since we are not interested in configurations over the special point ∈ (for which the two cylinders may be parallel), we may assume that the angle θ between the tangents t_1 and t_2 to the respective cylinders at Q is not 0 or π, so they determine a plane (t_1,t_2) with a specified normal direction N (towards the second cylinder, having ℓ_2 as its axis, say).This is just the linking vector (ℓ1,ℓ2)defined above. The pair (t_1,t_2) converges along the Cauchy sequence to a pair of intersecting lines [,](,ℓ) and we define φ̂():=ℓ∈_⊆ extending the original homeomorphism [.]φ:1→∖_Note that any ≠ℓ∈_ has two sources under φ̂, corresponding to the two possible sides of the plane (,ℓ) on which the cylinder around ℓ_2 could be.In fact, 1 is homotopy equivalent to its “new part”C:=1∖1 under the map ψ:1→ C sending [(,x,ϕ,θ)] to [.][(0,x,ϕ,θ)] Moreover, C is homeomorphic to a twice-pinched torus, and so we have shown:The reduced complete configuration space 1 of two lines in 3 is homotopy equivalent to [.]2∨2∨1 Moreover, the map φ̂:1→ is described up to homotopy by the map sending each of the two spheres to [,]2 with the pinch points(corresponding to the two oriented parallels to [)]ℓ1 sent to thetwo poles.For the general case of two oriented lines ℓ1 and ℓ2 in [,]3 assume first that in the linkage type 2 the line ℓ_1 is framed that is, equipped with a chosen normal direction . The space of framed oriented lines in 3 through the origin is [,]3 so the spaceof all framed oriented lines is given as forby [,](3×3)/ and thus: 2 = 3×3×1×([0,∞)×1×1×1)/∼ ,where ∼ is generated by the equivalence relation of 1 together with -translations along the first line [.]ℓ_1 To forget the orientations, we divide further by [,]0×0 and to forget the framing, we must further divide by the action of 1 on , so [.]1=2/1When Γ1 consists as above of two (oriented) lines, the completion 1 and the blow-up (Γ1) of the configuration spaceare homeomorphic. The virtual configurations in 1 are of two types, in which the two lines intersect or coincide. In the first case, the linking number of the two lines is the same constant in a tail of any two Cauchy sequences. In the second case, any Cauchy sequence is equivalent to one consisting only of parallel configurations, and therefore the linking numbers must be identified. <ref>.BOther pairs of generalized intervalsIn principle, all the other pairs of generalized intervals may be treated similarly. We shall merely point out the changes that need to be made in various special cases:Line and half-line First, consider a linkage 3=3 consisting of an oriented line and a half-line [.] For the pointed reduced embedding and configuration spaces, we may assume thatto be the positive direction of the x-axis, with endpoint [.]∈3We see that 3=3 is the subspace of ^(1) consisting of oriented lines ℓ=ℓ_2 in 3 which do not intersect [,] so in particular it is contained in [.]W:=^(1)∖{} Note that as in <ref>.A, the lines in W can be globally parameterized by [,](,x,ϕ,θ)∈[0,∞)×1×1×1 module the equivalence relation [.]θ∈{0,π}⇒1∼∗ (The case =0 allows the line to intersect ).If we let Z := {[(,x,ϕ,θ)]∈ W⊆([0,∞)×1×1×1)/∼ :=0 ⇒ x≤ 0} denote the subspace of W⊂^(1) consisting of lines which do not intersect [,](0,∞)⊆ we see that the completed configuration space 3=3 is the pushout:[]@ [drr]|<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<PO{[(,x,ϕ,θ)]∈1 | θ∉{0,π}⇒ 0<x}@^(->[d] @^(->[rr]^<<<<<<<<f Z @^(->[d] {[(,x,ϕ,θ)]∈1 | θ∉{0,π}⇒ 0<x}@^(->[rr] 3 , where f is just the inclusion (using the same coordinates for source and target).The quotient map q_3:3→^(3)=^(1) of <ref> is induced by the quotient map q_1:1→^(1) and the inclusion [.]Z^(1)For the unreduced case, as above we first let 4 consist of an oriented framed pair of line and half-line, so 4≅3×3×1 (with the natural basepoint), and 3 is obtained from 4 by dividing out by a suitable action of [.]1×1The discussion above carries over essentially unchanged to a linkage 5 consisting of an oriented line L_1 and a finite segment I: in the reduced case, we assume I=[0,a] and then replace the conditions x≤ 0 and 0<x in eqlinehalf and eqpushout with x∉(0,a) and [,]x∈(0,a) respectively.This is the first example where the moduli function λ_:(5)→1 is defined. However, all the embedding configuration spaces 5 are homoeomorphic to each other, and in fact [.](5)≅5×(0,∞) Two half-lines Let 6 be a linkage consisting of two oriented half-linesand . For the reduced pointed space of embeddings [,]6=6 we now assume as before thatis the positive half of the x-axis X, so each point ∈6 is determined by a choice of the vectorfrom the end point of theto the origin (i.e., the end point of ), and the direction vectorof the . Thus 6 embeds as (,) in [.]3×22The only virtual configurations we need to consider are whenandare not collinear, and span a plane E containingand . In this caseis determined by a single rotation parameter θ=θ (relative to ), and there areθ0<θ1 such thatintersectsif and only if [.]θ0<θ≤θ1 Thus we have an arc γ=γ in 22 of values forfor which we have two virtual configurations in [,](6) yielding an embedding of the compactification 22∖γ of a sphere with a slit removed in [.](6) Since all such compactifications are homeomorphic to [,]22∖γ say, we see that(6) ≅ X×22 ∪ (3∖ X)×22∖γ . Since 22∖γ is contractible, by collapsing the x axis X to the origin we see that [.](6)≃2Along the way our analysis showed that for all types of pairs of generalized intervals the maps :()→() and :→ of eqsummaryt-eqsummaryc are homeomorphismsthat is, the completed configuration space is identical with the blow-up. This does not hold for more general linkage typessee <ref> below. § LINES IN 3 In order to deal with more complex virtual configurations, we need to understand the completed configuration spaces of n lines in [.]3The configuration space of (non-intersecting) skew lines in 3 have been studied from several points of view (see <cit.> and the surveys in <cit.>). However, here we are mainly interested in the completion of this space, and in particular in the virtual configurations where the lines “intersect” (but still retain the information on their mutual position, as before). As above, let =0 denote the space of oriented lines in [,]3andthe space of all lines in 3 (so we have a double cover [).]An oriented line is determined by a choice of a basepoint in 3 and a vectorin [,]S^2 and since the basepoint is immaterial, the spaceof all oriented lines in 3 is [,](3× S^2)/ where acts by translation of the basepoint along ℓ. Alternatively, we can associate to each oriented line ℓ the pair [,](,) where ∈ S2 is the unit direction vector of ℓ, and ∈3 is the nearest point to the origin on ℓ, allowing us to identify: ≅ {(,)∈ S2×3 : ·=0} .If Γn=n is the linkage consisting of n oriented lines, we thus have an isometric embedding π of n=(n) in the product [,]n which extends to a surjection π̂:(n)→n (no longer one-to-one).Denote by n the subspace of n consisting of all those lines passing through the origin (that is, with [),]= so [,]n≅(S2)n and let [.]n:=π̂-1(n) Thus n consists of all (necessarily virtual) configurations of n lines passing through the origin.We denote by Σ the subspace of n⊆(S2×3)n for which at least two of the n unit vectors in S2 are parallel:Σ := {(1,1…,n,n)∈ S2)n :∃ 1≤ i<j≤ n∃ 0≠λ∈,i=λj} . Finally, let Σ̂:=π̂-1(Σ) denote the corresponding singular subspace of [.](n) There is a deformation retract [.]ρ:(n)→nFor each t∈[0,1] we may use eqorline to define a map ht:n→n by settinght((1,1),…,(n,n)) := ((1,t1),…,(n,tn)) . For [,]t>0ht is equivalent to applying the t-dilitation about the origin in 3 to each line in [.]n Thus it takes the subspace (n) of n to itself, and therefore extends to a map [.]ht:(n)→(n)Now consider a Cauchy sequence {Pi}_i=0∞ in [,](n) of the form Pi = ((1i,1i),…, (ni,ni)) ,converging to a virtual configuration [.]P∈(n) Choosing any sequence (ti)i=0∞ in (0,1] converging to 0, we obtain a new Cauchy sequence {hti(Pi)}_i=0∞ withhti(Pi) = ((1i,ti1i),…, (ni,tini)) , which is still a Cauchy sequence in [,](n) and furthermore lim_i→∞ tiji= for all 1≤ j≤ n since the vectors (1i,…,ni) have a common boundK for all [.]i∈ Thus {hti(Pi)}_i=0∞ represents a virtual configuration P' in (n) with [,]π(P')∈n and thus [.]P'∈n Moreover, choosing a different sequence (ti)i=0∞ yields the same [.]P' Thus if we set [,]h0(P):=P' we obtain the required map [,]ρ:=h0:(n)→n as well as a homotopy H:(n)×[0,1]→(n) with H(P,t)=ht(P) for t>0 and thus H(-,1)= and [.]H(-,0)=ρ The completed configuration space (n) of n oriented lines in 3 is homotopy equivalent to the completed space n of n oriented lines through the origin. § THREE LINES IN 3 Corollary <ref> allows us to reduce the study of the homotopy type of the completed configuration space of n (oriented) lines in 3 to the that of the simpler subspace of n lines through the origin (where we may fix ℓ1 to be the x-axis).The case of two lines again For [,]n=2 the remaining (oriented) line ℓ2 is determined by its direction vector [,]∈2 which is aligned with ℓ1 at the north pole, say, and reverse-aligned at the south pole. Since we need to take into account the linking number ±1∈/2of ℓ1 and [,]ℓ2 we actually have two copies of [.]2 However, the north and south poles of these spheres, corresponding to the cases when ℓ2 is aligned or reverse-aligned with [,]ℓ1 must be identified as in Figure <ref>, so we see that [,]2≃2∨2∨1 as in Proposition <ref>.The cell structure for three linesFor [,]n=3 we again fix ℓ1 to be the positively oriented x-axis. The remaining two lines ℓ2 and ℓ3 are determined by their two direction vectors [.](2,3)∈2(1)×2(2) Again all in all there are eight copies of [,]2×2 indexed by the triples of completed linking numbers δi,j:=ϕ(ℓi,ℓj)=± 1 for 1≤ i<j≤ 3 (see <ref>).As in <ref>, there are identifications among these products of two spheres, which occur when at least one pair of lines is aligned or reverse-aligned: this takes place either in the diagonal (2(1)×2(2)) (whenℓ2 and ℓ3 are aligned), in the anti-diagonal -(2(1)×2(2)) (whenℓ2 and ℓ3 are reverse-aligned), or in one of the four subspaces of the form 2(1)×{N} (whenℓ1 and ℓ2 are aligned), and so on. Thus we have:3 = (∐1≤ i<j≤ 3[2(1)×2(2)]δi,j)/∼ . The four special points [,](N,N), (N,S), (S,N), (S,S) each appearing in three of the identification spheres for each of the eight indices [,]δi,j must also be identified, as indicated by the arrows in Figure <ref>. This suggests the following cell structure for[,]3 in which we decompose each of the eight copies of 2×2 into 8 four-dimensional cells, as follows:Using cylindrical coordinates [,](θ,t) we think of 2 as a cylinder with the top and bottom identified to a point, (i.e., a square with top and bottom collapsed and vertical sides identified levelwise). As a result, 2×2 may be viewed as a product of the (t1,t2)-square with the (θ1,θ2)-square (with suitable identifications), and its eight cells are obtained by as products of their respective subdivisions, indicated in Figure <ref>. Note that it is convenient to replace the (θ1,θ2)-square by a parallelogram, so the opposite diagonal edges correspond to θ1=θ2 (identified with each other). The horizontal edges are also identified pointwise. Thus each of the eight copies of 2×2 decomposes into 84-dimensional cells: [.] A× K,B× K,C× K, D× K,A× L,B× L,C× L,D× LHowever, there are certain collapses in the lower-dimensional products, all deriving from the fact that when ti=±1 (at either end of the cylinder), the variable θi has no meaning, so under the quotient mapq = q(1)× q(2) : (1(1)×1(2))×([-1,1](1)×[-1,1](2)) 2(1)×2(2) any point (θ1,θ2,-1,t2) is sent to [,](S(1),q(2)(θ2,t2)) where S(1) is the south pole in the first sphere [,]2(1) and so on. Thus:*The ostensibly 3-dimensional cell K× s1 is collapsed horizontally under q to the 2-cell [.]S(1)×2(2) Note that the same 2-cell is also represented by a× s1 and [.]b× s1Similarly q(L× s1)=S(1)×2(2) and [.]q(K× n1)=q(L× n1)= N(1)×2(2)*On the other hand, K× s2 is collapsed horizontally to the 2-cell [,]c× s2 and similarly [,]q(K× n2)=q(c× n2)[,]q(L× s2)=q(d× s2) and [.]q(L× n2)=q(d× n2)*The sum (c∪ d)× s2 is identified under q with [,]2(1)× S(2) which is also represented by a× s2 or [.]b× s2Similarly, [.]q(c∪ d)× n2)= q(a× s2)=q(b× s2)=2(1)× N(2)*The two 2-cells c× s1 and d× s1 are both collapsed under q to the 1-cell [,]S(1)× H where H is the longitude θ2=0 in [.]2(2) Similarly, [.]q(c× n1)=q(d× n1)=N(1)× H[,]c× n1 and d× n1*Since V corresponds to the pair of south poles (S(1),S(2)) in [,]2(1)×2(2)K× V and L× V are collapsed to a single point [.](S(1),S(2))Similarly, [,]q(K× W)=q(L× W)=(S(1),N(2))[,]q(K× X)=q(L× X)=(N(1),N(2))and [.]q(K× Y)=q(L× Y)=(N(1),S(2)) In addition, there are identifications among cells associated to the eight 2-spheres indexed by [.](δ1,2, δ1,3, δ2,3)∈(/3)^3 These occur only for the 0-, 1-, and 2-cells, when at least two of [,]ℓ1[,]ℓ2 and ℓ3 are aligned, so [.]δi,j=0 The resulting identifications are as follows:*The two 2-cells [,]a× e and a× g consist of pairs of points (ti,θi)i=1,2 with t1=t2 and [,]θ1=θ2 so at all points in these cells ℓ2 and ℓ3 are aligned. Therefore, [,]δ2,3=0 or equivalently, the corresponding cells in the two products 2×2 indexed by the triples (δ1,2,δ1,3,+1) and (δ1,2,δ1,3,-1) are identified, for each of the four choices of [.](δ1,2,δ1,3)∈{±1}2*Similarly, [,]b× f and b× h consist of pairs of points (ti,θi)i=1,2 with t1=-t2 and [,]θ1=θ2 so ℓ2 and ℓ3 are reverse-aligned, and again the corresponding cells indexed by the triples (δ1,2,δ1,3,+1) and (δ1,2,δ1,3,-1) are identified. *The 2-cell a× n1 (identified with b× n1 consist of pairs of points with [,]t1=+1 so ℓ2 is aligned with ℓ1 and the corresponding cells indexed by the triples (+1,δ1,3,δ2,3) and (-1,δ1,3,δ2,3) are identified. *The 2-cell a× s1=b× s1 consist of pairs of points with [,]t1=-1 so ℓ2 is reverse-aligned with ℓ1 and the corresponding cells indexed by the triples (+1,δ1,3,δ2,3) and (-1,δ1,3,δ2,3) are identified. *The two 2-cells [,]c× n2 and d× n2 consist of pairs of points with [,]t2=+1 so ℓ3 is aligned with ℓ1 and the corresponding cells indexed by the triples (δ1,2,+1,δ2,3) and (δ1,2,-1,δ2,3) are identified. *The two 2-cells [,]c× s2 and d× s2 consist of pairs of points with [,]t2=-1 so ℓ3 is reverse-aligned with ℓ1 and the corresponding cells indexed by the triples (δ1,2,+1,δ2,3) and (δ1,3,-1,δ2,3) are identified. *From (a) we see that the 1-cell a× O indexed by(δ1,2,δ1,3,+1) and (δ1,2,δ1,3,-1) are identified, and similarly for P× e and [.]P× g*From (b) we see likewise that the 1-cells [,]b× O[,]Q× f and Q× h indexed by(δ1,2,δ1,3,+1) and (δ1,2,δ1,3,-1) are also identified. *From (c) and (d) we see that the 1-cells [,]P× s1=Q× s1[,]P× n1=Q× n1 indexed by (+1,δ1,3,δ2,3) and (-1,δ1,3,δ2,3) are identified. *From (e) and (f) we see that the 1-cells [,]P× s2[,]Q× s2[,]P× n2 and Q× n2 indexed by (δ1,2,+1,δ2,3) and (δ1,2,-1,δ2,3) are identified. *Finally, all six 0-cells [,]P× V=Q× V[,]P× W=Q× W[,]P× X=Q× X[,]P× Y=Q× Y[,]P× O and Q× O have all three lines [,]ℓ1[,]ℓ2 and ℓ3 aligned or reverse-aligned, so δ1,2=δ1,3=δ2,3=0 and all eight copies are identified.Using this cell decomposition, we can easily verify that [,]H43 ≅ ^8where for each of the eight products 2×2 indexed by (δ1,2,δ1,3,δ2,3)∈{± 1}^3 we have a copy ofgenerated by the fundamental 4-cycle[.]γ((δ1,2,δ1,3,δ2,3):= (K+L)×(A+B+C+D) It is also clear that 3 is connected, so [.]H03 ≅ We leave to the reader to verify that [,]H3(3;) ≅ 6[.]H2(3;) ≅ 12 and [.]H1(3;) ≅ 9§ CHAINS IN SPACE We can use the basic building blocks of Sections <ref>-<ref> to study some actual simple linkages, namely, those with a all vertices of valence ≤ 2,called chains.We begin with the simplest non-trivial example: <ref>.AClosed quadrilateral chainsLet Γ be a closed quadrilateral chain with vertices a, b, c, and d, and length vector [.]:= (ℓ1,ℓ2,ℓ3,ℓ4)=(|ab|, |bc|, |cd|, |ad|) See Figure <ref>(b) We naturally assume the feasibility inequalities on(generalized triangle inequalities), which guarantee thatis non-empty. In the generic case we have no equations of the form ℓ1=±ℓ2±ℓ3±ℓ1 (which would allow the quadrilateral to be fully aligned).In the reduced configuration space(cf. <ref>) we assume that a is fixed at the origin , the link ab is in the positive direction of the x-axis, so b is fixed at the point [.]:=(ℓ2,0) If we mod out by the 1-action rotating the link ad in space about the x-axis, we obtain the space [,] whose points are represented by embeddings of Γ for which the link ad lies in the closed upper half planein the x,yplane.Local description of the singularities in Consider the simpler linkage Δ consisting of the two links ab and ad of lengths ℓ1 and [,]ℓ4 respectively, and with the distance bd contained in the closed interval I=[r,R] for r:=|ℓ2-ℓ3| and [.]R=ℓ2+ℓ3 See Figure <ref>(a). The corresponding reduced planar configuration space, in which we require ab to lie on the x-axis and ad to lie in , is denoted by [,] and there is a “forgetful map”[.]ρ:→For a configurationin the angle ∠ bcd is determined by the locations of b and d, and thus by the corresponding configuration [,]=ρ() and given any [,]∈ there is a unique ∈ρ-1() in which a and c do not lie on the same side of the line through bd (unless a, b, and d are aligned).Given this , inthe elbow Λ formed by b, c and d can rotate freely about bd (unless it is aligned). However, when we rotate Λ fromby 180o back into the x,yplane, the resulting (non-convex) configuration ' may be self-intersecting, if the two opposite sides ab and [,]cd or else ad and [,]bc intersect in a point interior to one or the other.By the analysis of planar quadrilateral configurations in <cit.>, we see that given such a self-intersecting planar configuration ' of Γ, one may decrease one angle between adjacent links to obtain ”∈ (with angle [,]0o with the two links aligned), after which the self-intersection disappears. Therefore, to study the cases in which Λ cannot be fully rotated in [,]3 it suffices to consider the configurations ” where one link is folded onto an adjacent link. We call a case where bc is folded back on ab a collineation[.](acb) See Figure <ref>.In the full reduced configuration spacewe have two angles associated to each configuration ∈ρ-1() as above: θ determined by rotating Λ about [,]bd and ϕ by rotating the resulting rigid spatial quadrilateral (of [)] about the x-axis (assuming that a, b and d are not collinear).Note that for all spatial quadrilaterals inthe rotation by ϕ is possible, and yields different spatial configurations of Γ (since we are assuming Γ cannot be fully aligned, becauseis generic). Moreover, at a collineation ” the full rotation by θ about bd is still allowed.Any configuration ” representing a collineation [,](acb) say, has a neighborhood in the planar reduced configuration spacehomeomorphic to an open interval (,) (with ” itself identified with the midpoint 0) such that in one half [0,) the full rotation by ϕ is allowed, while in the other half (-,0) the rotation by ϕ=0o is distinct in the completion (or blowup)from the rotation by [,]ϕ=360o in which the link cd touches ab on opposite sides. Similarly for [,](acd)[,](cab) and [.](cad)However, when ” represents the collineation (abd) or [,](adb) the rotation by ϕ about the x-axis is the same as the rotation by θ about [.]bd In this case we simply exchange the roles of bd and ac in defining [,] and then the same analysis holds. Similarly for (bdc) and [.](dbc)Therefore, in the reduced completed spatial configuration space(in which the only restriction is that side ab lies on the x axis, with a at the origin) a collineation configuration ” has a neighborhood U diffeomorphic (as a manifold with corners) to the union of the thickened torus U1:=[0,)×1×1 and split thickened torus [,]U2:=(-,0]× [0o,360o]×1 with (0,θ,ϕ) in U1 identified with (0,θ,ϕ) in U2(and thus in particular (0,0o,ϕ) and (0,360o,ϕ) in U1 identified with (0,0o,ϕ)=(0,360o,ϕ) in [).]U2 The singular configuration ” is parameterized by the point [,](0,0o,0o) while the corresponding convex quadrilateralis parameterized by [.](0,180o,0o)If one link is folded onto an adjacent link, we obtain a triangle with sides [,]ℓi[,]ℓj and [,]|ℓk-ℓl| respectively (for [).]{i,j,k,l}={1,2,3,4} For this to be possible, the three sides must satisfy the triangle inequalities. If we take into account the feasibility inequalities, these reduce to two cases: ℓi+ℓk>ℓj+ℓl ,ℓj+ℓk>ℓi+ℓl ,andℓk>ℓl or ℓj+ℓl>ℓi+ℓk ,ℓi+ℓl>ℓj+ℓk ,andℓl>ℓk . Global description ofAs in the classical analysis of <cit.>,may be identified with the intersectionof the half-annulus A(d) inaboutof radii r,R with the half circle B(d) of radius ℓ4 aboutin the upper half-plane(both describing possible locations for d). See Figure <ref>. We may assume without loss of generality that ℓ4<ℓ1>ℓ2>ℓ3 ,so the collineations [,](dba)[,](dbc) or (cab) are impossible.Thusis an arc of [,]B(d) which is: *The full half circle B(d) when ℓ1+ℓ4<ℓ2+ℓ3 (so the leftmost pointof [,]B(d) corresponding the links cb and cd being aligned in opposite directions, is in the annulus), and |ℓ2-ℓ3|<ℓ1-ℓ4 (so the rightmost pointof [,]B(d) corresponding to the collineation [,](bdc) is in the annulus).*A proper closed arc of B(d) ending at theon the positive x-axis when eqfirst is reversed and eqsecond holds.*A proper closed arc of B(d) beginning aton the negative x-axis wheneqfirst holds and eqsecond is reversed. *A closed arc of B(d) not intersecting the x-axis when eqfirst and eqsecond are reversed (soandhave positive y-coordinates).Given a point (d) in this arc, the pointsand ' in ρ-1()⊆ are determined by the respective locations (c) and '(c) of c, which are obtained by intersecting the circle E of radius ℓ3 about (d) with the circle C(c) of radius ℓ2 about .Using the analysis in <ref> we see that a singular point ” corresponding to the collineation (bdc) occurs in cases (iii) or (iv) above, when =”(d) lies on the inner circle of the half annulus [.]A(d) At this ” the rotation by θ about bd is trivial.On the other hand, the collineation (bda) occurs when ”(d) lies in the positive half of the x-axis, which is possible only in cases (i) or (ii). The collineation (acb) occurs atwhen the circle G of radius ℓ3 about the point (c):=(ℓ1-ℓ2,0) intersectsat a point(which is [).](d)To determine when the two remaining mutually exclusive collineations (acd) or (cad) occur, we must interchange the roles of d and c and study the intersection of the circle C(c) of radius ℓ2 about =(ℓ1,0) with the inner circle of the annulus A(c) aboutthat is with the circle K of radius |ℓ4-ℓ3| about the origin. Assume that these intersect at the two points [,]{0(c),1(c)}=K∩ C(c) with corresponding lines X0 and X1 through the origin. The intersections 0 and 1 of these lines withyield the locations 0(d) and [.]1(d)Thus we can in principle obtain a full description of the completed configuration space [.] This will depend on the particular chamber of the moduli space of all length vectorsin 4+ that is, which set of inequalities of the form [,]eqfirstt[,]eqsecondt[,]eqfirstandeqsecond occur (subject to [).]eqwlogA particularly simple type of quadrilateral Γ is one which has “three long sides” (cf. <cit.>). For instance, if we assume ℓ1=ℓ2=ℓ4=5 and [,]ℓ3=1 the arcis parameterized by the angle α=∠ dabWhen α is maximal, we are at the aligned configuration , where the rotation θ about bd has no effect, so the fiber of ρ:→ is a single point (see Figure <ref>(a)). Decreasing α slightly as in Figure <ref>(b) yields a generic configuration, with fiber [.]1 Figure <ref>(c) represents the pointcorresponding to the collineation [,](acd) still with fiber [.]1 Further decreasing α yields the self-intersection of Figure <ref>(d), with fiber [.][0o,360o] This continues until the minimal α in Figure <ref>(e), corresponding to the collineation [,](bdc) with trivial fiber. Thusis homeomorphic to the slit sphere 2∖ (including the two edges of the cut), and the full reduced configuration spaceis [.]2∖×1 Finally, the unreduced configuration space is≅ 2∖ × 1 × 2 × 3 , since the quadrilateral can never be full aligned. <ref>.BOpen chainsFor an open chain 2 with two links (cf. <ref>), we have _2=_2 and [,]2=(2) since no self-intersections exist in our model. Moreover, _2≅2×(0,∞)×(0,∞) and [,]2≅2 and we can choose spherical coordinates (θ,ϕ) for [,]2 where θ is the rotation of the second edge about the first. The coordinates (ℓ1,ℓ2)∈(0,∞)×(0,∞) are the link lengths.For an open chain 3 with three links, we first consider the simplified case where the first and third link have infinite length: that is, the linkage 3 has two half-linesand , whose ends are joined by an interval I.If we do not specify the length of I, this linkage type is equivalent to 6 of <ref>. The specific mechanism Γ with |I|=ℓ2 corresponds to choosing the vectorin <ref> to be of length [,]ℓ2 thus replacing 3 there by a sphere [,]2 and replacing X by its two poles N and S. Thus we see that3 ≃ 3 ≃ 2∨2∨2 . The case where ℓ1 is finite and ℓ3 is infinite is analogous. When both are finite, we must distinguish the case when ℓ1+ℓ3>ℓ2 (again analogous to the infinite case) from that in which ℓ1+ℓ3≤ℓ2 (in which case [).]3≅2×2The analysis of open chains with more links requires a more complicated analysis of the moduli space of link lengths (see below). § APPENDIX:SPACES OF PATHS There is yet another construction which can be used to describe the virtual configurations of a linkage Γ, which we include for completeness, even though it is not used in this paper.Given a linkage typewith [,]()⊆(3)V let P() denote the space of paths γ:[0,1]→(3)V such that γ((0,1])⊆() (cf. <cit.>), and let 0:P()→(3)V send [γ] to [.]γ(0) Let E() denote the set of homotopy classes of such paths relative to [.]=γ(0) This is a quotient space of [,]P() with 0:E()→(3)V induced by [.]0 We shall call E() the path space of embeddings of [.]Similarly, let P(Γ)⊆ P() denote the space of paths γ:[0,1]→(3)V such that [,]γ((0,1])⊆ with E(Γ) the corresponding set of relative homotopy classes (a quotient of [).]P(Γ) We call E(Γ) the path space of configurations of Γ. The completed space of embeddings () is a quotient of the path space [,]E() andis a quotient of [.]E(Γ)First note that when [,]γ(0)∈() the path γ is completely contained in the open subspace () of [,](3)V so we may represent any homotopy [γ] by a path contained wholly in an open ball around γ(0) inside [,]() and any two such paths are linearly homotopic. Thus 0 restricted to ()) is a homeomorphism.In any completion X̂ of a metric space [,](X,d) the new points can be thought of as equivalence classes of Cauchy sequences in X. Since we can extract a Cauchy sequence(in the path metric) from any path γ as above, and homotopic paths have equivalent Cauchy sequences, this defines a continuous map[.]ϕ:E()→()In our case, X=() also has the structure of a manifold, and given any Cauchy sequence (xi)i=1∞ in X, choose an increasing sequence of integers nk such that (xi,xj)<2-k for all [.]i,j≥ nk By definition of [,] we have a path γk from xnk to xnk+1 of length [.]≤ 2-k By concatenating these and using Remark <ref>, we obtain a smooth path γ along which all (xi)i=1∞ lies. Thus we may restrict attention to Cauchy sequences (x_i)_i=1^∞=(γ(t_i))_i=1^∞ lying on a smooth path γ in X. We can parameterize γ so that [,]γ((0,1])⊆() and let [,]γ(0):=limi→∞ x_i∈(3)V which exists since (3)V is complete.Thus ϕ is surjective.We may thus summarize the constructions in this paper in the following diagram, generalizing eqsummaryt:[] () @^(->[lld] @^(->[d] @^(->[rrd] @^(->[rr]^Φ=(3)V×(/3)@->>[rd]q P()⊇E() @->>[rr]ϕ[rrd]0() @->>[rr]<<<<<<<<<<<[d]π() [lld]@^(->[r][llld]^(3)VSimilarly for the various types of configuration spaces shown in [.]eqsummaryc If we allow a linkage type =∞ consisting of a countable number of lines, we show that the maps ϕ:E()→() and ϕ:E(Γ)→ need not be one-to-one, in general:Consider the set S of configurationsofin which the first line (ℓ0) is the x-axis in [,]3 and all other lines (ℓn)n=1,2,… are perpendicular to the x,yplane, and thus determined by their intersections (ℓn) with the x,yplane, with (ℓn)=(0,1/n) for [.]n≥ 2 Thus the various configurations in S differ only in the location of [.](ℓ1)Now define the following two Cauchy sequences ('k)k=1∞ and (”k)k=1∞ in [:]S⊆(∞)=(∞) we let 'k(ℓ1):=(1/k,1/k) for [,]'k while ”k(ℓ1):=(-1/k,1/k) for [.]”k Since we can define a path in S between 'k and ”k of length <3/k, the two Cauchy sequences are equivalent, and thus define the same point in the completions [.](∞)=∞ However,if we embed ('k)k=1∞ in a path γ':[0,1]→ S defined by the intersection [,]'t(ℓ1):=(t,t) and similarly for [,](”k)k=1∞ it is easy to see that γ' and γ” cannot be homotopic (relative to endpoints), so they define different points in E() or [.]E(Γ)ABCDE [CEGSS]CEGSStolL B. Chazelle, H. Edelsbruner, K.J. Guibas, M. Sharir, & J. Stolfi, “Lines in Space:Combinatorics and Algorithms”,Algorithmica15 (1996), pp. 428-447.[CP]CPennC H. Crapo & R.J. Penne, “Chirality and the isotopy classification of skew lines in projective 3-space”, Adv. Math.103 (1994), pp. 1-106.[CF]CFoI R.H. Crowell & R.H. Fox", Introduction to Knot Theory, Springer, Berlin-New York, 1963.[DV]DViroC Yu.V. Drobotukhina & O.Ya. Viro, “ Configurations of skew-lines”,Algebra i Analiz1 (1989), pp. 222-246.[Fa]FarbT M.S. Farber, Invitation to Topological Robotics, European Mathematical Society, Zurich, 2008.[FTY]FTYuzvT M.Š. Farber, S. Tabachnikov, & S.A. Yuzvinskiĭ, “Topological robotics: motion planning in projective spaces”, Int. Math. Res. Notices34 (2003), 1853-1870.[Fr]FrieM A. Friedman, Foundations of Modern Analysis, Dover, New York, 1970.[G]GotRF D.H. Gottlieb, “Robots and fibre bundles”, Bull. Soc. Math. Belg. Sér. A38 (1986), 219-223.[Hal]HallK A.S. Hall, Jr., Kinematics and Linkage Design, Prentice-Hall, Englewood Cliffs, NJ, 1961.[Hau]HausT J.-C. Hausmann, “Sur la topologie des bras articulés”, in S. Jackowski, R. Oliver, & K. Pawałowski, eds., Algebraic Topology - Poznán 1989, Lect. Notes Math.1474, Springer Verlag, Berlin-New-York, 1991, pp. 146-159.[HK]HKnutC J.-C. Hausmann & A. Knutson, “The cohomology ring of polygon spaces”, Ann. Inst. Fourier (Grenoble)48 (1998), pp. 281-321.[Ho]HolcM M. Holcomb, “On the Moduli Space of Multipolygonal Linkages in the Plane”, Topology & Applic.154 (2007), pp. 124-143.[JS]JSteiC D. Jordan & M. Steiner, “Compact surfaces as configuration spaces of mechanical linkages”, Israel J. Math.122 (2001), pp. 175-187.[KM1]KMillM M. Kapovich & J. Millson, “On the moduli space of polygons in the Euclidian plane”,J. Diff. Geom.42 (1995), pp. 430-464.[KM2]KMillS M. Kapovich & J. Millson, “The symplectic geometry of polygons in Euclidean space”,J. Diff. Geom.44 (1996), pp. 479ג513.[K]KamiT Y. Kamiyama, “Topology of equilateral polygon linkages in the Euclideanplane modulo isometry group”,Osaka J. Math.36 (1999), pp. 731-745.[KT]KTsuC Y. Kamiyama & S. Tsukuda, “The configuration space of the n-arms machine in the Euclidean space”, Topology & Applic.154 (2007), pp. 1447-1464.[L]LeeRM J.M. Lee, Riemannian manifolds. An introduction to curvature, Springer-Verlag, Berlin-New York, 1997.[Me]MerlP J. P. Merlet, Parallel Robots, Kluwer Academic Publishers, Dordrecht, 2000.[MT]MTrinG R.J. Milgram & J. Trinkle, “The Geometry of Configuration spaces of Closed Chains in Two and Three Dimensions”,Homology, Homotopy & Applic.6 (2004), pp. 237-267.[Mu]MunkrTF J.R. Munkres, Topology, A First Course, Prentice-Hall, Englewood, NJ, 1975.[OH]OHaraM J. O'Hara, “The configuration space of planar spidery linkages”, Topology & Applic.154 (2007), pp. 502-526.[P]PennC R.J. Penne, “Configurations of few lines in 3-space: Isotopy, chirality and planar layouts”,Geom. Dedicata45 (1993), pp. 4982.[RR]RRimoI G. Rodnay & E. Rimon, “Isometric visualization of configuration spaces of two-degrees-of-freedom mechanisms”,Mechanism and machine theory36 (2001), pp. 523-545.[Se]SeliG J.M. Selig, Geometric Fundamentals of Robotics, Springer-Verlag Mono. Comp. Sci., Berlin-New York, 2005.[Sh]ShafB1 I.R. Shafarevich, Basic algebraic geometry, 1. Varieties in projective space, Springer-Verlag, Berlin-New York, 1994.[SSB]SSB N. Shvalb, M. Shoham & D. Blanc, “The Configuration Space of Arachnoid Mechanisms”, Fund. Math.17 (2005), 1033-1042.[T]TsaiR L.W. Tsai, Robot Analysis - The mechanics of serial and parallel manipulators, Wiley interscience Publication - John Wiley & Sons, New York, 1999.[Va]VassK V.A. Vassiliev, “Knot invariants and singularity theory”, in Singularity theory (Trieste, 1991), World Sci. Publ., River Edge, NJ, 1995, pp. 904-919.[Vi]ViroT O.Ya. Viro, “Topological problems on lines and points of three-dimensional space”,Dokl. Akad. Nauk SSSR284 (1985), pp. 10491052. | http://arxiv.org/abs/1705.09092v1 | {
"authors": [
"David Blanc",
"Nir Shvalb"
],
"categories": [
"math.AT",
"math.GT"
],
"primary_category": "math.AT",
"published": "20170525082739",
"title": "Configuration spaces of spatial linkages: Taking Collisions Into Account"
} |
The first author gratefully acknowledges supportby the FWF-grants p26736 and Y782, the third author gratefully acknowledges support by the GermanResearchFoundationthroughtheHausdorff Center for Mathematics and the Collaborative Research Center 1060. The Skorokhod Embedding Problem (SEP) is one of the classical problems in the study of stochastic processes, with applications in many different fields (cf. the surveys <cit.>). Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem (MSEP). Some of the first papers to consider this problem are <cit.>. However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in <cit.> establishing an extension of the Root construction, while other instances are only partially answered or remain wide open. In this paper, we extend the theory developed in <cit.> to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem.As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases.Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem. Keywords: optimal transport, Skorokhod embedding, multiple marginals, martingale optimal transport, peacocks.Mathematics Subject Classification (2010): Primary 60G42, 60G44; Secondary 91G20.The geometry of multi-marginal Skorokhod Embedding Martin Huesmann December 30, 2023 ==================================================§ INTRODUCTIONThe Skorokhod Embedding problem (SEP) is a classical problem in probability, dating back to the 1960s (<cit.>). Simply stated, the aim is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Recently, motivated by applications in probability, mathematical finance, and numerical methods, there has been renewed, sustained interest in solutions to the SEP (cf. the two surveys <cit.>) and its multi-marginal extension, the multi-marginal SEP: Given marginal measures μ_0,…,μ_n of finite variance and a Brownian motion with B_0∼μ_0, construct stopping times τ_1≤…≤τ_n s.t. MSEPB_τ_i∼μ_i for all 1≤ i≤ n and [τ_n]<∞.It is well known that a solution to (<ref>) exists iff the marginals are in convex order (μ_0≼_c…≼_c μ_n) and have finite second moment; under this condition Skorokhod's original results give the existence of solutions of the induced one period problems, which can then be pasted together to obtain a solution to (<ref>).It appears to be significantly harder to develop genuine extensions of one period solutions: many of the classical solutions to the SEP exhibit additional desirable characteristics and optimality properties which one would like to extend to the multi-marginal case.However the original derivations of these solutions make significant use of the particular structure inherent to certain problems, often relying on explicit calculations, which make extensions difficult if not impossible. The first paper which we are aware of to attempt to extend a classical construction to the multi-marginal setting is <cit.>, who generalised the Azéma-Yor embedding (<cit.>) to the case with two marginals. This work was further extended by Henry-Labordère, Obłój, Spoida, and Touzi <cit.>, who were able to extend to arbitrary (finite) marginals, under a non-trivial assumption on the measures.Using an extension of the stochastic control approach in <cit.> Claisse, Guo, and Henry-Labordère <cit.> constructed a two marginal extension of the Vallois embedding.Recently, Cox, Obloj, and Touzi <cit.> were able to characterise the solution to the general multi-marginal Root embedding through the use of an optimal stopping formulation. §.§ Mass transport approach and general multi-marginal embedding In this paper, we develop a new approach to the multi-marginal Skorokhod problem, based on insights from the field of optimal transport.Following the seminal paper of Gangbo and McCann <cit.> the mutual interplay of optimality and geometry of optimal transport plans has been a cornerstone of the field. As shown for example in <cit.> this in not limited to the two-marginal case but extends to the multi-marginal case where it turns out to be much harder though.Recently, similar ideas have been shown to carry over to a more probablistic context, to optimal transport problems satisfying additional linear constraints <cit.> and in fact to the classical Skorokhod embedding problem <cit.>.Building on these insights, we extend the mass transport viewpoint developed in <cit.> to themulti-marginal Skorokhod embedding problem.This allows us to give multi-marginal extensions of all the classical optimalsolutions to the Skorokhod problem in full generality, which we exemplify by several examples. In particular the classical solutions of Azéma-Yor, Root, Rost, Jacka, Perkins, and Vallois can be recovered as special cases.In addition, the approach allows us to derive a number of new solutions to (<ref>) which have further applications to e.g. martingale optimal transport and the peacock problem.A main contribution of this paper is that in many different cases, solutions to the multi-marginal SEP share a common geometric structure. In all the cases we consider, this geometric information will in fact be enough to characterise the optimiser uniquely, which highlights the flexibility of our approach. Furthermore, our approach to the Skorokhod embedding problem is very general and does not rely on fine properties of Brownian motion. Therefore, exactly as in <cit.> the results of this article carry over to sufficiently regular Markov processes,geometric Brownian motion, three-dimensional Bessel process and Ornstein-Uhlenbeck processes, and Brownian motion in ^d for d >1. As the arguments are precisely the same as in<cit.>, we refer to<cit.> for details. §.§ Related WorkInterest in the multi-marginal Skorokhod problem comes from a number of directions and we describe some of these here:* Maximising the running maximum: the Azéma-Yor embeddingSuppose (M_t)_t≥0 is a martingale and write M̅_t:=sup_s≤ tM_s.The relationship between the laws of M_1 and M̅_1 has been studied by Blackwell and Dubins <cit.>, Dubins and Gilat <cit.> and Kertz and Rösler <cit.>, culminating in a complete classification of all possible joint laws by Rogers <cit.>. In particular given the law of M_1, the set of possible laws of M̅_1 admits a maximum w.r.t. the stochastic ordering, this can be seen through the Azéma-Yor embedding.Given initial and terminal laws of the martingale, Hobson <cit.> gave a sharp upper bound on the law of the maximum based on an extension of the Azéma-Yor embedding to Brownian motion started according to a non-trivial initial law. These results are further extended in <cit.> to the case of martingales started in 0 and constrained to a specified marginal at an intermediate time point, essentially based on a further extension of the Azéma-Yor construction. The natural aim is to solve this question in the case of arbitrarily many marginals. Assuming that the marginals have ordered barycenter functions this case is included in the work of Madan and Yor <cit.>, based on iterating the Azéma-Yor scheme. More recently, the stochastic control approach of <cit.> (for one marginal) is extended by Henry-Labordère, Obłój, Spoida, and Touzi <cit.> to marginals in convex order satisfying an additional assumption (<cit.>[As shown by an example in <cit.> this condition is necessary to carry out their explicit construction.]). Together with the Dambis-Dubins-Schwarz Theorem,Theorem <ref> below provides a solution to this problem in full generality. * Multi-Marginal Root embeddingIn a now classical paper, Root <cit.> showed that for any centred distribution with finite second moment, μ, there exists a (right) barrier ℛ, i.e. a Borel subset of _+× such that (t,x)∈ℛ implies (s,x)∈ℛ for all s≥ t, and for which B_τ_ℛ∼μ, τ_ℛ=inf{t: (t,B_t)∈ℛ}. This work was further generalised to a large class of Markov processes by Rost <cit.>, who also showed that this construction was optimal in that it minimised [h(τ)] for convex functions h.More recent work on the Root embedding has focused on attempts to characterise the stopping region. A number of papers do this either through analytical means (<cit.>) or through connections with optimal stopping problems (<cit.>). Recently the connection to optimal stopping problems has enabled Cox, Obłój, and Touzi <cit.> to extend these results to the multi-marginal setting Moreover, they prove that this solution enjoys a similar optimality property to the one-marginal Root solution. The principal strategy is to first prove the result in the case of locally finitely supported measures by means of a time reversal argument. The proof is then completed in the case of general measures by a delicate limiting procedure. As a consequence of the theoretical results in this paper, we will be able to prove similar results. In particular, the barrier structure as well as the optimality properties are recovered in Theorem <ref>. Indeed, as we will show below, the particular geometric structure of the Root embedding turns out to be archetypal for a number of multi-marginal counterparts of classical embeddings. * Model-independent FinanceAn important application field for the results in this paper, and one of the motivating factors behind the recent resurgence of interest in the SEP, relates to model-independent finance. In mathematical finance, one models the price process S as a martingale under a risk-neutral measure, and specifying prices of call options at maturity T is equivalent to fixing the distribution μ of S_T. Understanding no-arbitrage price bounds for a functional γ, can often be seen to be equivalent to finding the range of [γ(B)_τ] among all solutions to the Skorokhod embedding problem for μ. This link between SEP and model-independent pricing and hedging was pioneered by Hobson <cit.> and has been an important question ever since. A comprehensive overview is given in <cit.>.However, the above approach uses only market data for the maturity time T, while in practice market data for many intermediate maturities may also be available, and this corresponds to the multi-marginal SEP. While we do not pursue this direction of research in this article we emphasize that our approach yields a systematic method to address this problem.In particular, the general framework of super-replication results for model-independent framework now includes a number of important contributions, see <cit.>, and most of these papers allow for information at multiple intermediate times. * Martingale optimal transportOptimal transport problems where the transport plan must satisfy additional martingale constraints have recently been investigated,in <cit.>. Besides having a natural interpretation in finance, such martingale transport problems are also of independent mathematical interest, for example – similarly to classical optimal transport – they have consequences for the investigation of martingale inequalities (see<cit.>).As observed in <cit.> one can gain insight into the martingale transport problem between two probabilities μ_1 and μ_2 by relating it to a Skorokhod embedding problem which may be considered as continuous time version of the martingale transport problem. Notably this idea can be used to recover the known solutions of the martingale optimal transport problem in a unified fashion (<cit.>). It thus seems natural that an improved understanding of an n-marginal martingale transport problem can be obtained based on the multi-marginal Skorokhod embedding problem. Indeed this is exemplified in Theorem <ref> below, where we use a multi-marginal embedding to establish an n-period version of the martingale monotone transport plan, and recover similar results to recent work of Nutz, Stebegg, and Tan <cit.>. * Construction of peacocksDating back to the work of Madan–Yor <cit.>, and studied systematically in the book of Hirsch, Profeta, Roynette and Yor <cit.>, given a family of probability measures (μ_t)_t ∈ [0,T] which are increasing in convex order, a peacock (from the acronym PCOC “Processus Croissant pour l'Ordre Convexe”) is a martingale such that M_t ∼μ_t for all t ∈ [0,T].The existence of such a process is granted by Kellerer's celebrated theorem, and typically there is an abundance of such processes. Loosely speaking, the peacock problem is to give constructions of such martingales.Often such constructions are based on Skorokhod embedding or particular martingale transport plans, and often one is further interested in producing solutions with some additional optimality properties; see for example the recent works <cit.>.Given the intricacies of multi-period martingale optimal transport and Skorokhod embedding, it is necessary to make additional assumptions on the underlying marginals and desired optimality properties are in general not preserved in a straight forward way during the inherent limiting/pasting procedure.We expect that an improved understanding of the multi-marginal Skorokhod embedding problem will provide a first step to tackle these range of problems in a systematic fashion.§.§ Outline of the Paper We will proceed as follows. In Section <ref>, we will describe our main results. Our main technical tool is a `monotonicity principle', Theorem <ref>. This result allows us to deduce the geometric structure of optimisers. Having stated this result, and defined the notion of `stop-go pairs', which are important mathematical embodiment of the notion of `swapping' stopping rules for a candidate optimiser, we will be able to deduce our main consequential results. Specifically, we will prove the multi-marginal generalisations of the Root, Rost and Azéma-Yor embeddings, using their optimality properties as a key tool in their construction. The Rost construction is entirely novel, and the solution to the Azéma-Yor embedding generalises existing results, which have only previously been given under a stronger assumption on the measures. We also give a multi-marginal generalisation of an embedding due to Hobson & Pedersen; this is, in some sense, the counterpart of the Azéma-Yor embedding; classically, this is better recognised as the embedding of Perkins <cit.>, however for reasons we give later, this embedding has no multi-marginal extension.Moreover the proofs of these results will share a common structure, and it will be clear how to generalise these methods to provide similar results for a number of other classical solutions to the SEP.In Section <ref>, we also use our methods to give a multi-marginal martingale monotone transport plan, using a construction based on a SEP-viewpoint.The remainder of the paper is then dedicated to proving the main technical result, Theorem <ref>. In Section <ref>, we introduce our technical setup, and prove some preliminary results. As in <cit.>, it will be important to consider the class of randomised multi-stopping times, and we define these in this section, and derive a number of useful properties. It is technically convenient to consider randomised multi-stopping times on a canonical probability space, where there is sufficient additional randomisation, independent of the Brownian motion, however we will prove in Lemma <ref> that any sufficiently rich probability space will suffice. A key property of the set of randomised multi-stopping times embedding a given sequence of measures is that this set is compact in an appropriate (weak) topology, and this will be proved in Proposition <ref>; an important consequence of this is that optimisers of the multi-marginal SEP exist under relatively mild assumptions on the objective (Theorem <ref>).In Section <ref> we introduce the notions of color-swap pairs, and multi-colour swap pairs. These will be the fundamental constituents of the set of `bad-pairs', or combinations of stopped and running paths that we do not expect to see in optimal solutions. In this section we define these pairs, and prove some technical properties of the sets.In Section <ref> we complete the proof of Theorem <ref>. In spirit this follows the proof of the corresponding result in <cit.>, and we only provide the details here where the proof needs to adapt to account for the multi-marginal setting.§.§ Frequently used notation * The set of (sub-)probability measures on a space 𝖷 is denoted by 𝒫(𝖷) / 𝒫^≤ 1(𝖷).* Ξ^d={(s_1,…,s_d):0≤ s_1≤…≤ s_d} denotes the d-dimensional simplex. * The d-dimensional Lebesgue measure will be denoted by ^d.* For a measure ξ on 𝖷 we write f(ξ) for the push-forward of ξ under f:𝖷→𝖸.* We use ξ(f) as well as ∫ f dξ to denote the integral of a function f against a measure ξ.*denotes the continuous functions starting in x; =⋃_x∈.For ω∈ we write θ_sω for the path indefined by (θ_sω)_t≥ 0=(ω_t+s-ω_s)_t≥ 0.*denotes Wiener measure; _μ denotes law of Brownian motion started according to a probability μ; ^0(^a) the natural (augmented) filtration on .* For d∈ we set =×[0,1]^d, =⊗^d, and =(_t)_t≥ 0 the usual augmentation of (_t^0⊗ℬ([0,1]^d))_t≥ 0. To keep notation manageable, we suppress d from the notation since the precise number will always be clear from the context.*is a Polish space equipped with a Borel probability measure m. We set :=×, =m⊗, ^0=(_t^0)_t≥ 0=(ℬ()⊗_t^0)_t≥ 0, ^a the usual augmentation of ^0. * For d∈ we set =× [0,1]^d, =⊗^d, and =(_t)_t≥ 0 the usual augmentation of (_t^0⊗ℬ([0,1]^d))_t≥ 0. Again, we suppress d from the notation since the precise number will always be clear from the context.* The set of stopped paths started at 0 is denoted byS ={(f,s): f:[0,s] →} and we define r:×_+→ S by r(ω, t):= (ω_↾ [0,t],t). The set of stopped paths started inis S_ =(,S)={(x,f,s): f:[0,s] →, x∈} and we define r_:××_+→ S_ by r_(x,ω, t):= (x,ω_↾ [0,t],t), i.e. r_=(𝕀,r). * We use ⊕ for the concatenation of paths: depending on the context the arguments may be elements of S,or ×_+. Specifically, ⊕: 𝖸×𝖹→𝖹, where 𝖸 is either S or ×_+, and 𝖹 may be any of the three spaces. For example, if (f,s) ∈ S and ω∈, then (f,s) ⊕ω is the path ω'(t) = f(t) t ≤ sf(s) + ω(t-s) t > s . * As well as the simple concatenation of paths, we introduce a concatenation operator which keeps track of the concatenation time: if (f,s),(g,t) ∈ S, then (f,s)(g,t) = (f⊕ g,s,s+t).We denote the set of elements of this form as 2, and inductively, i in the same manner.* Elements of i will usually be denoted by (f,s_1,…,s_i) or (g,t_1,…,t_i). We define r_i:×Ξ^i→i by r_i(ω, s_1,…,s_i)):= (ω_↾ [0,s_i],s_1,…,s_i). Accordingly, the set of i-times stopped paths started inis i_=(,i). Elements of i_ are usually denoted by (x,f,s_1,…,s_i) or (y,g,t_1,…,t_i). In case of = we often simply write (f,s_1,…,s_i) or (g,t_1,…,t_i) with the understanding that f(0),g(0)∈. In case that there is no danger of confusion we will also sometimes write i_=i.The operators ⊕, generalise in the obvious way to allow elements of i_ to the left of the operator.* For (x,f,s_1,…,s_i)∈i_, (h,s)∈ S we often denote their concatenation by (x,f,s_1,…,s_i)|(h,s) which is the same element as (x,f,s_1,…,s_i)⊗ (h,s) but comes with the probabilistic interpretation of conditioning on the continuation of (f,s_1,…,s_i) by (h,s). In practice, this means that we will typically expect the (h,s) to be absorbed by a later ⊕ operation.* The map ×Ξ^i∋(x,ω,s_1,…,s_i)↦ (x,ω_⌞ [0,s_i],s_1,…,s_i)∈i_ will (by slight abuse of notation) also be denoted by r_i. * We set r̃_i: ×Ξ^i →i_× C(_+), (x,ω,s_1,…,s_i)↦ ((x,ω_⌞ [0,s_i],s_1,…,s_i), θ_s_iω). Then r̃_i is clearly a homeomorphism with inverse mapr̃^-1_i:((x,f,s_1,…,s_i),ω)↦ (x,f⊕ω,s_1,…,s_i).Hence, ξ^i=r̃_i^-1(r̃_i(ξ^i)) . For 1≤ i<d we can extend r̃_i to a map r̃_d,i: ×Ξ^d→i_× C(_+)×Ξ^d-i by settingr̃_d,i(x,ω,s_1,…,s_d)=((x,ω_⌞ [0,s_i],s_1,…,s_i),θ_s_iω, (s_i+1-s_i,…,s_d-s_i)).* For Γ_i ⊆iwe set Γ_i^<:={(f,s_1,…, s_i-1,s_i): ∃ (f̃,s_1,…,s_i-1, s̃)∈Γ, s_i-1≤ s_i< s̃}, where we set s_0=0.* For (f,s_1,…,s_i) ∈i we write f = sup_r ≤ s_i f(r), and f = inf_r ≤ s_i f(r).* For 1≤ i<n and Fa function on n resp. ×Ξ^n and (f,s_1,…,s_i)∈i we set F^(f,s_1,…,s_i) (η,t_i+1,…,t_n) := F(f⊕η,s_1,…,s_i,s_i+t_i+1,…,s_i+t_n) = F( (f,s_1,…,s_i) (η,t_i+1,…,t_n)),where (η,t_i+1,…,t_n) may be an element of n-i, or ×Ξ^n-i. We similarly defineF^(f,s_1,…,s_i)⊕ (η,t_i+1,…,t_n) := F(f⊕η,s_1,…,s_i-1,s_i+t_i+1,…,s_i+t_n) = F( (f,s_1,…,s_i)⊕ (η,t_i+1,…,t_n)),where (η,t_i,…,t_n) may be an element of n-i+1, or ×Ξ^n-i+1.* For any j-tuple 1≤ i_1 <… < i_j≤ d we denote by _× (i_1,…,i_j)the projectionfrom ×^d to ×^j defined by(x,ω,y_1,…,y_d)↦ (x,ω,y_i_1,…,y_i_j)and correspondingly, for ξ∈𝒫(×^d), ξ^(i_1,…,i_j)=_× (i_1,…,i_j)(ξ).When j=0, we understand this as simply the projection onto . If (i_1,…,i_j)=(1,…,j) we simply write ξ^(1,…,j)=ξ^i. § MAIN RESULTS§.§ Existence and Monotonicity PrincipleIn this section we present our key results and provide an interpretation in probabilistic terms. To move closer to classical probabilistic notions, in this section, we slightly deviate from the notation used in the rest of the article. We consider a Brownian motion B on some generic probability space and recall that, for each 1≤ i ≤ n,i:={(f,s_1,…,s_i):0≤ s_1≤…≤ s_i, f∈ C([0,s_i])}.We note that i carries a natural Polish topology.For a function γ:n→ which is Borel and a sequence (μ_i)_i=0^n of centered probability measures on , increasing in convex order, we are interested in the optimization problemP_γ= inf{[γ( (B_s)_s≤τ_n, τ_1, …, τ_n)]: τ_1, …, τ_nsatisfy (<ref>)}𝖮𝗉𝗍𝖬𝖲𝖤𝖯.We denote the set of all minimizers of (<ref>) by _γ. Take another Borel measurable function γ_2:n→. We will be also interested in the secondary optimization problemP_γ_2|γ=inf{[γ_2( (B_s)_s≤τ_n, τ_1, …, τ_n)]: (τ_1,…,τ_n)∈_γ}. 𝖮𝗉𝗍𝖬𝖲𝖤𝖯_2Both optimization problems, (<ref>) and (<ref>),will not depend on the particular choice of the underlying probability space, provided that (Ω,,(_t)_t≥ 0,) is sufficiently rich that it supports a Brownian motion (B_t)_t ≥ 0 starting with lawμ_0, and an independent, uniformly distributed random variable Y, which is _0-measurable (see Lemma <ref>). We will from now on assume that we are working in this setting. On this space, we denote the filtration generated by the Brownian motion by ^B.We will usually assume that (<ref>) and (<ref>) are well-posed in the sense that [γ((B_s)_s≤τ_n,τ_1,…,τ_n)] and [γ_2((B_s)_s≤τ_n,τ_1,…,τ_n)] exist with values in (-∞,∞] for all τ=(τ_1,…,τ_n) which solve (<ref>) and is finite for one such τ.Let γ,γ_2:n→ be lsc and bounded from below. Then, there exists a minimizer to (<ref>). The condition that the functions γ, γ_2 are bounded below can easily be relaxed (see <cit.>). We will prove this result in Section <ref>.Our main result is the monotonicity principle, Theorem <ref>, which is a geometric characterisation of optimizers τ̂=(τ̂_1,…,τ̂_n) of (<ref>). The version we state here is weaker than the result we will prove in Section <ref> but easier to formulate and still sufficient for our intended applications.For two families of increasing stopping times (σ_j)_j=i^n and (τ_j)_j=i^n with τ_i=0 we definek:=inf{j≥ i: τ_j+1≥σ_j}and stopping timesσ̃_j = {[ τ_jifj≤ k;σ_jifj>k ].and analogously τ̃_j = {[ σ_jifj≤ k;τ_jifj>k ]..Note that (σ̃_j)_j=i^n and (τ̃_j)_j=i^n are again two families of increasing stopping times, since τ̃_i=σ_i ≤ τ̃_i+1=_τ_i+1≥σ_iτ_i+1 + _τ_i+1<σ_iσ_i+1 ≤ τ̃_i+2=_τ_i+1≥σ_iτ_i+2 + _τ_i+1<σ_i (_τ_i+2≥σ_i+1τ_i+2 + _τ_i+2<σ_i+1σ_i+2)≤ τ̃_i+3=…, and similarly for σ̃_j. A pair ((f,s_1,…, s_i-1,s), (g,t_1,…,t_i-1,t))∈i×i constitutes an i-th stop-go pair, written ((f,s_1,…, s_i-1,s), (g,t_1,…, t_i-1,t))∈_i, if f(s)=g(t) and for all families of ^B-stopping times σ_i≤…≤σ_n, 0=τ_i≤τ_i+1≤…≤τ_nsatisfying 0<[σ_j]<∞ for all i≤ j≤ n and 0≤[τ_j]<∞ for all i<j≤ n[γ(( (f⊕ B)_u)_u≤ s + σ_n, s_1, …, s_i-1, s+ σ_i,s+ σ_i+1,…, s+ σ_n)] +[γ(( (g⊕ B)_u)_u≤ t + τ_n, t_1 , …, t_i-1 , t+ σ_i , t + τ_i+1 ,…, t + τ_n )] >[γ(( (f⊕ B)_u)_u≤ s + σ̃_n, s_1, …, s_i-1, s + σ_i,s+ σ̃_i+1,…, s+ σ̃_n)] +[γ(( (g⊕ B)_u)_u≤ t + τ̃_n, t_1 , …, t_i-1 , t + τ̃_i , t + τ̃_i+1 ,…, t + τ̃_n )],whenever both sides are well defined and the left hand side is finite. (See Figure <ref>.)A pair ((f,s_1,…, s_i-1,s), (g,t_1,…,t_i-1,t)) ∈i×i constitutes a secondary i-th stop-go pair, written ((f,s_1,…, s_i-1,s), (g,t_1,…,t_i-1,t)) ∈_2,i, if f(s)=g(t) and for all families of ^B-stopping times σ_i≤…≤σ_n, 0=τ_i≤τ_i+1≤…≤τ_nsatisfying 0<[σ_j]<∞ for all i≤ j≤ n and 0≤[τ_j]<∞ for all i<j≤ n the inequality (<ref>) holds with ≥ and if there is equality we have[γ_2(( (f⊕ B)_u)_u≤ s + σ_n, s_1, …, s_i-1, s+ σ_i,s+ σ_i+1,…, s+ σ_n)] +[γ_2(( (g⊕ B)_u)_u≤ t + τ_n, t_1 , …, t_i-1 , t+ σ_i , t + τ_i+1 ,…, t + τ_n )] >[γ_2(( (f⊕ B)_u)_u≤ s + σ̃_n, s_1, …, s_i-1, s + σ_i,s+ σ̃_i+1,…, s+ σ̃_n)] +[γ_2(( (g⊕ B)_u)_u≤ t + τ̃_n, t_1 , …, t_i-1 , t + τ̃_i , t + τ̃_i+1 ,…, t + τ̃_n )], whenever both sides are well defined and the left hand side (of (<ref>)) is finite. For 0≤ i<j≤ n we define _i:j→i by (f,s_1,…,s_j)↦ (f_⌞ [0,s_i],s_1,…,s_i) where we take s_0=0, 0=, and f_⌞ [0,0]:=f(0)∈.A set Γ=(Γ_1,…,Γ_n) with Γ_i⊂i for each i is called γ_2|γ-monotone if for each 1≤ i ≤ n _2,i∩ (Γ_i^<×Γ_i)=∅,whereΓ_i^<={(f,s_1,…,s_i-1,u):there exists(g,s_1,…,s_i-1,s)∈Γ_i, s_i-1≤ u <s, g_⌞ [0,u]=f},and _i-1(Γ_i)⊂Γ_i-1.Let γ,γ_2:n→ be Borel measurable, B be a Brownian motion on some stochastic basis (Ω,,(_t)_t≥ 0,) with B_0 ∼μ_0 and let τ̂=(τ̂_1,…,τ̂_n) be an optimizer of (<ref>). Then there exists a γ_2|γ-monotone set Γ=(Γ_1,…,Γ_n) supporting τ̂ in the sense that - a.s. for all 1≤ i≤ n ((B_s)_s≤τ_i,τ_1,…,τ_i)∈Γ_i. We will also consider ternary or j-ary optimization problems given j Borel measurable functions γ_1,…,γ_j:n→ leading to ternary or j-ary i-th stop-go pairs _i,3,…,_i,j for 1≤ i ≤ n, the notion of γ_j|…|γ_1-monotone sets and a corresponding monotonicity principle. To save (digital) trees we leave it to the reader to write down the corresponding definitions. §.§ New n-marginal embeddings §.§.§ The n-marginal Root embedding The classical Root embedding <cit.> establishes the existence of a barrier (or right-barrier) ℛ⊂_+× such that the first hitting time of ℛ solves the Skorokhod embedding problem. A barrier ℛ is a Borel set such that (s,x)∈ℛ⇒ (t,x)∈ℛ for all t>s. Moreover, the Root embedding has the property that it minimises [h(τ)] for a strictly convex function h:_+→ over all solutions to the Skorokhod embedding problem, cf. <cit.>. We will show that there is a unique n- marginal Root embedding in the sense that there are n barriers (ℛ^i)_i=1^n such that for each i≤ n the first hitting time of ℛ^i after hitting ℛ^i-1 embeds μ_i.Put γ_i:n→, (f,s_1,…,s_n)↦ h(s_i) for some strictly convex function h:_+→ and assume that (<ref>) is well posed. Then there exist n barriers (ℛ^i)_i=1^n such that definingτ^Root_1(ω)=inf{t≥ 0: (t,B_t(ω))∈ℛ^1}and for 1<i≤ nτ^Root_i(ω)=inf{t≥τ^Root_i-1(ω) : (t,B_t(ω))∈ℛ^i}the multi-stopping time (τ^Root_1,…,τ^Root_n) minimises[h(τ̃_i)] simultaneously for all 1≤ i≤ n among all increasing families of stopping times (τ̃_1,…,τ̃_n) such that B_τ̃_j∼μ_j for all 1≤ j≤ n. This solution is unique in the sense that for any solution τ̃_1, …, τ̃_n of such a barrier-type we have τ^Root_i=τ̃_i a.s. Fix a permutation κ of {1,…,n}. We consider the functions γ̃_1=γ_κ(1),…,γ̃_n=γ_κ(n) on nand the corresponding family of n-ary minimisation problems, (𝖮𝗉𝗍𝖬𝖲𝖤𝖯_n). Let (τ^Root_1,…,τ^Root_n) be an optimiser of P_γ̃_n|…|γ̃_1. By the n-ary version of Theorem <ref>, choose an optimizer (τ^Root_1,…,τ^Root_n) of (𝖮𝗉𝗍𝖬𝖲𝖤𝖯_n) and, by the corresponding version of Theorem <ref>,a γ̃_n|…|γ̃_1-monotone family of sets(Γ_1,…,Γ_n) supporting (τ^Root_1,…,τ^Root_n). Hence for every i≤ n we have-a.s.((B_s)_s≤τ_i,τ^Root_1,…,τ^Root_i)∈Γ_i, and (Γ^<_i×Γ_i)∩_i,n=∅.We claim that, for all 1≤ i≤ n we have_i,n⊃{((f,s_1,…,s_i),(g,t_1,…,t_i)): f(s_i)=g(t_i), s_i>t_i}.Fix (f,s_1,…,s_i),(g,t_1,…,t_i)∈i satisfying s_i>t_i and consider two families of stopping times (σ_j)_j=i^n and (τ_j)_j=i^n on some probability space (Ω,,) together with their modifications (σ̃_j)_j=i^n and (τ̃_j)_j=i^n as in Section <ref>. Putj_1:=inf{m≥ 1 : κ(m) ≥ i}and inductively for 1< a≤ n-i+1j_a:=inf{m≥ j_a-1 : κ(m)≥ i}.Let l={a: [σ_j_a≠σ̃_j_a]>0}. By the definition of σ̃_j and τ̃_j we have in case of j_l=i the equality {σ_j_l≠σ̃_j_l} = Ω and for j_l>i it holds that{σ_j_l≠σ̃_j_l} = ⋂_i≤ k <j_l{σ_k > τ_k+1}.As τ_k≤τ_k+1, in particular, we have on {σ_j_l≠σ̃_j_l} the inequality σ_k > τ_k for every i ≤ k≤ j_l. The strict convexity of h and s>t implies[h(s+σ_j_l)] + [h(t+τ_j_l)]>[h(s+σ̃_j_l)] + [h(t+τ̃_j_l)] .Hence, we get a strict inequality in (the corresponding κ^-1(j_l)-ary version of) (<ref>) and the claim is proven.Then we define for each 1≤ i≤ nℛ_^i:={(s,x)∈_+×: ∃ (g,t_1,…,t_i)∈Γ_i, g(t_i)=x, s≥ t_i}andℛ_^i:={(s,x)∈_+×: ∃ (g,t_1,…,t_i)∈Γ_i, g(t_i)=x, s > t_i}.Following the argument in the proof of Theorem 2.1 in <cit.>, we defineτ^1_ and τ^1_ to be the first hitting times of ℛ^1_ and ℛ^1_ respectively to see that actually τ^1_≤τ^Root_1≤τ_^1 and τ^1_=τ_^1 a.s. by the strong Markov property. Then we can inductively proceed and defineτ^i_:=inf{t≥τ^i-1_ : (t,B_t)∈ℛ^i_}andτ^i_:=inf{t≥τ^i-1_ : (t,B_t)∈ℛ^i_}.By the very same argument we see that τ^i_≤τ_i^Root≤τ^i_ and in fact τ^i_=τ^i_. Finally, we need to show that the choice of the permutation κ does not matter. This follows from a straightforward adaptation of the argument of Loynes <cit.> (see also <cit.> and <cit.>) to the multi-marginal set up. Indeed, the first barrier ℛ^1 is unique by Loynes original argument. This implies that the second barrier is unique because Loynes argument is valid for a general starting distribution of the process (t,B_t) in _+× and we can conclude by induction.* In the last theorem, the result stays the same if we take different strictly convex functions h_i for each i.*Moreover, it is easy to see that the proof is simplified if one starts with the objective ∑_i=1^n h_i(τ_i), which removes the need for taking an arbitrary permutation of the indices at the start. Of course, to get the more general conclusion, one needs to consider these permutations. Let h:_+→ be a strictly convex function and let γ:n→, (f,s_1,…,s_n)↦∑_i=1^n h(t_i). Let τ^Root=(τ^Root_1,…,τ^Root_n) be the minimizer of Theorem <ref>. Then it also minimizes [γ(τ̃_1,…,τ̃_n)]among all increasing families of stopping times τ̃_1≤…≤τ̃_n satisfying B_τ̃_i∼μ_i for all 1≤ i≤ n.§.§.§ The n-marginal Rost embedding The classical Rost embedding <cit.> establishesthe existence of an inverse barrier (or left-barrier) ℛ⊂_+× such that the first hitting time of ℛ solves the Skorokhod embedding problem. An inverse barrier ℛ is a Borel set such that (t,x)∈ℛ⇒ (s,x)∈ℛ for all s<t. Moreover, the Rost embedding has the property that it maximises [h(τ)] for a strictly convex function h:_+→ over all solutions to the Skorokhod embedding problem, cf. <cit.>. Similarly to the Root embedding it follows that Put γ_i:n→, (f,s_1,…,s_n)↦ -h(s_i) for some strictly convex function h:_+→ and assume that (<ref>) is well posed. Then there exist ninverse barriers (ℛ^i)_i=1^n such that definingτ^Rost_1(ω)=inf{t≥ 0: (t,B_t(ω))∈ℛ^1}and for 1<i≤ nτ^Rost_i(ω)=inf{t≥τ^Rost_i-1(ω) : (t,B_t(ω))∈ℛ^i}the multi-stopping time (τ^Rost_1,…,τ^Rost_n) maximises[h(τ_i)] simultaneously for all 1≤ i≤ n among all increasing families of stopping times ( τ_1,…,τ_n) such that B_τ_j∼μ_j for all 1≤ j≤ n. Moreover, it also maximises∑_i=1^n[ h(τ_i)].This solution is unique in the sense that for any solution τ̃_1, …, τ̃_n of such a barrier-type we have τ^Rost_i=τ̃_i. The proof of this theorem goes along the very same lines as the proof of Theorem <ref>. The only difference is that due to the maximisation we get_i,n⊃{(f,s_1,…,s_i),(g,t_1,…,t_i): f(s_i)=g(t_i), s_i<t_i}leading to inverse barriers. We omit the details. §.§.§ The n-marginal Azéma-Yor embedding For (f,s_1,…,s_n)∈n we will use the notation f̅_s_i:=max_0≤ s≤ s_i f(s). There exist n barriers (ℛ^i)_i=1^n such that definingτ^AY_1=inf{t≥ 0: (B̅_t,B_t)∈ℛ^1}and for 1<i≤ nτ^AY_i=inf{t≥τ^AY_i-1 : (B̅_t,B_t)∈ℛ^i}the multi-stopping time (τ^AY_1,…,τ^AY_n) maximises[∑_i=1^nB̅_τ_i] among all increasing families of stopping times ( τ_1,…,τ_n) such that B_τ_j∼μ_j for all 1≤ j≤ n. This solution is unique in the sense that for any solution τ̃_1, …, τ̃_n of such a barrier-type we have τ^AY_i=τ̃_i. We emphasise that this result has not appeared previously in the literature in this generality; previously the most general result was due to <cit.> and <cit.>, which proved a closely related result under an additional condition on the measures, which is not necessary here.In fact, similarly to the n-marginal Root and Rost solutions τ^AY simultaneously solves the optimization problemssup{ [B̅_τ̃_i]: τ̃_1≤…≤τ̃_n, B_τ̃_1∼μ_1, …,B_τ̃_n∼μ_n}for each i which of course impliesTheorem <ref> (see also Remark <ref>.<ref>). To keep the presentation readable, we only prove the less general version.Fix a bounded and strictly increasing continuous function ϕ:_+→_+ and consider thecontinuous functions γ(f,s_1,…,s_n)=-∑_i=1^n f̅_s_i and γ̃(f,s_1,…,s_n)=∑_i=1^n ϕ(f̅_s_i)f(s_i)^2 defined on n.Pick, by Theorem <ref>, a minimizer τ^AY of (<ref>) and, by Theorem <ref>, a γ̃|γ-monotone family of sets (Γ_i)_i=1^nsupporting τ^AY=(τ^AY_i)_i=1^n such that for all i≤ n_i,2∩(Γ_i^<×Γ_i)=∅.We claim that_i,2⊃{ ((f,s_1,…,s_i),(g,t_1,…,t_i))∈i×i: f(s_i)=g(t_i), f̅_s_i>g̅_t_i}.Indeed, pick ((f,s_1,…,s_i),(g,t_1,…,t_i))∈i×i with f(s_i)=g(t_i) and f̅_s_i>g̅_t_i and take two families of stopping times (σ_j)_j=i^n and (τ_j)_j=i^ntogether with their modifications (σ̃_j)_j=i^n and (τ̃_j)_j=i^n as in Section <ref>. We assume that they live on some probability space (Ω,,) additionally supporting a standard Brownian motion W. Observe that (as written out in the proof of Theorem <ref>) on {σ_j≠σ̃_j} it holds that σ_j>τ_j. Hence, on this set we have W̅_σ_j≥W̅_τ_j. This implies that forω∈{σ_j≠σ̃_j} (and hence σ̃_j=τ_j,τ̃_j=σ_j)f̅_s_i ∨(f(s_i) + W̅_σ_j(ω))+ g̅_t_i∨(g(t_i) + W̅_τ_j(ω))≤f̅_s_i∨(f(s_i) + W̅_σ̃_j(ω))+ g̅_t_i∨(g(t_i) + W̅_τ̃_j(ω)),with a strict inequality unless either W̅_σ_j(ω)≤g̅_t_i-g(t_i) or W̅_τ_j≥f̅_s_i-f(s_i).Onthe set {σ_j=σ̃_j} we do not change the stopping rule for the j-th stopping time and hence we get a (pathwise) equality in (<ref>). Thus, we always have a strictinequality in (<ref>) unless a.s. either W̅_σ_j(ω)≤g̅_t_i-g(t_i) orW̅_τ_j≥f̅_s_i-f(s_i) for all j. However, in that case we have for all j such that [σ_j≠σ̃_j]>0 (there is at least one such j, namely j=i)[ϕ(f̅_s_i)(f(s_i)+W_σ_j)^2]+[ϕ(g̅_t_i)(g(t_i) +W_τ_j)^2] >[ϕ(f̅_s_i)(f(s_i) +W_σ̃_j)^2]+[ϕ(g̅_t_i)(g(t_i) + W_τ̃_j)^2].Hence, ((f,s_1,…,s_i),(g,t_1,…,t_i))∈⊂_2 in the first case and in the second case we have((f,s_1,…,s_i),(g,t_1,…,t_i))∈_2proving (<ref>). For each i≤ n we define ℛ^i_:={(m,x):∃ (f,s_1,…,s_i)∈Γ_i, f(s_i)=x, f̅_s_i<m}andℛ^i_:={(m,x):∃ (f,s_1,…,s_i)∈Γ_i, f(s_i)=x, f̅_s_i≤ m}with respective hitting times (τ^0=0)τ^i_:=inf{t≥τ^i-1_: (B̅_t,B_t)∈ℛ^i_} and τ^i_:=inf{t≥τ^i-1_: (B̅_t,B_t)∈ℛ^i_}.We will show inductively on i that firstly τ^i_≤τ_i^AY≤τ^i_ a.s. and secondly τ^i_=τ^i_ a.s. proving the theorem. The case i=1 has been settled in <cit.>. So let us assume τ^i-1_=τ^i-1_ a.s. Then τ^i_≤τ_i^AY follows from the definition of τ^i_.To show that τ^AY_i≤τ_^i pick ω satisfying ((B_s(ω))_s≤τ_i^AY,τ_1^AY(ω),…,τ_i^AY(ω))∈Γ_i and assume that τ^i_(ω)<τ_i^AY(ω). Then there exists s∈[τ^i_(ω),τ_i^AY(ω)) such that f:=(B_r(ω))_r≤ s satisfies (f̅, f(s))∈ℛ^i_. Since τ^i-1_(ω)≤τ^i_(ω)≤ s < τ_i^AY(ω) we have (f,τ^1_(ω),…,τ^i-1_(ω),s)∈Γ_i^<. By definition of ℛ^i_, thereexists (g,t_ 1,…,t_i)∈Γ_i such that f(s)= g(t_i) and g̅_t_i < f̅_s, yielding a contradiction to (<ref>).Finally, we need to show that τ^i_=τ^i_ a.s. Before we proceed we give a short reminder of the case i=1 from <cit.>. We defineψ̃^1_0(m) = sup{x: ∃ (m,x) ∈ℛ^1_}.From the definition of ℛ^1_, we see that ψ̃^1_0(m) is increasing, and we define the right-continuous function ψ^1_+(m) = ψ̃^1_0(m+), and the left-continuous function ψ^1_-(m) = ψ̃^1_0(m-). It follows from the definitions of τ^1_ and τ^1_ that:τ_+ := inf{t ≥ 0: B_t ≤ψ^1_+(B_t)}≤τ^1_≤τ^1_≤inf{t ≥ 0: B_t < ψ^1_-(B_t)} =: τ_-.As ψ̃^1_0 has at most countably many jump points (discontinuity points) it is easily checked that τ_- = τ_+ a.s. and hence τ^1_=τ^1_= τ_1^AY. Note also that the law μ̅^1 of B̅_τ_1^AY can have an atom only at the rightmost point of its support. Hence, with π^1:= (B̅_τ_1^AY, B_τ_1^AY), the measure π^1_⌞{(x,y): y<x} has continuous projection onto the horizontal axis. Defining these quantities in obvious analogy for j∈{2, …, n}, we need to prove τ^i+1_=τ^i+1_= τ_i+1^AY assumingthatπ^ihas continuous projection onto the horizontal axis. To do so, we decompose π^i into free and trapped particlesπ^i_f:=π^i_⌞{(m,x):x > ψ^i_-(m)},π^i_t:=π^i_⌞{(m,x):x ≤ψ^i_-(m)}. Here π^i_f refers to particles which are free to reach a new maximum, while π^i_t refers to particles which are trapped in the sense that they will necessarily hit ℛ _^i (and thus also ℛ _^i) before they reach a new maximum. For particles started in π^i_f it follows precisely as above that the hitting times of ℛ _^i+1 and ℛ _^i+1 agree.For particles started in π^i_t this is a consequence of Lemma <ref>. Additionally, as above we findthatπ^i+1_⌞{(x,y): y<x}has continuous projection onto the horizontal axis.<cit.> Let μ be a probability measure on ^2 such that the projection onto the horizontal axis _x μ is continuous (in the sense of not having atoms) and let ψ: → be a Borel function. Set R^:= {(x,y): x>ψ(y)}, R^:= {(x,y): x≥ψ(y)}.Start a vertically moving Brownian motion B in μ and defineτ^:= inf{ t≥ 0: (x, y+B_t)∈ R^}, τ^:= inf{ t≥ 0: (x, y+B_t)∈ R^}.Then τ^=τ^ almost surely. §.§.§ The n-marginal Perkins/Hobson-Pedersen embedding For (f,s_1,…,s_n)∈n we will use the notation f_s_i:=min_0≤ s≤ s_i f(s) to denote the running minimum of the path up to time s_i. (Recall also that f̅_s_i is the maximum of the path). In this section we will consider a generalisation of the embeddings of Perkins <cit.> and Hobson and Pedersen <cit.>. The construction of Perkins to solve the one-marginal problem with a trivial starting law can be shown to simultaneously minimise [h(B̅_τ)] for any increasing function h, and maximise [k(B_τ)] for any increasing function k, over all solutions of the embedding problem. Later Hobson and Pedersen <cit.> described a closely related construction which minimised [h(B̅_τ)] over all solutions to the SEP with a general starting law. The solution of Perkins took the form:τ^P := inf{ t ≥ 0: B_t ≤γ_-(B̅_t)orB_t ≥γ_+(B_t)}for decreasing functions γ_+, γ_-. Hobson and Pedersen constructed, for the case of a general starting distribution, a stopping timeτ^HP := inf{ t ≥ 0: B_t ≤γ_-(B̅_t)or B_t ≥ G}where G was an appropriately chosen, _0-measurable random variable. They showed that τ^HP minimised [h(B̅_τ)] for any increasing function, but it is clear that the second minimisation does not hold in general. In <cit.>, the existence of a version of Perkins' construction for a general starting law is conjectured. Below we will show that the construction of Hobson and Pedersen can be generalised to the multi-marginal case, and sketch an argument that there are natural generalisations of the Perkins embedding to this situation, but argue that there is no `canonical' generalisation of the Perkins embedding. To be more specific, for given increasing functions h,k, the embedding(s) which maximise [k(B_τ_n)] over all solutions to the multi-marginal embedding problem which minimise [h(B̅_τ_n)] will typically differ from the embeddings which minimise [h(B̅_τ_n)] over all maximisers of [k(B_τ_n)].There exist n left-barriers (ℛ^i)_i=1^n and stopping times τ_1^*≤τ_2^* ≤…≤τ_n^* where τ_i^*<∞ implies B_τ^*_i = B̅_τ_i^* such thatτ^HP_1 = inf{t≥ 0: (B̅_t, B_t)∈ℛ^1}∧τ_1^*and for 1<i≤ nτ^HP_i=inf{t≥τ^HP_i-1 : (B̅_t, B_t)∈ℛ^i}∧τ_i^*the multi-stopping time (τ^HP_1,…,τ^HP_n) minimises[∑_i=1^nB̅_τ_i] among all increasing families of stopping times (τ_1,…,τ_n) such that B_τ_j∼μ_j for all 1≤ j≤ n.Fix a bounded and strictly increasing continuous function ϕ:_+→_+ and consider thecontinuous functions γ(f,s_1,…,s_n)=∑_i=1^n f̅_s_i, γ_2(f,s_1,…,s_n)=∑_i=1^n ϕ(f̅_s_i) f(s_i)^2 defined on n. Pick, by Theorem <ref>, a minimizer τ^HP of (<ref>) and, by Theorem <ref>,a γ_2|γ-monotone family of sets (Γ_i)_i=1^n supporting τ^HP=(τ^HP_i)_i=1^n such that for all i≤ n_i,2∩(Γ_i^<×Γ_i)=∅.By an essentially identical argument to that given in Theorem <ref>, we have_i,2⊃{ ((f,s_1,…,s_i),(g,t_1,…,t_i))∈i×i: f(s_i)=g(t_i), f̅_s_i < g̅_t_i}.Note that, given τ^HP_i, we can define stopping times τ^*_i:= τ^HP_i if B_τ^HP_i = B̅_τ_i^HP and to be infinite otherwise. For each i≤ n we define ℛ^i_:={(m,x):∃ (f,s_1,…,s_i)∈Γ_i, f(s_i)=x, f̅_s_i >m, x<m }andℛ^i_:={(m,x):∃ (f,s_1,…,s_i)∈Γ_i, f(s_i)=x, f̅_s_i≥ m, x<m }with respective hitting times (τ^0=0)τ^i_:=inf{t≥τ^i-1_: (B̅_t, B_t)∈ℛ^i_}and τ^i_:=inf{t≥τ^i-1_: (B̅_t, B_t)∈ℛ^i_}.It can be shown inductively on i that firstly τ^i_∧τ^*_i≤τ_i^HP≤τ^i_∧τ_i^* a.s., and secondly τ^i_∧τ^*_i=τ^i_∧τ^*_i a.s., proving the theorem. The proofs of these results are now essentially identical to the proof of Theorem <ref>. Of course, as before, a more general version of the statement (without the summation) can be proved, at the expense of a more complicated argument. The result above says nothing about the uniqueness of the solution. However the following argument (also used in <cit.>) shows that any optimal solution (to both the primary and secondary optimisation problem in the proof of Theorem <ref>) will have the same barrier form: specifically, suppose that (τ^i) and (σ^i) are both optimal. Define a new stopping rule which, at time 0, chooses either the stopping rule (τ^i), or the stopping rule (σ^i), each with probability 1/2. This stopping rule is also optimal (for both the primary and secondary rules), and the arguments above may be re-run to deduce the corresponding form of the optimal solution.In fact, a more involved argument would appear to give uniqueness of the resulting barrier among the class of all such solutions; the idea is to use a Loynes-style argument as before, but applied both to the barrier and the rate of stopping at the maximum. The difficulty here is to argue that any stopping times of the form given above are essentially equivalent to another stopping time which simply stops at the maximum according to some rate which will be dependent only on the choice of the lower barrier (that is, in the language above, (H_x^i < τ^*_i = τ^HP_i ≤ H_x+^i) is independent of the choice of τ^HP_i for any x and >0, where H_x^i:= inf{t ≥τ^HP_i-1: B_t ≥ x). By identifying each of the possible optimisers with a canonical form of the optimiser, and using a Loynes-style argument which combines two stopping rules of the form above by taking the maximum of the left-barriers, and the fastest stopping rate of the rules, one can deduce that there is a unique sequence of barriers and stopping rate giving rise to an embedding of this form. We leave the details to the interested reader. We conclude by considering informally the `Perkins'-type construction implied by our methods. Recall that in the single marginal case, where B_0 = 0, the Perkins embedding simultaneously both maximises the law of the minimum, and minimises the law of the maximum. A slight variant of the methods above would suggest that one could adapt the arguments above to consider the optimiser which has the same primary objective as above, and also then aims to minimise the law of the minimum. In this case the arguments may be run to give stopping regions (for each marginal) which are barriers in the sense that it is the first hitting time of a left-barrier ℛ which is left-closed in the sense that if (for a fixed x) a path with f̅_s = m, f_s = j is stopped, then so too are all paths with g̅_s = m',g_s = j', where (m',-j') ≺ (m,-j) and ≺ denotes the lexicographical ordering. With this definition, the general outline argument given above can proceed as usual, however we do not do this here since the final stage of the argument — showing that the closed and open hitting times of such a region are equal — would appear to be much more subtle than previous examples, and so we leave this as an open problem for future work.However, more notable is that in the multiple marginal case (and indeed, already to some extent in the case of a single marginal with a general starting law), the Perkins optimality property is no longer strictly preserved. To see why this might be the case (see also <cit.>) we note that, in the case of a single marginal, with trivial starting law, the embedding constructed via the double minimisation problems always stops at a time when the process sets a new minimum or a new maximum. At any given possible stopping point, the decision to stop should depend both on the current minimum, and the current maximum; however when the process is at a current maximum, both the current position and the current maximum are the same. In consequence, the decision to stop at e.g.a new maximum will only depend on the value of the minimum, and the optimisation problem relating to maximising a function of the maximum will be unaffected by the choice. In particular, it is never important which optimisation is the primary optimisation problem, and which is the secondary optimisation problem: in terms of the barrier-criteria established above, this can be seen by observing that in lexicographic ordering, (m',-j') ≺ (m,-j) is equivalent to (-j',m') ≺ (-j,m) if either m=m' or j=j'.On the other hand, with multiple marginals, we may have to consider possible stopping at times which do not correspond to setting a new maximum or minimum. Consider for example the case with μ_0 = δ_0, μ_1 = (δ_1 + δ_-1)/2, μ_2 = 2(δ_2 + δ_-2)/5+δ_0/5. In particular, the first stopping time, τ_1 must be the first hitting time of {-1,1}, and if the process stops at 0 at the second stopping time, then to be optimal, it must stop there the first time it hits 0 after τ_1. If we consider the probability that we return to 0 after τ_1, before hitting {-2,2}, then this is larger than 1/5, and we need to choose a rule to determine which of the paths returning to 0 we should stop. It is clear that, if the primary optimisation is to minimise the law of the maximum, then this decision would only depend on the running maximum, while it will depend only on the running minimum if the primary and secondary objectives are switched. In particular, the two problems give rise to different optimal solutions. The difference here arises from the fact that we are not able to assume that all paths have either the same maximum, or the same minimum. As a consequence, we do not, in general, expect to recover a general version of the Perkins embedding, in the sense that there exists a multi-marginal embedding which minimises the law of the maximum, and maximises the law of the minimum simultaneously. §.§.§ Further “classical” embeddings and other remarksBy combining the ideas and techniques from the previous sections and the techniques from <cit.> we can establish the existence of n-marginal versions of the Jacka and Vallois embeddings and their siblings (replacing the local time with a suitably regular additive functional) as constructed in <cit.>.We leave the details to the interested reader. We also remark that it is possible to get more detailed descriptions of the structure of the different barriers. At this point we only note that all the embeddings presented above have the nice property that their n-marginal solution restricted to the first n-1 marginals is in fact the n-1 marginal solution. This is a direct consequence of the extension of the Loynes argument to n-marginals as shown in the proof of Theorem <ref>. For a more detailed description of the barriers for the n-marginal Root embedding we refer to <cit.>.We also observethat, as in <cit.>, it is possible to deduce multi-marginal embeddings of some of the embeddings presented in the previous sections, e.g. Root and Rost, in higher dimensions. We leave the details to the interested reader. §.§.§ A n-marginal version of the monotone martingale coupling. We next discuss the embedding giving rise to a multi-marginal version of the monotone martingale transport plan. Note that we need an extra assumption on the starting law μ_0, but on μ_0 only. Assume that μ_0 is continuous (in the sense that μ_0(a)=0 for all a∈). Let c:×→ be three times continuously differentiable with c_xyy<0. Put γ_i:n→, (f,s_1,…,s_n)↦ c(f(0),f(s_i)) and assume that (<ref>) is well posed. Then there exist n barriers (ℛ^i)_i=1^n such that definingτ_1=inf{t≥ 0: (B_t-B_0, B_t)∈ℛ^1}and for 1<i≤ nτ_i=inf{t≥τ_i-1: (B_t-B_0, B_t)∈ℛ^i}the multi-stopping time (τ_1,…,τ_n) minimises[c(B_0,B_τ_i)] simultaneously for all 1≤ i≤ n among all increasing families of stopping times (τ̃_1,…,τ̃_n) such that B_τ̃_j∼μ_j for all 1≤ j≤ n. This solution is unique in the sense that for any solution τ̃_1, …, τ̃_n of such a barrier-type we have τ_i=τ̃_i. In the final stage of writing this article we learned of the work of Nutz, Stebegg, and Tan <cit.> on multi-period martingale optimal transport which (among various further results) provides an n-marginal version of the monotone martingale transport plan. Their methods are rather different from the ones employed in this article and in particular not related to the Skorokhod problem.The overall strategy of the proof, and in particular the first steps follow exactly the arguments encountered above. Fix a permutation κ of {1,…,n}. We consider the functions γ̃_1=γ_κ(1),…,γ̃_n=γ_κ(n) on nand the corresponding family of n-ary minimisation problems. Pick, by the n-ary version of Theorem <ref>, an optimizer (τ_1,…,τ_n) and, by the n-ary version of Theorem <ref>,a γ̃_n|…|γ̃_1-monotone family of sets(Γ_1,…,Γ_n) supporting (τ_1,…,τ_n), i.e. for every i≤ n we have-a.s.((B_s)_s≤τ_i,τ_1,…,τ_i)∈Γ_i, and (Γ_i^<×Γ_i)∩_i,n=∅.We claim that for all 1≤ i≤ n we have_i,n⊃{(f,s_1,…,s_i),(g,t_1,…,t_i): f(s_i)=g(t_i), g(0)>f(0)}.To this end, we have to consider (f,s_1,…,s_i),(g,t_1,…,t_i)∈i satisfying f(s_i)=g(t_i), f(s_i)-f(0)>g(t_i)-g(0) and consider two families of stopping times (σ_j)_j=i^n and (τ_j)_j=i^n together with their modifications (σ̃_j)_j=i^n and (τ̃_j)_j=i^n as in Section <ref>. However, since the modification of stopping times consists only of repeated swapping of the two stopping times what is effectively sufficient to prove is the following: For f(s)-f(0)>g(t)-g(0) and anystopping times ρ, σ, τ, where ρ≤σ, we have forσ̃:= σ_ρ≤τ +τ_ρ >τ, τ̃:= τ_ρ≤τ +σ_ρ >τthe inequality [ c(f(0), f(s) + B_σ)] +[c(g(0),g(t)+B_τ) ]> [c(f(0),f(s) + B_σ̃) ]+ [ c(g(0),g(t)+B_τ̃)],and that this inequality is strict, provided that the set ρ>τ has positive probability. To establish this inequality, of course only the parts were ρ> τ matters. Otherwise put, the inequality remains equallyvalid if we replace all of σ, τ, σ̃, τ̃ by τ∨σ on the set ρ≤τ, in which case wehave σ̃= τ, τ̃=σ, σ≥τ. Hence to prove (<ref>) it is sufficient to show for α:= (B_σ), β:= (B_τ) and a:= f(s)=g(t) that ∫ c(f(0),a+x)dα(x) + ∫ c(g(0),a+x)dβ(x) > ∫ c(f(0),a+x)dβ(x) + ∫ c(g(0),a+x)dα(x).To obtain this, we claim that t↦∫ c(t,a+x)dα(x) - ∫ c(t,a+x)dβ(x)is decreasing in t: This holds true since c_x is concave and β precedesαin the convex order (strictly if (ρ>τ)>0).Having established the claim, we define for each 1≤ i≤ nℛ_^i:={(d,x)∈_+×: ∃ (g,t_1,…,t_i)∈Γ_i, g(t_i)=x, d≥ g(t_i)-g(0)}andℛ_^i:={(d,x)∈_+×: ∃ (g,t_1,…,t_i)∈Γ_i, g(t_i)=x, d> g(t_i)-g(0)}.Following the argument used above, we defineτ^1_ and τ^1_ to be the first times the process (B_t-B_0, B_t)_t≥ 0 hits ℛ^1_ and ℛ^1_ respectively to see that actually τ^1_≤τ_1≤τ_^1. It remains to show thatτ^1_=τ_^1 (This has already been shown in <cit.>; we present the argument for completeness). To this end, note that the hitting time of (B_t-B_0, B_t)_t≥ 0 into a barrier can equally well be interpreted as the hitting time of (-B_0, B_t)_t≥ 0 into a transformed (i.e. sheared through the transformation (d,x)↦ (d-x,x) ) barrier.The purpose of this alteration is that the process (-B_0,B_t)_t≥ 0 moves only vertically and we can now apply Lemma <ref> to establish that indeedτ^1_=τ_^1. Observe that at this stage the continuity assumption on μ_0 is crucial. We then proceed by induction. As above, uniqueness and the irrelevance of the permutation follow from Loynes' argument. A very natural conjecture is then that Theorem <ref> would give rise to a solution to the peacock problem. The set of martingales (S_t)_t∈ [0,T] (more precisely the set of corresponding martingale measures) carries a natural topology and given D⊆ [0,T] with T∈ D the set of martingaleswith prescribed marginals (μ_t)_t∈ D is compact (cf.<cit.>). By taking limits of the solutions provided above along appropriate finite discretisations D⊆ [0,T], one obtains a sequence of optimisers to the discrete problem whose limit (S_t)_t∈ [0,T] satisfies S_t∼μ_t, t∈ [0,T] andminimizes[(S_t-S_0)^3] simultaneously for all t∈ [0,T] among all such martingales. However, since this is not the scope of the present article we leave details for future work. We note that this also provides a continuous time extension of the martingale monotone coupling rather different from the constructions given by Henry-Labordère, Tan, and Touzi <cit.> and Juillet <cit.>. § STOPPING TIMES AND MULTI-STOPPING TIMES For a Polish spaceequipped with a probability measure m we define a new probability space (,^0,(^0_t)_t≥ 0,) with :=×, ^0:= ℬ()⊗^0, ^0_t:=ℬ()⊗^0_t, := m ⊗, where ℬ() denotes the Borel σ-algebra on ,denotes the Wiener measure, and (^0_t)_t≥ 0the natural filtration. We denote the usual augmentation of ^0 by ^a. Moreover, for *∈{0,a} we set ^*_0-:=ℬ()⊗^*_0. If we want to stress the dependence on (,m) we write^a(,m), ^a_t(,m),….The natural coordinate process onwill be denoted by Y, i.e. for t≥ 0 we setY_t(x,ω)=(x,ω_t).Note that under , in the case where =, the process Y can be interpreted as a Brownian motion with starting law m. In particular, t↦ Y_t(x,ω) is continuous and ^0_t=σ(Y_s, s≤ t). We recallS:= { (f,s) : f∈ C[0,s], f(0)=0},S_:=(, S)and introduce the mapsr: ×_+→ S,(ω,t)↦ (ω_⌞ [0,t],t),r_: ×_+→ S_, (x,ω,t)↦ (x,r(ω,t)).We equip C_0(_+) with the topology of uniform convergence on compacts and S_ with the final topology inherited from ×_+ turning it into a Polish space. This structure is very convenient due to the following proposition which is a particular case of <cit.>. Optional sets / functions on ×_+ correspond to Borel measurable sets / functions on S_. More precisely we have: * A set D⊆×_+ is ^0-optional iffthere is a Borel set A⊆ S_with D=r_^-1(A). * A process Z=(Z_t)_t∈_+ is ^0-optional iff there is a Borel measurable H:S_→ such that Z=H∘ r. A ^0-optional set A⊆×_+ is closed in ×_+ iff the corresponding set r_(A) is closed in S_. A ^0-optional process Z=H∘ r_ is called S_- continuous (resp. l./u.s.c.) iff H:S_→ is continuous (resp. l./u.s.c.).Since the process t↦ Y_t(x,ω) is continuous the predictable and optional σ- algebras coincide (<cit.>). Hence, every ^0-stoppingtime τ is predictable and, therefore, foretellable on the set {τ > 0}.Let Z:→ be a measurable function which is bounded or positive. Then we define [Z|_t^0] to be the unique _t^0-measurable function satisfying [Z|_t^0](x,ω):=Z^M_t(x,ω):=∫ Z(x,ω_⌞[0,t]⊕ω')d(ω').Let Z∈ C_b(). Then Z^M_t defines an S_- continuous martingale (see Definition <ref>), Z^M_∞=lim_t→∞Z^M_t exists and equals Z. Up to a minor change of the probability space this is <cit.>.§.§ Randomised stopping times We set𝖬:={ξ∈𝒫^≤ 1(×_+): ξ(d(x,ω),ds)=ξ_x,ω(ds) (d(x,ω)), ξ_x,ω∈𝒫^≤ 1(_+)}and equip it with the weak topology induced by the continuous and bounded functions on ×_+. Each ξ∈𝖬 can be uniquely characterized by its cumulative distribution function A^ξ_x,ω(t):=ξ_x,ω([0,t]).A measure ξ∈𝖬 is called randomized stopping time, written ξ∈, iff the associated increasing process A^ξ is optional. If we want to stress the Polish probability space (,ℬ(), m) in the background, we write (,m). We remark that randomized stopping times are a subset of the so called 𝐏-measures introduced by Doleans <cit.> (for motivation and further remarks see <cit.>).In the sequel we will mostly be interested in a representation of randomized stopping times on an enlarged probability space. We will be interested in (',',(_t')_t≥ 0,') where ':=× [0,1], '(A_1× A_2)=(A)(A_2) ( denoting Lebesgue measure on ), ' is the completion of ^0⊗ℬ([0,1]), and (_t')_t≥ 0 the usual augmentation of (_t^0⊗ℬ([0,1]))_t≥ 0. The following characterization of randomized stopping times is essentially Theorem 3.8 of <cit.>. The only difference is the presence of thein the starting position, however it is easily checked that this does not affect the proof. Let ξ∈𝖬. Then the following are equivalent: * There is a Borel function A:S_→[0,1] such that the process A∘ r_ is right-continuous increasing andξ_x,ω([0,s]):=A∘ r_(x,ω,s)defines a disintegration of ξ wrt to . * We have ξ∈(,m), i.e. given adisintegration (ξ_x,ω)_(x,ω)∈ of ξ wrt , the random variable Ã_t(x,ω)=ξ_x,ω([0,t]) is ^a_t-measurable for all t∈_+. * For all f∈ C_b(_+) supported on some[0,t], t≥ 0 and all g∈ C_b() ∫ f(s) (g-[g|_t^0])(x,ω)ξ(dx,dω, ds)=0 * On the probability space (',',(_t')_t≥ 0,'), the random timeρ(x,ω,u) :=inf{ t ≥ 0 : ξ_x,ω([0,t]) ≥ u}defines an '-stopping time.An immediate consequence of (<ref>) is the closedness ofwrt to the weak topology induced by the continuous and bounded functions on ×_+ (cf. <cit.> andLemma <ref>).§.§ Randomised multi-stopping times In this section, we extend the results of the last section to the case of multiple stopping. Recall the notation defined in Section <ref>. In particular, for d≥ 1, recall that Ξ^d:={(s_1,…,s_d)∈^d_+,s_1≤…≤ s_d}and define 𝖬^d to consist of all ξ∈𝒫^≤ 1(×Ξ^d) such thatξ(d(x,ω),ds_1,…,ds_d)=ξ_x,ω(ds_1,…,ds_d) (d(x,ω)), ξ_x,ω∈𝒫^≤ 1(Ξ^d) .Recall that (,,(_t)_t≥ 0,) is defined by =× [0,1]^d,(A_1× A_2)=(A_1)^d(A_2), where ^d denotes the Lebesgue measure on ^d and _t is the usual augmentation of ^0_t⊗ℬ([0,1]^d). We mostly denote ^d(du)by du. For (u_1,…,u_d)∈[0,1]^d we often just write (u_1,…,u_d)=u. We suppress the d- index in the notation for the extended probability space. It will either be clear from the context which d we mean or we explicitly write down the corresponding spaces.A measure ξ∈^d is called randomised multi-stopping time, denoted by ξ∈_d, if for all 0≤ i≤ d-1 r̃_i+1,i(ξ^(i+1))∈(i_,r_i(ξ^i)).We denote the subset of all randomised multi-stopping times with total mass 1 by _d^1. If we want to stress the dependence on (,m) we write _d(,m) or _d^1(,m). Unlike for the randomised stopping times, there is no obvious analogue of (1), (2) or (3) of Theorem <ref> in the multi-stopping time setting. However below we prove a representation result for randomised multi-stopping times in a similar manner to (4). The following lemma (c.f. <cit.>) then enables us to conclude that, on an arbitrary probability space, all sequences of increasing stopping times can be represented as a randomised multi-stopping time on our canonical probability space. Let B be a Brownian motion on some stochastic basis (Ω, ℋ, (ℋ_t)_t≥ 0,) with right continuous filtration. Let τ_1,…,τ_n be an increasing sequence of ℋ-stopping times and considerΦ:Ω→×Ξ^d, ω̅↦ ((B_t)_t≥ 0,τ_1(ω̅),…,τ_n(ω̅)).Then ξ:=Φ() is a randomized multi-stopping time and for any measurable γ:n→ we have∫γ(f,s_1,…,s_n) r_n(ξ)(d(f,s_1,…,s_n))=_[γ((B_t)_t≤τ_n,τ_1,…,τ_n)].If Ω is sufficiently rich that it supports a uniformly distributed random variable which is ℋ_0-measurable, then for ξ∈ we can find an increasing family (τ_i)_1≤ i ≤ n of ℋ-stopping times such that ξ=Φ() and (<ref>) holds.For notational convenience we show the first part for the case n=2. Let B_0∼ m. It then follows by <cit.> that r̃_1,0(ξ)∈(,m). Hence, we need to show that r̃_2,1(ξ)∈(S_,r_1(ξ)), i.e. we have to show that ξ^2_(f,s) is r_1(ξ)–a.s. a randomized stopping time, where (ξ^2_(f,s))_(f,s) denotes a disintegration of ξ^2 wrt r_1(ξ). (Here and in the rest of the proof we assume f(0)∈ and suppress the “x” from the notation).First we show that r̃_1(ξ^1)(d(f,s),dω)=r_1(ξ^1)(d(f,s))(dω). Take a measurable and bounded F:S_×→. Then,using the strong Markov property in the last step, we have ∫ F((f,s),ω) r̃_1(ξ^1)(d(f,s),dω) = ∫ F((r_(ω̃,s),θ_sω̃) ξ^1(dω̃,ds)= ∫_Ω F(r_(B(ω),τ_1(ω)),θ_τ_1(ω)B(ω)) (dω)= ∫ F((f,s),ω̃) r_1(ξ^1)(d(f,s))(dω̃) .Let q be the projection from S_××_+ to S_×, and p be the projection from ×Ξ^2→×_+, p(ω,s_1,s_2)=(ω,s_1). Then, q∘r̃_2,1=r̃_1∘ p. Recalling that ξ^1=p(ξ^2) there is a disintegration of r̃_2,1(ξ^1,2) wrt r̃_1(ξ^1) which we denote byξ^2_(f,s_1),ω(ds_2)∈𝒫^≤ 1(_+).Then, we set ξ^2_(f,s_1)(dω,ds_2):=ξ^2_(f,s_1),ω(ds_2)(dω). Since dr̃_1(ξ^1)=dr_i(ξ^1)d the measures ξ^2_(f,s_1) define a disintegration of r̃_2,1(ξ^2) wrt r_1(ξ^1). We have to show that r_1(ξ^1) a.s. ξ^2_(f,s_1) is a randomized stopping time. We will show property (2) in Theorem <ref>, where now 𝖷=S_, m=r_1(ξ) and accordingly ^0_t=ℬ(S_)⊗^0_t with usual augmentation _t^a (cf. Section <ref>).To this end, fix t≥ 0 and let g:S_×→ be bounded and measurable and set h=_m[g|_t^a]. Then, it holds that _[g(r_1(B,τ_1),θ_τ_1B)|ℋ_τ_1+t]=h(r_1(B,τ_1),θ_τ_1B). Using rightcontinuity of the filtration ℋ in the third step to conclude that τ_2-τ_1 is an (ℋ_τ_1+t)_t≥ 0 stopping time, this implies∫ g((f,s),ω) ξ^2_(f,s),ω([0,t]) r_1(ξ^1)(d(f,s))(dω)=_[g(r_1(B,τ_1),θ_τ_1 B)_τ_2-τ_1≤ t]=_[_[g(r_1(B,τ_1),θ_τ_1 B)|ℋ_τ_1+t]_τ_2-τ_1≤ t]=_[h(r_1(B,τ_1),θ_τ_1 B)_τ_2-τ_1≤ t]= ∫ h((f,s),ω) ξ^2_(f,s),ω([0,t]) r_1(ξ^1)(d(f,s))(dω).This shows the first part of the lemma. To show the second part of the lemma we start by constructing an increasing sequence of stopping times on the extended canonical probability space(,,(_t)_t≥ 0,). By Theorem <ref> and the assumption that ξ^1∈(,m) there is astopping time ρ^1(x,ω,u)=ρ^1(x,ω,u_1) defining a disintegration of ξ^1 wrtvia∫δ_ρ^1(ds_1) du .By assumption, r̃_2,1(ξ^2)∈(S_,r_1(ξ^1)). Hence, writing s_2'=s_2-s_1 we can disintegrate ξ^2(d(x,ω),ds_1,ds_2)=∫ξ^2_r_(x,ω,ρ^1(x,ω,u_1))(θ_ρ^1(x,ω,u_1)ω,ds_2')δ_ρ^1(x,ω,u_1)(ds_1) du_1such that for r_1(ξ^1) a.e. (x,f,s_1) thedisintegration ξ^2_(x,f,s_1) is a randomized stopping time. Again by Theorem <ref> there is a stopping time ρ̃^2_x,f,s_1(ω̃,u_2) representing ξ^2_(x,f,s_1) as in (<ref>). Then,ρ^2(x,ω,u_1,u_2):= ρ^1(x,ω,u_1)+ρ̃^2_r_(x,ω,ρ_1(x,ω,u_1))(θ_ρ_1(x,ω,u_1)ω,u_2)defines astopping time such that (x,ω)↦∫_[0,1]^dδ_ρ^1(x,ω,u)(dt_1) δ_ρ^2(x,ω,u)(dt_2) dudefines a ^a- measurable disintegration of ξ^2 w.r.t. . We proceed inductively. To finish the proof, let U be the [0,1]^d–valued uniform ℋ_0–measurable random variable. Then τ_i:=ρ^i(B,U) define the required increasing family of ℋ stopping times. Lemma <ref> shows that optimizing over an increasing family of stopping times on a rich enough probability space in(<ref>) is equivalent to optimizing over randomized multi-stopping times on the Wiener space. Let ξ∈. On the extended canonical probability space(,,(_t)_t≥ 0,) there exists an increasing sequence (ρ^i)_i=1^d of - stopping times such that * for u=(u_1,…,u_d)∈ [0,1]^d and for each 1≤ i≤ d we have ρ^i(x,ω,u)=ρ^i(x,ω,u_1,…,u_i);*(x,ω)↦∫_[0,1]^dδ_ρ^1(x,ω,u)(dt_1)⋯δ_ρ^d(x,ω,u)(dt_d) dudefines a ^a- measurable disintegration of ξ w.r.t. . Next we introduce some notation to stateanother straightforward corollary. It is easy to see that q^d,i∘r̃_d,i = r̃_i∘ p^d,i, where q^d,i is the projection from i_× C(_+) ×Ξ^d-i to i_× C(_+), and p^d,i is the projection from ×Ξ^d to ×Ξ^i defined by (x,ω,s_1,…,s_d)↦ (x,ω,s_1,…,s_i). Recalling that ξ^i = p^d,i(ξ), it follows that there exists a disintegration of r̃_d,i(ξ) with respect to r̃_i(ξ^i), which we denote by:ξ_(x,f,s_1,…,s_i),ω(ds_i+1,…,ds_d) ∈𝒫( Ξ^d-i).Moreover, we setξ_(x,f,s_1,…,s_i)(dω,ds_i+1,…,ds_d):=ξ_(x,f,s_1,…,s_i),ω(ds_i+1,…,ds_d)(dω) ∈𝒫(C(_+) ×Ξ^d-i).The map (x,f,s_1,…,s_i)↦ξ_(x,f,s_1,…,s_i) inherits measurability from the joint measurability of ((x,f,s_1,…,s_i),ω)↦ξ_(x,f,s_1,…,s_i),ω. In particular, ξ_(x,f,s_1,…,s_i) defines a disintegration of r̃_d,i(ξ) w.r.t. r_i(ξ^i), since dr̃_i(ξ^i) = d dr_i(ξ^i) by the same calculation as (<ref>). Following exactly the line of reasoning as in the first part of the proof of Lemma <ref> yields Let ξ∈_d(,m) and 1≤ i<d. Then, for r_i(ξ^i) a.e. (x,f,s_1,…,s_i) we have ξ_(x,f,s_1,…,s_i)∈_d-i({0},δ_0).We note that the last Corollary still holds for i=0 by setting 0_=, r_0(ξ^j)=m. Then, the result says that for a disintegration (ξ_x)_x of ξ w.r.t. m for m-a.e. x∈ we have ξ_x∈_d. Of course this can also trivially be seen as a consequence of =m⊗. An important property ofis the following Lemma. is closed w.r.t. the weak topology induced by the continuous and bounded functions on ×Ξ^d.We fix 0≤ i≤ d-1 and considerthe Polish space =i_ with corresponding =× and =r_i(ξ^i)⊗. To show the defining property (<ref>) in Definition <ref> we consider condition (2) in Theorem <ref>; the goal is to express measurability of Z_t(x,ω):= ξ^i+1_x,ω(f), f∈C_b([0,t]), x∈i_, ω∈ in a different fashion. Note that a bounded Borel function h is _t^0-measurable iff for all bounded Borel functions G:→ [h G]=[h [G|_t^0]],of course this does not rely on our particular setup.By a functional monotone class argument, for _t^0-measurability of Z_t it is sufficient to checkthat [Z_t (G-[G|_t^0])]=0for all G∈ C_b().In terms of ξ^i+1, (<ref>) amounts to 0=[Z_t (G-[G|_t^0])]=∫(dx,dω) ∫ξ^i+1_x,ω(ds) f(s) (G-[G|_t^0])(x,ω) =∫ f(s) (G-[G|_t^0])(x,ω)r̃_i+1,i(ξ^i+1)(dx,dω, ds),which is a closed condition by Proposition <ref>. Given ξ∈^d and s≥ 0 we define the random measure ξ∧ son Ξ^d by setting for A⊂Ξ^d and each (x,ω)∈(ξ∧ s)_x,ω(A)= ∫_A(s_1∧ s,…,s_d∧ s) ξ_x,ω(ds_1,…,ds_d). Assume that (M_s)_s≥ 0 is a process on . Then(M_s^ξ)_s≥ 0 is defined to be the probability measure on ^d+1 such that for all bounded and measurable functions F:^d+1→∫_^d+1 F(y) M_s^ξ(dy)=∫_×Ξ^d F(M_0(x,ω),M_s_1(x,ω),…,M_s_d(x,ω)) (ξ∧ s)(dx,dω,ds_1,…,ds_d).This means that M_s^ξ is the image measure of ξ∧ s under the map M:×Ξ^d→^d+1 defined by(x,ω,s_1,…,s_d)↦ (M_0(x,ω),M_s_1(x,ω),…,M_s_d(x,ω)).We write lim_s→∞M^ξ_s=M_ξ if it exists.§.§ The set (μ_0,μ_1,…,μ_n), compactness and existence of optimisers.In this subsection, we specialise our setup to =, m=μ_0∈𝒫() and d=n. Let μ_0,μ_1,…,μ_n∈𝒫() be centered, in convex order and with finitesecond moment[It is possible to relax this, see <cit.>] ∫ x^2 μ_i(dx)=V_i<∞ for all i≤ n. In particular V_i≤ V_i+1. For t≥ 0 we set B_t(x,ω)=x+ω_t . We extend B to the extended probability spaceby setting B̅(x,ω,u)=B(x,ω). By considering the martingale B̅_t^2-t we immediately get (see the proof of Lemma 3.12 in <cit.> for more details)Let ξ∈_n and assume that B_ξ=(μ_0,μ_1,…,μ_n). Let (ρ_1,…,ρ_n) be any representation of ξ granted by Lemma <ref>. Then, the following are equivalent * [ρ^i]<∞ for all 1≤ i≤ n* [ρ^i]=V_i-V_0 for all 1≤ i≤ n* (B̅_ρ^i∧ t)_t≥0 is uniformly integrable for all 1≤ i≤ n.Of course it is sufficient to test any of the above quantities for i=n.We denote by (μ_0,μ_1,…,μ_n) the set of all randomised multi-stopping times satisfying one of the conditions in Lemma <ref>.By pasting solutions to the one marginal Skorokhod embedding problem one can see that the set (μ_0,μ_1,…,μ_n) is non-empty. However, the most important property isThe set (μ_0,μ_1,…,μ_n) is compact wrt to the topology induced by the continuous and bounded functions on ×Ξ^d.This is a direct consequence of the compactness of (μ_n) established in <cit.> as the set (μ_0,μ_1,…,μ_n)is closed.This result allows us to deduce one of the critical results for our optimisation problem:This follows from Proposition <ref> and the Portmanteau theorem.§.§ Joinings of Stopping timesWe now introduce the notion of a joining; these will be used later to define new stopping times which are candidate competitors for our optimisation problem.Let (,σ) be a Polish probability space. The set (m,σ) of joinings between =m⊗ and σ is defined to consist of all subprobability measures π∈𝒫^≤ 1(×_+×) such that* _×_+(π_⌞×_+× B)∈(,m) for all B∈ℬ();* _(π)=* _(π)≤σ .An important example in the sequel will be the probability space (,) constructed from=i_ and m=r_i(ξ^i) for ξ∈^1_n(,μ_0) and 0≤ i<n, where we set 0=, r_0(ξ^0)=μ_0 leading to =i_× C(_+) and =r_i(ξ^i)=r̃_i(ξ^i) (cf. Corollary <ref>).§ COLOUR SWAPS, MULTI-COLOUR SWAPS AND STOP-GO PAIRS In this section, we will define the general notion of stop-go pairs which was already introduced in a weaker form in Section <ref>. We will do so in two steps. First we define colour swap pairs and then we combine several colour swaps to get multi-colour swaps. Together, they build the stop-go pairs.Our basic intuition for the different swapping rules comes from the following picture. We imagine that each of the measures μ_1,…,μ_n carries a certain colour, i.e. the measure μ_i carries colour i. The Brownian motion will be thought of being represented by a particle of a certain colour: at time zero the Brownian particle has colour 1 and when it is stopped for the i-th time it changes its colour from i to i+1 (cf.Figure <ref> in Section <ref>).In identifying a stop-go pair, we want to consider two sub-paths, (f, s_1, …, s_i) and (g,t_1, …, t_i), and imagine the future stopping rules, which will now be a sequence of colour changes, obtained by concatenating a path ω onto the two paths. The simplest way of creating a new stopping rule is simply to exchange the coloured tails. This will preserve the marginal law of the stopped process, while generating a new multi-stopping time. A generalisation of this rule would be to try and swap back to the original colour rule at the jth colour change, where i < j. In this case, one would swap the colours until the first time one of the paths would stop for the jth time, after which one attempts to revert to the previous stopping rule. Note however that this may not be possible: if the other path has not yet reached the j-1st colour change, then the rules cannot be stopped, since one would have to switch from the jth colour to the j-1st colour, which is not allowed. Instead, in such a case, we simply keep the swapped colourings. We call recolouring rules of this nature colour swaps (or i ↔ j colour swaps). We will define such colour swap pairs in Section <ref>.After consideration of these colour swaps, it is clear that the determination of when to revert to the original stopping rule could be determined in a more sophisticated manner. For example, instead of trying to revert only on the jth colour change, one could instead try to revert on every colour change, and revert the first time it is possible to revert. This recolouring rule gives us a second set of possible path swaps, and we call such pairs multi-colour swaps. We will define these recolouring rules in Section <ref>. Of course, a multitude of other rules can easily be created. For our purposes, colour swaps and multi-colour swaps will be sufficient, but other generalisations could easily be considered. An important aspect of the recolouring rules are that they provide a recipe to map from one stopping rule to another, and an important aspect that needs to be verified is that the new stopping rule does indeed define a randomised multi-stopping time.We fix ξ∈_n^1(,μ_0) and γ:n_→. As in the previous section, we denote ξ^i=ξ^(1,…,i)=_× (1,…,i)(ξ). For (x,f,s_1,…,s_i)∈i_ we write (f,s_1,…,s_i) and agree on f(0)=x∈. For (f,s_1,…,s_i)∈i_ and (h,s)∈ S we will often write (f,s_1,…,s_i)|(h,s) instead of (f,s_1,…,s_i) (h,s)∈i+1_ to stress the probabilistic interpretation of conditioning the continuation of (f,s_1,…,s_i) on (h,s). §.§ Coloured particles and conditional randomised multi-stopping times. By Corollary <ref> and Remark <ref> (for i=0), for each 0≤ i≤ n the measure ξ_(f,s_1,…,s_i) is r_i(ξ^i)-a.s. a randomised multi-stopping time. For each 0≤ i≤ n-1 wefix a disintegration (ξ_(f,s_1,…,s_i),ω)_(f,s_1,…,s_i),ωof r̃_n,i(ξ) w.r.t. r̃_i(ξ^i) and set ξ_(f,s_1,…,s_i)=ξ_(f,s_1,…,s_i),ω(dω). We will need to consider randomised multi-stopping times conditioned on not yet having stopped the particle of colour i+1. To this end, observe thatξ^i+1_(f,s_1,…,s_i):= _C(_+)×Ξ^1(ξ_(f,s_1,…,s_i))defines a disintegration of r̃_i+1,i(ξ^i+1) wrt r_i(ξ^i). By Definition <ref>, ξ^i+1_(f,s_1,…,s_i)∈ a.s. and we set A^ξ_(f,s_1,…,s_i)(ω,t):= A^ξ_(f,s_1,…,s_i)(ω_⌞[0,t],t):=(ξ^i+1_(f,s_1,…,s_i))_ω([0,t])which is well defined for r_i(ξ^i)-almost every (f,s_1,…,s_i) by Theorem <ref>. For (f,s_1,…,s_i)∈i_ we define the conditional randomised multi-stopping time given (h,s)∈ S to be the (sub) probability measure ξ_(f,s_1,…,s_i)|(h,s)on C(_+)×Ξ^n-i given by(ξ_(f,s_1,…,s_i)|(h,s))_ω([0,T_i+1]×…×[0,T_n])=(ξ_(f,s_1,…,s_i))_h⊕ω((s,s+T_i+1]×…×(s,s+T_n]) ifA^ξ_(f,s_1,…,s_i)(h,s)<1 Δ A^ξ_(f,s_1,…,s_i)(h,s) (ξ_(f⊕ h,s_1,…,s_i,s_i+s))_ω([s,s+T_i+2]×…× [s,s+T_n]) ifA^ξ_(f,s_1,…,s_i)(h,s)=1,where Δ A^ξ_(f,s_1,…,s_i)(h,s)= A^ξ_(f,s_1,…,s_i)(h,s)- A^ξ_(f,s_1,…,s_i)(h,s-). The second case in (<ref>) corresponds to (ξ^i+1_(f,s_1,…,s_i))_h⊕ω having an atom at time s eating up `all the remaining (positive) mass' which is of course independent of ω. This causes a δ_0 to appear in (<ref>) below. Moreover, in this caseit is possible that also all particles of colour j∈{i+2,…,n} are stopped at time s by (ξ^i+1_(f,s_1,…,s_i))_h⊕ω. This is the reason for the closed intervals in the second line on the right hand side of (<ref>). Using Lemma <ref> resp. Corollary <ref> it is not hard to see that (<ref>) indeed defines a randomized multi-stopping times (you simply have to consider the stopping times ρ^l(ω,u_1,…,u_l) representingξ_(f,s_1,…,s_i) with u_1> A^ξ_(f,s_1,…,s_i) for the first case and the second case is immediate).Accordingly, we define the normalised conditional randomised multi-stopping times, byξ̅_(f,s_1,…,s_i)|(h,s):=1/1-A^ξ_(f,s_1,…,s_i)(h,s)·ξ_(f,s_1,…,s_i)|(h,s)ifA^ξ_(f,s_1,…,s_i)(h,s)<1, δ_0ξ_(f⊕ h,s_1,…,s_i,s_i+s)ifA^ξ_(f,s_1,…,s_i)(h,s-)<1and A^ξ_(f,s_1,…,s_i)(h,s)=1, δ_0⋯δ_0 else.We emphasize that the construction of ξ̅_(f,s_1,…,s_i)|(h,s) and ξ_(f,s_1,…,s_i)|(h,s) only relies on the fixed disintegration of r̃_n,i(ξ) w.r.t. r̃_i(ξ). In particular, the map((f,s_1,…,s_i),(h,s))↦ξ̅_(f,s_1,…,s_i)|(h,s)is measurable. Recall the connection of Borel sets of S_ and optional sets in ×_+ given by Proposition <ref>.Let (,m) be a Polish probability space. A set F⊂ S_ is called m-evanescent iff r_^-1(F)⊂×_+ is evanescent (wrt the probability space (,)) iff there exists A⊂ such that (A)=(m⊗)(A)=1 and r_(A×_+)∩ F=∅. By Corollary <ref>, ξ_(f,s_1,…,s_i)∈_n-i for r_i(ξ^i) a.e. (f,s_1,…,s_i)∈i. The next lemma says that for typical (f,s_1,…,s_i)|(h,s)∈i+1 this still holds for ξ̅_(f,s_1,…,s_i)|(h,s).Let ξ∈_n^1 and fix 0≤ i <n.* ξ̅_(f,s_1,…,s_i)|(h,s)∈^1_n-i outside a r_i(ξ^i)-evanescent set.* If F:n→ satisfies ξ(F∘ r_n)<∞, then the set {(f,s_1,…,s_i)|(h,s) : ξ̅_(f,s_1,…,s_i)|(h,s)(F^(f,s_1,…,s_i)|(h,s)⊕∘ r_n-i)=∞} is r_i(ξ^i)-evanescent. In particular, this applies to F(f,s_1,…,s_n)=s_n if ξ∈(μ_0,…,μ_n).Observe that a direct consequence of Corollary <ref>, assuming ξ(F∘ r_n)<∞, is that {(f,s_1,…,s_i) : ξ_(f,s_1,…,s_i)(F^(f,s_1,…,s_i)⊗∘ r_n-i)=∞} is a r_i(ξ^i) null set. It is apparent that ξ_(f,s_1…,s_i)|(h,s)∈.By Corollary <ref>, (<ref>), and Remark <ref> it is sufficient to show the claims under the additional hypothesis that A^ξ_(f,s_1…,s_i)(h,s)<1. Hence, considerU_1 ={(f,s_1,…,s_i)|(h,s) : A^ξ_(f,s_1…,s_i)(h,s)<1, ξ̅_(f,s_1,…,s_i)|(h,s)∉^1_n-i}, U_2 ={(f,s_1,…,s_i)|(h,s) : A^ξ_(f,s_1…,s_i)(h,s)<1, ξ̅_(f,s_1,…,s_i)|(h,s)(F^(f,s_1,…,s_i)|(h,s)⊕∘ r_n-i)=∞}.Set A^ξ_(f,s_1,…,s_i)(ω):=lim_s→∞ A^ξ_(f,s_1,…,s_i)(r(ω,s)). Then, (f,s_1,…,s_i)|(h,s)∈ U_1 is equivalent to ∫ A^ξ_(f,s_1,…,s_i)(h⊕ω) (dω)<1. Set X=i and m=r_i(ξ^i) and recall that the natural coordinate process onis denoted by Y. Given a ^0-stopping time τ on (,,) we have r_i(ξ^i) a.s. by the strong Markov property and the fact that ξ is almost surely a finite stopping time:1=∫ A^ξ_(f,s_1,…,s_i)(ω) (dω) =∫[_τ(ω)=∞ A^ξ_(f,s_1,…,s_i)(ω)+ ∫_τ(ω)<∞ A^ξ_(f,s_1,…,s_i)(ω_⌞ [0,τ]⊕ω̃) (ω̃)] (dω),implying that [((Y_s)_s≤τ,τ)∈ U_1]=0. Hence, the first part follows from the optional section Theorem.Additionally, setting α(d(x,ω),dt)=δ_τ(x,ω)(dt)(d(x,ω)) we have∫_U_2 dr_(α)(x,h,s) (1-A^ξ_x(h,s)) ξ̅_x|(h,s)(F^x|(h,s)⊕)≤ξ(F)<∞,implying r_(α)(U_2)=0. Hence, we have [((Y_s)_s≤τ,τ)∈ U_2]=0 proving the claim by the optional section theorem, e.g. <cit.> (see also Remark <ref>). §.§ Colour swaps As a first step towards the definition of stop-go pairs we introduce an important building block, the colour swap pairs.By Corollary <ref> and Corollary <ref>, for r_i(ξ^i) a.e. (g,t_1,…,t_i) there is an increasing sequence (ρ^j_(g,t_1,…,t_i))_j=i+1^n of ^a-stopping times such thatω↦∫_[0,1]^n-iδ_ρ_(g,t_1,…,t_i)^i+1(ω,u)(dt_i+1)⋯δ_ρ_(g,t_1,…,t_i)^n(ω,u)(dt_n) dudefines an ^a- measurable disintegration of ξ_(g,t_1,…,t_i) w.r.t. _μ_0. Similarly, by Lemma <ref>, outside an r_i-1(ξ^i-1) evanescent set, for (f,s_1,…,s_i-1)|(h,s) ∈i_ such that ξ̅_(f,s_1,…,s_i-1)|(h,s)≠δ_0 ⋯δ_0there is an increasing sequence (ρ^j_(f,s_1,…,s_i-1)|(h,s))_j=i^n of ^a-stopping times such thatω↦∫_[0,1]^n-i+1δ_ρ_(f,s_1,…,s_i-1)|(h,s)^i(ω,u)(ds_i)⋯δ_ρ_(f,s_1,…,s_i-1)|(h,s)^n(ω,u)(ds_n) dudefines an ^a- measurable disintegration of ξ̅_(f,s_1,…,s_i-1)|(h,s) w.r.t. _μ_0. We make the important observation that, if A^ξ_(f,s_1,…,s_i-1)(h,s)=1 (hence in this situation Δ A^ξ_(f,s_1,…,s_i-1)(h,s)>0), we have ρ_(f,s_1,…,s_i-1)|(h,s)^i≡δ_0. This representation allows us to couple the two stopping rules by taking realizations of the ρ^j_(g,t_1,…,t_i) stopping times and ρ_(f,s_1,…,s_i)|(h,s)^k stopping times on the same probability space Ω̅^f⊗ h,g:=C(R_+)× [0,1]^n-i+1 where of course one of the u-coordinates is superfluous for the ρ^j_(g,t_1,…,t_i) stopping times.For (f,s_1,…,s_i-1),(h,s) and (g,t_1,…,t_i) as above and n>j≥ i we defineΛ_j^f⊗ h,g:={ (ω,u)∈Ω̅^f⊗ h,g :ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u) ∨ρ^j_(g,t_1,…,t_i)(ω,u) ≤ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u) ∧ρ^j+1_(g,t_1,…,t_i)(ω,u)}.Note that this is the set where it is possible to swap the stopping rules from colour i up to colour j and not swap the stopping rule for colours greater than j.The set of colour swap pairs between colour i and j, i ≤ j < n, denoted by ^ξ_i↔ j is defined to consist of all (f,s_1,…,s_i-1)∈i-1_, (h,s)∈ S and (g,t_1,…,t_i)∈i_ such that f⊕ h(s_i-1+s)=g(t_i), 1-A^ξ_(f,s_1,…,s_i-1)(h,s)+Δ A^ξ_(f,s_1,…,s_i-1)(h,s)>0, and∫γ^(f,s_1,…,s_i-1)|(h,s)⊕(ω,s_i,…,s_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,ds_i,…,ds_n) + ∫γ^(g,t_1,…,t_i)⊗(ω,t_i+1,…,t_n) ξ_(g,t_1,…,t_i)(dω,dt_i+1,…,dt_n)> ∫(dω)du_Λ_j^f⊗ h,g(ω,u) [ ∫γ^(f,s_1,…,s_i-1)|(h,s)⊗(ω,t_i+1,…,t_j,s_j+1,…,s_n). δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dt_i+1)⋯δ_ρ^j_(g,t_1,…,t_i)(ω,u)(dt_j) δ_ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_j+1)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_n)+ ∫γ^(g,t_1,…,t_i)⊕(ω,s_i,…,s_j,t_j+1,…,t_n)δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_i)⋯δ_ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_j) δ_ρ^j+1_(g,t_1,…,t_i)(ω,u)(dt_j+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dt_n) ]+ ∫(dω)du (1-_Λ_j^f⊗ h,g(ω,u) ) [ ∫γ^(f,s_1,…,s_i-1)|(h,s)⊗(ω,t_i+1,…,t_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dt_i+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dt_n) + ∫γ^(g,t_1,…,t_i)⊕(ω,s_i,…,s_n) δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_i)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_n) ]. Moreover, we agree that (<ref>) holds in each of the following cases* ξ̅_(f,s_1,…,s_i-1)|(h,s)∉^1_n-i+1, ξ_(g,t_1,…,t_i)∉^1_n-i;* the left hand side is infinite;* any of the integrals appearing is not well-defined. Then we set _i^ξ = ⋃_j ≥ i_i↔ j^ξ. * In case that Λ_j^f⊕ h,g≠Ω̅^f⊗ h,g it is not sufficient to only change the colours/stopping rules from colour i to j. On the complement of Λ^f⊕ h,g, we have to switch the whole stopping rule from colour i up to colour n in order to stay within the class of randomised multi-stopping times. This is precisely the reason for the two big integrals appearing on the right hand side of the inequality.* Recall that ρ^i_(f,s_1,…,s_i-1)|(h,s)=δ_0 is possible so that it might happen that on both sides of (<ref>) the stopping rule of colour i is in fact the same and we only change the stopping rule from colour i+1 onwards.* In case of ^ξ_i↔ i the condition 1-A^ξ_(f,s_1,…,s_i-1)(h,s)+Δ A^ξ_(f,s_1,…,s_i-1)(h,s)>0 is not needed since there is no colour swap pair (with finite well defined integrals) not satisfying this condition. §.§ Multi-colour swaps Having introduced colour swap pairs we can now proceed and combine different colour swaps into multi-colour swap pairs.As described above, the aim is now to swap back as soon as possible. To this end, we consider for fixed i<n the following partition of Ω̅^f⊗ h,g defined in such a way that modifications of stopping rules in accordance to this partition transform randomised multi-stopping times into randomised multi-stopping times (c.f. (<ref>)).Ω̅^f⊗ h,g=⋃_j=i^n(Λ_j^f⊗ h,g∖∪_k=i^j-1Λ_k^f⊗ h,g),where Λ_n^f⊗ h,g:={ρ_(g,t_1,…,t_i)^i+1<ρ^i_(f,s_1,…,s_i-1)|(h,s), ρ_(g,t_1,…,t_i)^n < ρ^n_(f,s_1,…,s_i-1)|(h,s)}.This is indeed a partition: The different sets are disjoint by construction. Hence, the right hand side of (<ref>) is contained in the left hand side. We need to show that also the converse conclusion holds. Take (ω,u)∈Ω̅^f⊗ h,g. If ρ^i+1_(g,t_1,…,t_i)(ω,u)≥ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u),then (ω,u)∈Λ^f⊕ h,g_i. Otherwise, it holds thatρ^i+1_(g,t_1,…,t_i)(ω,u) < ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)≤ρ^i+2_(f,s_1,…,s_i-1)|(h,s)(ω,u) and eitherρ^i+2_(g,t_1,…,t_i)(ω,u)≥ρ^i+1_(f,s_1,…,s_i-1)|(h,s)(ω,u)orρ^i+2_(g,t_1,…,t_i)(ω,u)< ρ^i+1_(f,s_1,…,s_i-1)|(h,s)(ω,u) ≤ρ^i+3_(f,s_1,…,s_i-1)|(h,s)(ω,u) .In the former case, we have (ω,u)∈Λ^f⊕ h,g_i+1∖Λ^f⊕ h,g_i and in the latter case we have eitherρ^i+3_(g,t_1,…,t_i)(ω,u)≥ρ^i+2_(f,s_1,…,s_i-1)|(h,s)(ω,u)orρ^i+3_(g,t_1,…,t_i)(ω,u)< ρ^i+2_(f,s_1,…,s_i-1)|(h,s)(ω,u) ≤ρ^i+4_(f,s_1,…,s_i-1)|(h,s)(ω,u) .By induction, the claim follows.We put Λ̅_j^f⊗ h,g=Λ_j^f⊗ h,g∖∪_k=i^j-1Λ_k^f⊗ h,g. Then, the set of all multi-colour swap pairs starting at colour i, denoted by ^ξ_i, is defined to consist of all (f,s_1,…,s_i-1) ∈i-1_,(h,s)∈ S, (g,t_1,…,t_i)∈i_ such that f⊕ h(s_i-1+s)=g(t_i) and∫γ^(f,s_1,…,s_i-1)|(h,s)⊕(ω,s_i,…,s_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,ds_i,…,ds_n) + ∫γ^(g,t_1,…,t_i)⊗(ω,t_i+1,…,t_n) ξ_(g,t_1,…,t_i)(dω,dt_i+1,…,dt_n)> ∫(dω)du ∑_j=i^n-1[ _Λ̅_j^f⊗ h,g(ω,u) .∫γ^(f,s_1,…,s_i-1)|(h,s)⊗(ω,t_i+1,…,t_j,s_j+1,…,s_n) δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dt_i+1)⋯δ_ρ^j_(g,t_1,…,t_i)(ω,u)(dt_j) δ_ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_j+1)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_n)+ ∫γ^(g,t_1,…,t_i)⊕(ω,s_i,…,s_j,t_j+1,…,t_n)δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_i)⋯δ_ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_j) δ_ρ^j+1_(g,t_1,…,t_i)(ω,u)(dt_j+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dt_n) ]+ ∫(dω)du (1-∑_j=i^n-1_Λ̅_j^f⊗ h,g(ω,u) ) [ ∫γ^(f,s_1,…,s_i-1)|(h,s)⊗(ω,t_i+1,…,t_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dt_i+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dt_n)+ ∫γ^(g,t_1,…,t_i)⊕(ω,s_i,…,s_n) δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_i)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(ds_n) ].As in the case of colour swaps we agree that (<ref>) holds in each of the following cases* ξ̅_(f,s_1,…,s_i-1)|(h,s)∉^1_n-i+1, ξ_(g,t_1,…,t_i)∉^1_n-i;* the left hand side is infinite;* any of the integrals appearing is not well-defined. * Note that when ρ^i_(f,s_1,…,s_i-1)|(h,s)≡δ_0 we have Ω̅^f⊗ h,g=Λ_i^f⊗ h,g. Inserting this case into (<ref>) we see that both sides agree so that there are no multi-colour swap pairs satisfying ρ^i_(f,s_1,…,s_i-1)|(h,s)≡δ_0.* Observe that in the definition of ^ξ_i we do not need to impose the condition 1-A^ξ_(f,s_1,…,s_i-1)(h,s)+Δ A^ξ_(f,s_1,…,s_i-1)(h,s)>0 by Remark<ref>.§.§ Stop-go pairsFinally, we combine the previous two notions.Let ξ∈_n^1(,μ_0). The set of stop-go pairs of colour i relative to ξ, ^ξ_i, is defined by ^ξ_i = _i^ξ∪_i^ξ. We define the stop-go pairs of colour i in the wide sense by ^ξ_i=^ξ_i∪{(f,s_1,…,s_i-1)|(h,s)∈i_:A^ξ_(f,s_1,…,s_i-1)(h,s)=1}×i_. The set of stop-go pairs relative to ξ is defined by ^ξ := ⋃_1≤ i≤ n_i^ξ. The stop-go pairs in the wide sense are ^ξ:= ⋃_1≤ i≤ n_i^ξ.Every stop-go pair is a stop-go pair in the wide sense, i.e._i⊂_i^ξ for any 1≤ i≤ n. By loading notation, this follows using exactly the same argument as for the proof of <cit.>.As in <cit.>, we observe that the sets ^ξ and ^ξ are both Borel subsets of i×i, since the maps given in e.g. (<ref>) are measurable. In contrast, the setis in general just co-analytic.§ THE MONOTONICITY PRINCIPLE The aim of this section is to prove the main results, Theorem <ref> and the closely related Theorem <ref>. The structure of this section follows closely the structure of the proof of the corresponding results, Theorem 5.7 (resp. Theorem 5.16), in <cit.>. For the benefit of the reader, and to keep our presentation compact, we concentrate on those aspects of the proof where additional insight is needed to account for the multi-marginal aspects of the problem. We refer the reader to <cit.> for other details.The essence of the proof is to first show that if we have a candidate optimiser ξ, and a joining rule π which identifies stop-go pairs, we can construct an infinitesimal improvement ξ^π, which will also be a candidate solution, but which will improve the objective. It will follow that the joining π will place no mass on the set of stop-go pairs. The second part of the proof shows that we can strengthen this to give a pointwise result, where we can exclude any stop-go pair from a set related to the support of the optimiser. Important convention: Throughout this section, we fix a function γ:n→ and a measure ξ∈^1(μ_0,μ_1,…,μ_n). Moreover, for each 0≤ i≤ n-1 we fix a disintegration (ξ_(f,s_1,…,s_i),ω)_(f,s_1,…,s_i),ωof r̃_n,i(ξ) w.r.t. r̃_i(ξ^i) and set ξ_(f,s_1,…,s_i)=ξ_(f,s_1,…,s_i),ω(dω). Recall the map _i from Section <ref>.A family of Borelsets Γ=(Γ_1,…,Γ_n) with Γ_i⊂i_ is called (γ,ξ)-monotone iff for all 1≤ i ≤ n^ξ_i∩(Γ_i^<×Γ_i)=∅,whereΓ_i^<={(f,s_1,…,s_i-1,s):there exists(g,s_1,…,s_i-1,t)∈Γ_i, s_i-1≤ s <t, g_⌞ [0,s]=f},and _i-1(Γ_i)⊂Γ_i-1.Assume that γ:n→ is Borel measurable. Assume that (<ref>) is well posed and that ξ∈(μ_0,…,μ_1) is an optimizer. Then there exists a(γ,ξ)-monotone family of Borel sets Γ=(Γ_1,…,Γ_n) such that r_i(ξ)(Γ_i)=1 for each 1≤ i≤ n.The proof of Theorem <ref> is based on the following two propositions.Let γ:n_→ be Borel. Assume that (<ref>) is well posed and that ξ∈(μ_0,…,μ_1) is an optimizer. Fix 1≤ i ≤ n and set =i-1_, m=r_i-1(ξ^i-1). Then(r_⊗𝕀)(π)(_i^ξ)=0for all π∈(r_i-1(ξ^i-1),r_i(ξ^i)).Let (,m) and (, ν) bePolish probability spaces andE⊆ S_× a Borel set. Then the following are equivalent: * (r_⊗𝕀)(π)(E)=0 for all π∈^1(m, ν ).* E ⊆ (F ×) ∪ (S_× N) for some evanescent set F⊂ S_ and a ν-null set N⊆. This is a straightforward modification of <cit.> to the case of a general starting law (see also the proof of <cit.>). Note that Proposition <ref> is closely related to the classical section theorem (cf. <cit.>) which in our setup implies the following statement:Let (X,ℬ, m) be a Polish probability space. E⊂ S_ be Borel. Then the following are equivalent:* r_(α)(E)=0 for all α∈(,m)* E is m-evanescent* (((Y_s)_s≤τ,τ)∈ E)=0 for every ^0-stopping time τ.Fix 1≤ i≤ n. Set =i-1_, m=r_i-1(ξ^i-1) and consider the corresponding probability space (,). By Proposition <ref> (r_⊗𝕀)(π)(_i^ξ)=0 for all π∈^1(r_i-1(ξ^i-1),r_i(ξ^i)). Applying Proposition <ref> with =i_, ν=r_i(ξ^i) we deduce that there exists a r_i-1(ξ^i-1)-evanescent set F̃_i and a r_i(ξ^i)-null set N_i such that _i^ξ⊆ (F̃_i ×i_) ∪ (i_× N_i).Put F_i:={(g,t_1,…,t_i)∈i_ : ∃ (f,t_1,…,t_i-1,s_i)∈F̃_i,t_i≥ s_i, g≡ fon[0,s_i]}. Then, F_i is r_i-1(ξ^i-1)-evanescent and_i^ξ⊆ (F_i ×i_) ∪ (i_× N_i).Setting Γ̃_i=i_∖ (F_i∪ N_i) we have r_i(ξ^i)(Γ̃_i)=1 as well as ^ξ_i∩ (Γ̃_i^<×Γ̃_i)=∅. DefineΓ_i:=Γ̃_i∩{(g,t_1,…,t_i)∈i_ : A^ξ_(g,t_1,…,t_i-1)(θ_t_i-1(g)_⌞ [0,s],s)<1for alls<t_i-t_i-1},where θ_u(g)(·)=g(·+u)-g(u) as usual. Then r_i(ξ^i)(Γ_i)=1 and Γ_i^<∩{(g,t_1,…,t_i)∈i_ : A^ξ_(g,t_1,…,t_i-1)(θ_t_i-1(g)_⌞ [0,t_i-t_i-1],t_i-t_i-1)=1}=∅ so that ^ξ_i∩(Γ_i^< ×Γ_i)=∅. Finally, we can take a Borel subset of Γ_i with full measure and taking suitable intersections we can assume that _i-1_(Γ_i)⊆Γ_i-1.§.§ Proof of Proposition <ref> For notational convenience we will only prove the statement for the colour swap pairs _i^ξ. As the colour swap pairs are the main building block for the multi-colour swap pairs ^ξ_i it will be immediate how to adapt the proof for the general case. Moreover, it is clearly sufficient to show that for every j ≥ i we have (r_⊗𝕀)(π)(^ξ_i↔ j)=0 for each π∈(r_i-1(ξ^i-1),r_i(ξ^i)).Working towards a contradiction, we assume that there is an index i≤ j≤ n and π∈(r_i-1(ξ^i-1),r_i(ξ^i)) such that (r_⊗𝕀)(π)(^ξ_i↔ j)>0. By the definition of joinings (Definition <ref>) also π_⌞ (r_⊗𝕀)^-1(E)∈(r_i-1(ξ^i-1),r_i(ξ^i)) for any E⊂i-1_×i_. Hence, by considering (r_⊗𝕀)(π)_⌞^ξ_i ↔ j we may assume that (r_⊗𝕀)(π) is concentrated on ^ξ_i↔ j. Recall that (by definition) there is no colour swap pair ((f,s_1,…,s_i-1)|(h,s),(g,t_1,…,t_i)) with A^ξ_(f,s_1,…,s_i-1)(h,s)=1 and Δ A^ξ_(f,s_1,…,s_i-1)(h,s)=0. Hence,π[((f,s_1,…,s_i-1)|(h,s),(g,t_1,…,t_i)) : A^ξ_(f,s_1,…,s_i-1)(h,s)=1and Δ A^ξ_(f,s_1,…,s_i-1)(h,s)=0]=0. We argue by contradiction and define two modifications of ξ, ξ^E and ξ^L, based on the definition of _i ↔ j^ξ such that their convex combination yields a randomised multi-stopping time embedding the same measures as ξ and leading to strictly less cost. The stopping time ξ^E will stop paths earlier than ξ, and ξ^L will stop paths later than ξ.By Lemma <ref> and Corollary <ref>, for (f,s_1,…,s_i-1)|(h,s) outside an r_i-1(ξ^i-1) - evanescent set and (g,t_1,…,t_i) outside an r_i(ξ^i) null set there areincreasing sequences of ^a-stopping times(ρ^j_(f,s_1,…,s_i-1)|(h,s))_j=i^n and (ρ^j_(g,t_1,…,t_i))_j=i+1^n defining ^a-measurable disintegrations of ξ_(f,s_1,…,s_i-1)|(h,s) and ξ_(g,t_1,…,t_i) as in (<ref>).For B⊂ C(_+)×Ξ^n and (g,t_1,…,t_i)∈i_ we set B^(g,t_1,…,t_i)⊕:= {(ω,T_i,…,T_n)∈ C(_+)×Ξ^n-i+1 : (g⊕ω,t_1,…,t_i-1,t_i+T_i,…,t_i+T_n)∈ B}andB^(g,t_1,…,t_i)⊗:={(ω,T_i+1,…,T_n)∈ C(_+)×Ξ^n-i : (g⊕ω,t_1,…,t_i,t_i+T_i+1,…,t_i+T_n)∈ B}.Observe that both B^(g,t_1,…,t_i)⊕ and B^(g,t_1,…,t_i)⊗ are then Borel.Note that if we define F(g,t_1, …, t_n) = _B(g,t_1, …, t_n), then F^(f,s_1,…,s_i)⊕ (η,t_i+1,…,t_n) = _B^(f,s_1,…,s_i)⊕(η,t_i+1,…,t_n), and similarly for B^(g,t_1,…,t_i)⊗.Observe, that for (f,s_1,…,s_i-1)|(h,s) with A^ξ_(f,s_1,…,s_i-1)(h,s)=1 and Δ A^ξ_(f,s_1,…,s_i-1)(h,s)>0 it follows from (<ref>) thatξ_(f,s_1,…,s_i-1)|(h,s)(B^(g,t_1,…,t_i) ⊕) = Δ A^ξ_(f,s_1,…,s_i-1)(h,s)ξ_(f⊕ h,s_1,…,s_i-1,s_i-1+s)(B^(g,t_1,…,t_i)⊗).Then, we define the measure ξ^E by setting for B⊂ C(_+)×Ξ^n (recall Λ_j^f⊗ h,g from (<ref>)) ξ^E(B):= ξ(B)- ∫ξ_(f,s_1,…,s_i-1)|(h,s)(B^(f⊕ h,s_1,…,s_i-1,s_i-1+s)⊕) (r_⊗𝕀)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i)) + ∫(r_⊗𝕀)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i))(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s))[∫_C(_+)∫_[0,1]^n-i+1_Λ_j^f⊗ h,g(ω,u) _B^(f⊕ h,s_1,…,s_i-1,s_i-1+s)⊗(ω,ρ^i+1_(g,t_1,…,t_i)(ω,u),…,.. . ρ^j_(g,t_1,…,t_i)(ω,u),ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u),…,ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)) (dω) du+∫_C(_+)∫_[0,1]^n-i+1(1- _Λ_j^f⊗ h,g(ω,u) ) _B^(f⊕ h,s_1,…,s_i-1,s_i-1+s)⊗(ω,ρ^i+1_(g,t_1,…,t_i)(ω,u),…,ρ^n_(g,t_1,…,t_i)(ω,u))(dω) du].Similarly we define the measure ξ^L bysetting for B⊂ C(_+)×Ξ^nξ^L(B):= ξ(B)- ∫(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s)) ξ_(g,t_1,…,t_i)( B^(g,t_1,…,t_i)⊗) (r_⊗𝕀)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i)) + ∫(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s))[∫_C(_+)∫_[0,1]^n-i+1_Λ_j^f⊗ h,g(ω,u) _B^(g,t_1,…,t_i)⊕(ω,ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u),…,.. . ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u),ρ^j+1_(g,t_1,…,t_i)(ω,u),…,ρ^n_(g,t_1,…,t_i)(ω,u)) (dω) du+∫_C(_+)∫_[0,1]^n-i+1(1- _Λ_j^f⊗ h,g(ω,u) ). _B^(g,t_1,…,t_i)⊕(ω,ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u),…,ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u))(dω) du] (r_⊗𝕀)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i)). Then, we define a competitor of ξ byξ^π:=1/2(ξ^E+ξ^L). We will show that ξ^π∈(μ_0,…,μ_n) and ∫γ dξ > ∫γ dξ^π which contradicts optimality of ξ. First of all note that from the definition of Λ_j^f⊕ h, g in (<ref>) both ξ^E,ξ^L∈ (also compare (<ref>)). Hence, also ξ^π∈. Next we show that ξ^π∈(μ_0,…,μ_n).For bounded and measurable F: C()×Ξ^n→ (<ref>) and (<ref>) imply by using (<ref>) and (<ref>)2∫ F d(ξ-ξ^π)= ∫ (r_⊗ r_i)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i))(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s)) ×(∫ F^(f,s_1,…,s_i-1)|(h,s) ⊕(ω,S_i,…,S_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,dS_i,…,dS_n). +∫ F^(g,t_1,…,t_i)⊗(ω,T_i+1,…,T_n) ξ_(g,t_1,…,t_i)(dω,dT_i+1,…,dT_n)- ∫_C(_+)∫_[0,1]^n-i+1(dω)du_Λ_j^f⊗ h,g(ω,u) [∫F^(f,s_1,…,s_i-1)|(h,s)⊗(ω,T_i+1,…, T_j,S_j+1,…,S_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dT_i+1)⋯δ_ρ^j_(g,t_1,…,t_i)(ω,u)(dT_j) δ_ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_j+1)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_n)- ∫ F^(g,t_1,…,t_i)⊕(ω,S_i,…,S_j,T_j+1,…,T_n)δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_i)⋯δ_ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_j) δ_ρ^j+1_(g,t_1,…,t_i)(ω,u)(dT_j+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dT_n) ]- ∫(dω)du (1-_Λ_j^f⊗ h,g(ω,u) ) [ ∫F^(f,s_1…,s_i-1|(h,s)⊗( ω,T_i+1,…,T_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dT_i+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dT_n) - ∫ F^(g,t_1,…,t_i)⊕(ω,S_i,…,S_n) δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_i)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_n) ]) . Next we show that (<ref>) extends to nonnegative F satisfying ξ(F)<∞ in the sense that it is well defined with a value in [-∞,∞). To this end, we will show that ∫ (r_⊗ r_i)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i))(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s)) ×[∫ F^(f,s_1,…,s_i-1)|(h,s) ⊕( ω,S_i,…,S_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,dS_i,…,dS_n). +∫ F^(g,t_1,…,t_i)⊗(ω,T_i+1,…,T_n) ξ_(g,t_1,…,t_i)(dω,dT_i+1,…,dT_n)]<∞Since π∈(r_i-1(ξ^i-1),r_i(ξ^i)) the integral of the second term in the square brackets in (<ref>) is bounded by ξ(F), and hence is finite. To see that the first term is also finite, write _×_+(π)=:π_1 and note that π_1∈(i-1_,r_i-1(ξ^i-1)). Hence, the disintegration (π_1)_(f,s_1,…,s_i-1)of π_1 wrt r_i-1(ξ^i-1) is a.s. in . Fix (f,s_1,…,s_i-1)∈i-1_ and assume α:=(π_1)_(f,s_1,…,s_i-1)∈ (which holds on a set of measure one). In case that α_ω(_+)<1 we extend it to a probability on [0,∞] by adding an atom at ∞. We denote the resulting randomized stopping time still by α.Then, we can calculate using the strong Markov property of the Wiener measure for the first equality and F≥ 0 for the first inequality∫ F^(f,s_1,…,s_i-1)⊗(ω,s_i,…,s_n) ξ_(f,s_1,…,s_i-1)(dω,ds_i,…,ds_n) = ∬ F^(f,s_1,…,s_i-1)⊗(ω_⌞ [0,t]⊕θ_tω,s_i,…,s_n) (ξ_(f,s_1,…,s_i-1))_ω_⌞ [0,t]⊕θ_tω(ds_i,…,ds_n)α_ω(dt) (dω) = ∬F^(f,s_1,…,s_i-1)⊗(ω_⌞ [0,t]⊕ω̃,s_i,…,s_n) (ξ_(f,s_1,…,s_i-1))_ω_⌞ [0,t]⊕ω̃(ds_i,…,ds_n)α_ω(dt) (dω) (dω̃)≥ ∬_{(ω,t) : t ≤ s_i<∞} F^(f,s_1,…,s_i-1)⊗(ω_⌞ [0,t]⊕ω̃,s_i,…,s_n) (ξ_(f,s_1,…,s_i-1))_ω_⌞ [0,t]⊕ω̃(ds_i,…,ds_n)α_ω(dt) (dω) (dω̃) = ∬_{(ω,t) : t <∞} F^(f,s_1,…,s_i-1)|r(ω,t)⊕( ω̃,s'_i,…,s'_n) ξ_(f,s_1,…,s_i-1)|r(ω,t)(dω̃,ds_i',…,ds_n')α(dω,dt) Hence,∫ (r_⊗ r_i)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i))(1-A^ξ_(f,s_1,…,s_i-1)(h,s) + _A^ξ_(f,s_1,…,s_i-1)(h,s)=1Δ A^ξ_(f,s_1,…,s_i-1)(h,s)) ×∫ F^(f,s_1,…,s_i-1)|(h,s) ⊕( ω,S_i,…,S_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,dS_i,…,dS_n)=∫ (r_⊗ r_i)(π)(d((f,s_1,…,s_i-1)|(h,s)),d(g,t_1,…,t_i)) ×∫ F^(f,s_1,…,s_i-1)|(h,s) ⊕( ω,S_i,…,S_n) ξ_(f,s_1,…,s_i-1)|(h,s)(dω,dS_i,…,dS_n)≤ ∬ F^(f,s_1,…,s_i-1)⊗(ω,s_i,…,s_n) ξ_(f,s_1,…,s_i-1)(dω,ds_i,…,ds_n) r_i-1(ξ^i-1)(d((f,s_1,…,s_i-1))= ξ(F) < ∞ .Applying (<ref>) to 𝔱_n(ω,t_1,…,t_n)=t_n, and observing that all the terms on the right-hand side cancel implies that ξ^π(𝔱_n)=ξ(𝔱_n)<∞. Taking F(ω,s_1,…,s_n)=G(ω(s_j)) for 0≤ j≤ n with s_0:=0 for bounded and measurable G:→ the right hand side vanishes. For j<i this follows since ξ^i-1=(ξ^π)^i-1 as we have not changed any stopping rule for colours prior to i. For j≥ i this follows from the fact thatπ is concentrated on pairs ((f,s_1,…,s_i-1)|(h,s),(g,t_1,…,t_i)) satisfying f⊕ h(s_i+s)=g(t_i). Hence, we have shown that ξ^π∈(μ_0,…,μ_n).Using that ξ^π(γ^-),ξ(γ^-)<∞, by well posedness of (<ref>), we can apply (<ref>) to F=γ to obtain by the Definition of ^ξ_i↔ j that ∫ (γ∘ r_n)^(f,s_1,…,s_i-1)|(h,s) ⊕(ω,S_i,…,S_n) ξ̅_(f,s_1,…,s_i-1)|(h,s)(dω,dS_i,…,dS_n) +∫ (γ∘ r_n)^(g,t_1,…,t_i)⊗(ω,T_i+1,…,T_n) ξ_(g,t_1,…,t_i)(dω,dT_i+1,…,dT_n)- ∫_C(_+)∫_[0,1]^n-i+1_Λ_j^f⊗ h,g(ω,u) [∫(γ∘ r_n)^(f,s_1,…,s_i-1)|(h,s)⊗(ω,T_i+1,…, T_j,S_j+1,…,S_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dT_i+1)⋯δ_ρ^j_(g,t_1,…,t_i)(ω,u)(dT_j) δ_ρ^j+1_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_j+1)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_n)- ∫ (γ∘ r_n)^(g,t_1,…,t_i)⊕(ω,S_i,…,S_j,T_j+1,T_n)δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_i)⋯δ_ρ^j_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_j) δ_ρ^j+1_(g,t_1,…,t_i)(ω,u)(dT_j+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dT_n) ](dω)du- ∫(1-_Λ_j^f⊗ h,g(ω,u) ) [ ∫(γ∘ r_n)^(f,s_1,…,s_i-1)|(h,s)⊗( ω,T_i+1,…,T_n) .δ_ρ^i+1_(g,t_1,…,t_i)(ω,u)(dT_i+1)⋯δ_ρ^n_(g,t_1,…,t_i)(ω,u)(dT_n)- ∫ (γ∘ r_n)^(g,t_1,…,t_i)⊕(ω,S_i,…,S_n) δ_ρ^i_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_i)⋯δ_ρ^n_(f,s_1,…,s_i-1)|(h,s)(ω,u)(dS_n) ](dω)duis (r_⊗𝕀)(π) a.s. strictly positive applying Lemma <ref>. Hence, we arrive at the contradiction ∫γ dξ^π < ∫γ dξ. §.§ Secondary optimization (and beyond) We setμ̅=(μ_0,…,μ_n) and denote by (γ,μ̅) the set of optimizers of (<ref>). If π↦∫γ dπ is lower semicontinuous then (γ,μ̅) is a closed subset of (μ_0,μ_1,…,μ_n) and therefore also compact. Let γ,γ':n_→ be Borel measurable. The set of secondary stop-go pairs of colour i relative to ξ, short ^ξ_2,i, consists of all (f,s_1,…,s_i-1)|(h,s)∈i_,(g,t_1,…,t_i)∈i_ such that f⊕ h(s_i-1+s)=g(t_i) and either ((f,s_1,…,s_i-1)|(h,s),(g,t_1,…,t_i))∈^ξ_i or equality holds in (<ref>) for γ, and strict inequality holds in (<ref>) for γ', or equality holds in (<ref>) for γ, and strict inequality holds in (<ref>) for γ'. As before we agree that ((f,s_1,…,s_i-1)|(h,s),(g,t_1,…,t_i))∈^ξ_i if either of the integrals in (<ref>) or (<ref>) is infinite or not well defined.We also define the secondary stop-go pairs of colour i relative to ξ in the wide sense, ^ξ_2,i, by ^ξ_2,i=^ξ_2,i∪{(f,s_1,…,s_i-1)|(h,s)∈i_:A^ξ_(f,s_1,…,s_i-1)(h,s)=1}×i_.The set of secondary stop-go pairs relative to ξ is defined by ^ξ := ⋃_1≤ i≤ n_i^ξ. The secondary stop-go pairs in the wide sense are ^ξ:= ⋃_1≤ i≤ n_i^ξ. Let γ,γ':n_→ be Borel measurable. Assume that (γ,μ̅)≠∅ and that ξ∈(γ,μ̅) is an optimiser for P_γ'|γ(μ̅)=inf_π∈(γ,μ̅)∫γ' dπ.Then, for any i≤ n there exists a Borel set Γ_i⊂i such that r_i(ξ^i)(Γ_i)=1 and _2,i^ξ∩(Γ_i^<×Γ_i)=∅. Theorem <ref> follows from a straightforward modification of Proposition <ref> by the same proof as for Theorem <ref> using Proposition <ref>. We omit further details. Of coursethe previous theorem can be applied repeatedly to a sequence of functions γ,γ',γ”,… and sets (γ,μ̅),(γ'|γ,μ̅),(γ”|γ'|γ,μ̅),… leading to ^ξ_3,^ξ_4,… .We omit further details.§.§ Proof of Main Result We are now able to conclude, by observing that our main result is now a simple consequence of previous results.Since any ξ∈(μ_0,…,μ_n) induces via Lemma <ref> and Corollary <ref> a sequence of stopping times as used for the definition of stop-go pairs in Section <ref> the result follows from Theorem <ref>. plain | http://arxiv.org/abs/1705.09505v1 | {
"authors": [
"Mathias Beiglboeck",
"Alexander Cox",
"Martin Huesmann"
],
"categories": [
"math.PR",
"q-fin.PR",
"60G42, 60G44, 91G20"
],
"primary_category": "math.PR",
"published": "20170526095938",
"title": "The geometry of multi-marginal Skorokhod Embedding"
} |
APS/123-QED Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, U.K. Studying the effects of dark energy and modified gravity on cosmological scales has led to a great number of physical models being developed.The effective field theory (EFT) of cosmic acceleration allows an efficient exploration of this large model space, usually carried out on a phenomenological basis. However, constraints on such parametrized EFT coefficients cannot be trivially connected to fundamental covariant theories.In this paper we reconstruct the class of covariant Horndeski scalar-tensor theories that reproduce the same background dynamics and linear perturbations as a given EFT action.One can use this reconstruction to interpret constraints on parametrized EFT coefficients in terms of viable covariant Horndeski theories. We demonstrate this method with a number of well-known models and discuss a range of future applications.Valid PACS appear here Reconstructing Horndeski models from the effective field theory of dark energy Andy Taylor December 30, 2023 ==============================================================================§ INTRODUCTIONIdentifying the nature of the observed late-time accelerated expansion of the Universe <cit.> is one of the major outstanding problems in physics.The cosmological constant provides the simplest explanation but it is associated with a range of theoretical challenges <cit.>.Traditionally, an additional dark energy component in the matter sector or modifications of General Relativity on cosmological scales <cit.> have therefore been invoked to address the observed cosmic acceleration.Large-scale modifications of gravity may be motivated by low-energy extra degrees of freedom that could arise as effective remnants of a more fundamental theory of gravity and couple to the metric non-minimally.Moreover, non-standard gravitational effects can also be of interest to address problems in the cosmological small-scale structure <cit.>.Cosmological observations provide a new laboratory for tests of gravity that differ by about fifteen orders of magnitude in length scale to the more conventional tests in the Solar System <cit.>. Therefore it is well worth studying the range of possible large-scale modifications that can arise and the independent constraints on them that can be inferred from cosmology.In the simplest case the modification is introduced by a universally non-minimally coupled scalar field. This is the scenario considered here.The most general scalar-tensor theory introducing at most second-order equations of motion to evade Ostrogradski instabilities is described by the Horndeski action <cit.>.Despite providing restrictions on the space of possible scalar-tensor models, there remains considerable freedom within Horndeski theory.As a result, testing any observational consequences of the free functions in the Horndeski action directly is inefficient. It is necessary to solve the equations of motion for each model that one wishes to test in turn, and then compare it with observations.The formalism of effective field theory (EFT) can address these issues. One starts from the bottom up, with minimal assumptions about the underlying theory, and then constrains a smaller set of functions that parametrize a much larger class of covariant theories. The application of EFT to cosmology was originally carried out in the context of inflation <cit.>, while later being applied to dark energy and modified gravity models <cit.>.It has proved to be a fruitful approach. For example, it was shown using EFT that Horndeski theories cannot yield an observationally compatible self-acceleration that is genuinely due to modified gravity, unless the speed of gravitational waves significantly differs from the speed of light <cit.>. The same techniques used in EFT were also utilized in the discovery that there exists a class of scalar-tensor theories that contain higher order time derivatives, yet still avoid ghost-like instabilities <cit.> (also see Ref. <cit.>). Further applications can be found in Ref. <cit.>. Despite the utility of EFT, some issues remain to be addressed. For instance, it is not clear whether the chosen parametrization of the EFT functions arises naturally in modified gravity models <cit.>.Moreover, constraints on parametrized EFT functions describing the cosmological background and perturbations around it, cannot be connected to the non-perturbative non-linear regime or to different backgrounds than the cosmological setting. This omits, for instance, constraints arising from the requirement of screening effects <cit.> in high-density regions. Hence, in order to connect the observational constraints and interpret them in terms of the allowed forms of the Horndeski functions, one requires a covariant description of the phenomenological modifications adopted.In this paper we present the reconstruction of a baseline covariant scalar-tensor action from the EFT functions of a second-order unitary gauge action, defined in Sec. <ref>, that shares the same cosmological background and linear perturbations around it. Variations can then be applied to this action to move to another covariant theory that is equivalent at the background and linear perturbation level.This reconstruction enables measurements of parametrized EFT functions to be related to a range of sources from the covariant Horndeski terms, which can then be used to address the theoretical motivation of the phenomenological parameterizations. It can also be employed to extend predictions to the non-linear sector or to non-cosmological environments and implement screening conditions on the theoretical parameter space. The paper is organized as follows. In Sec. <ref>, we briefly review Horndeski scalar-tensor theory and the unitary gauge formalism that provides the tools for an EFT approach to the cosmological perturbations.We then present in Sec. <ref> our covariant action that is constructed to reproduce the unitary gauge action up to second order in the perturbations and hence yield the equivalent cosmological background dynamics and the linear perturbations around it.In Sec. <ref>, the derivation of the reconstructed action is discussed, before applying it to a few simple example models in Sec. <ref>.Finally, we present conclusions of our work in Sec. <ref>.§ HORNDESKI GRAVITY AND EFFECTIVE FIELD THEORY Horndeski gravity <cit.> describes the most general local, Lorentz-covariant, four-dimensional theory of a single scalar field interacting with the metric that yields at most second-order equations of motion and hence avoids Ostrogradski instabilities. Its action is given by S=∑_i=2^5∫ d^4 x √(-g) ℒ_i ,where the four Lagrangian densities are defined as ℒ_2 ≡G_2(ϕ,X),ℒ_3 ≡G_3(ϕ, X)ϕ ,ℒ_4 ≡G_4(ϕ, X)R -2G_4X(ϕ, X) [(ϕ)^2-(∇^μ∇^νϕ)(∇_μ∇_νϕ)],ℒ_5 ≡G_5(ϕ, X)G_μν∇^μ∇^νϕ+1/3G_5X(ϕ, X) [(ϕ)^3 -3(ϕ) (∇_μ∇_νϕ)(∇^μ∇^νϕ) .. +2(∇_μ∇_νϕ)(∇^σ∇^νϕ)(∇_σ∇^μϕ) ] ,and we have defined X ≡ g^μν∂_μϕ∂_νϕ. We shall work in units where c=ħ=1 throughout.These Lagrangians have been studied in a variety of different systems including black holes <cit.>, neutron stars <cit.> and inflationary models <cit.>.For cosmological purposes, at the background and linear level, it has proven useful to adopt a unitary gauge description of Eq. (<ref>) <cit.>.In this EFT formalism the freedom in the cosmological background metric and each G_i(ϕ,X) reduces to five free time-dependent functions. One describes the background dynamics while the other four functions encompass the linear perturbations around it. In the following, we shall briefly discuss the principles that go into building this EFT for the cosmological dynamics in the unitary gauge (see Refs. <cit.> for more details).The general procedure invokes the Arnowitt-Deser-Misner (ADM) formalism of General Relativity on a Friedmann-Lemaître-Robertson-Walker (FLRW) background to foliate the spacetime with spacelike hypersurfaces. The ADM line element is given by <cit.> ds^2=-N^2dt^2+h_ij(dx^i+N^idt)(dx^j+N^jdt) ,where N is the lapse, N^i is the shift and h_ij is the induced metric on the spacelike hypersurface. The induced metric can also be written in four-dimensional notation as h_μν=g_μν+n_μn_ν ,by identifying h_00=N^iN_i and h_0i=N_i. This framework provides a natural motivation for the introduction of the scalar field by treating it as the pseudo-Nambu-Goldstone boson of spontaneously broken time translational symmetry <cit.>.By associating the time coordinate with the scalar field, the scalar perturbations are absorbed into the metric.One is free to choose the functional form of the spacetime foliation, as long as the scalar field is a smooth function with a time-like gradient.We can then simplify the calculations by settingϕ = t M_*^2 , where M_* is a mass scale to match the dimensions. It can be thought of as a bare Planck mass related to the physical Planck mass through corrections from the EFT parameters <cit.>. Note that as the coordinate time is related to the scale factor in the FLRW background metric a(t), and this in turn is related to the matter content of the universe through the Friedmann equations, the gravitational action and the matter action are now no longer independent after this identification has been made.In this unitary gauge, we furthermore have X = g^00ϕ̇^2 = (-1+δ g^00)M_*^4 ,where g^00 is related to the lapse via g^00=-N^-2. Here and throughout the paper dots denote time derivatives and primes will represent derivatives with respect to the scalar field ϕ. Another geometrical quantity that will be used in the EFT action is the extrinsic curvature K_μν defined as K_μν=h_μσ∇^σn_ν ,where n_μ is the normal vector on the uniform time hypersurface, n_μ=-δ^0_μ/√(-g^00) . On a spatially flat FLRW background K_μν=Hh_μν, where H≡ȧ/a is the Hubble parameter, and hence the perturbation of the extrinsic curvature becomes δ K_μν=K_μν-Hh_μν. The final geometrical quantity that will be used is the three dimensional Ricci scalar R^(3), defined in the usual way but with the metric h_μν.The full unitary gauge action that describes the background and linear dynamics of Horndeski gravity is then given by <cit.> S = S^(0,1)+S^(2)+S_M[g_μν,ψ],whereS^(0,1) =M_*^2/2∫ d^4x √(-g)[ Ω(t) R -2Λ(t)-Γ(t)δ g^00],andS^(2) =∫ d^4x √(-g)[ 1/2M^4_2(t)(δ g^00)^2-1/2M̅^3_1(t) δ K δ g^00. .-M̅^2_2(t) ( δ K^2-δK^μνδ K_μν - 1/2δ R^(3)δ g^00) ]. For the zeroth and first-order action S^(0,1) we have adopted the notation of Ref. <cit.>. S^(2) is the action at second order and S_M is the matter action with minimal coupling between metric and matter fields.Note here that R^(3) is itself a perturbation on flat FLRW. Although everything in this work assumes flat space we keep the above notation of δ R^(3) to emphasize that it is a first- order quantity throughout.The EFT action (<ref>) separates out the background dynamics and the perturbations around it in a systematic way. We have six free functions of time, where a seventh free function of time enters through the FLRW metric with the scale factor a(t) or equivalently H(t).Four free functions are introduced at the background level, while another three enter the dynamics of the linear perturbations.Note, however, that two of the background EFT functions in Eq. (<ref>), including H(t), will be fixed by the Friedmann equations with a specified matter content. Given H(t), this leaves a degenerate background function which is only fixed at the level of the linear perturbations. The separation of the background and linear perturbations is more manifest in the notation introduced in Ref. <cit.>, in which there is one free function H(t) that determines the background evolution and four free functions describing the perturbations.More specifically, the background equations that follow from the EFT action, providing the two constraints, are given by <cit.> Γ +Λ= 3(Ω H^2+Ω̇H)-ρ_m/M_*^2 , Λ= 2ΩḢ+3Ω H^2+2Ω̇H+Ω̈ , where we assumed a matter-only universe with pressureless dust.Finally, an important aspect of the unitary gauge action (<ref>) for our discussion in Secs. <ref> and <ref> is that, at the level of linear theory, no new EFT functions appear in the description of ℒ_5 in addition to those introduced for ℒ_1-4 <cit.>. Hence, it will be sufficient to consider the reconstruction of a baseline covariant action for ℒ_1-4 only.§ RECONSTRUCTED HORNDESKI ACTIONSo far, much work has been devoted to representing specific theories in terms of the unitary gauge EFT parameters and devising parametrizations of the time-dependent EFT functions (see, e.g. Ref. <cit.>). Here we are interested in the inverse procedure. That is, the class of covariant theories that a set of EFT functions corresponds to.While a previous reconstruction was presented in Ref. <cit.>, the resulting general covariant action is not of the Horndeski type. Therefore it is not guaranteed to be theoretically stable.We shall now present a covariant formulation of a scalar-tensor theory that is embedded in the Horndeski action (<ref>) and is reconstructed from the free EFT functions of the second-order unitary gauge action (<ref>) such that they share the same cosmological background and linear dynamics. Given that it is not possible to specify a unique covariant theory based on its background and linear theory only, the reconstructed action will serve as a foundation upon which variations can then be applied to move between different covariant theories that are equivalent at the background and linear perturbation level.The basis of this reconstruction is the correspondence between the covariant formalism and the particular unitary gauge adopted, specified by Eq. (<ref>). The covariant Horndeski action that reproduces the same dynamics of the cosmological background and linear perturbations as the EFT action (<ref>) is given by (see Sec. <ref> for a derivation) G_2(ϕ, X) = -M_*^2U(ϕ) - 1/2M_*^2 Z(ϕ)X+a_2(ϕ)X^2+Δ G_2 , G_3(ϕ,X) =b_0(ϕ)+b_1(ϕ)X+Δ G_3 , G_4(ϕ, X) = 1/2M_*^2F(ϕ)+c_1(ϕ)X+Δ G_4 , G_5(ϕ, X)= Δ G_5 ,where the functional forms of the coefficients of X^n are presented in Table <ref>. The notation in Eqs. (<ref>) through (<ref>) is motivated such that Eq. (<ref>) reduces to the scalar-tensor action of Ref. <cit.> in the limit that a_2=b_0,1=c_1=0.The variations Δ G_i characterize the changes that can be performed on the baseline action (Δ G_i=0) to move between different covariant actions that are degenerate at the level of background and linear cosmology.For example, one may add terms to G_2 which are 𝒪[(1+X/M_*^4)^3]. In the unitary gauge these terms will be at least of order (δ g^00)^3 and hence do not affect linear theory.Similarly, after one takes into account an integration by parts relating terms in b_0(ϕ) and Z(ϕ) the variations Δ G_3 are 𝒪[(1+X/M_*^4)^3]. In fact, any non-zero contribution in b_0(ϕ) can be absorbed into Z(ϕ) in this way. Given this freedom, we have set b_0 to zero by default.The Δ G_4 term must be 𝒪[(1+X/M_*^4)^4], which is due to the presence of G_4X in Eq. (<ref>), changing anything of 𝒪[(1+X/M_*^4)^4] to 𝒪[(1+X/M_*^4)^3] with the variation having no effect on linear theory.Finally, as emphasized in Sec. <ref>, at the linear level contributions from G_5 can be absorbed into G_2, G_3, and G_4, and so the first term that appears in G_5 only affects non-linear scales. As ℒ_5 in the unitary gauge has at most one X derivative acting on G_5 <cit.>, as with Δ G_4, Δ G_5 starts at 𝒪[(1+X/M_*^4)^4]. Importantly, note that the coefficients in Eqs. (<ref>) through (<ref>) are not independent since there are only five free independent EFT functions in Eq. (<ref>). Hence, this leads to constraint equations between the coefficients.Another aspect worth noting is that due to the variations of the form (1+X/M_*^4)^n around the baseline covariant theory expressed in orders of X^n, the variations introduce well defined changes to all orders of each G_i in Eqs. (<ref>) through (<ref>).The functional form of each Δ G_i is specified by Δ G_2,3 = ∑_n>2ξ^(2,3)_n(ϕ)(1+X/M_*^4)^n, Δ G_4,5 = ∑_n>3ξ^(4,5)_n(ϕ)(1+X/M_*^4)^n,where ξ_n^(i)(ϕ) are a set of n free functions for each Δ G_i.Note that, using the reconstruction, one can build a model with a non-zero constant EFT function Λ and all the other EFT functions set to zero. As the addition of a Δ G_i term does not affect linear theory, by adding these extra terms one can construct a theory that can only be discriminated from ΛCDM on non-linear scales. Given a set of unitary gauge EFT functions Ω, Γ, Λ, M_2^4, M̅_1^3,M̅_2^2 and H, one can plug them into the relations given in Table <ref> and Eqs. (<ref>) through (<ref>) and derive the corresponding baseline covariant action.However, it is important to stress again that the action obtained in the process is not unique. Indeed, it may require the addition of specific Δ G_i as well as several field redefinitions to recover a recognizable form for a given theory. Examples of this are given in Sec. <ref>.Finally, for ease of use, we present in Table <ref> the relation of the EFT functions we have adopted to different parameterizations that are frequently used in the literature. These expressions can be thought of as consistency relations. For example, we have the relationship between the background conformal factor Ω, the mass scale M and the speed of gravitational waves c_T^2 Ω(t)=M^2/M_*^2c_T^2 . As discussed in Ref. <cit.>, a cosmological self-acceleration that is genuinely due to modified gravity implies a significant evolution in Ω departing from the value Ω=1 of General Relativity. The relation (<ref>) makes it explicit that this requires a deviation of the Planck mass from its bare value M_*, or a speed of gravitational waves that differs from that of light.It hence tests the consistency of a self-acceleration effect between the cosmological background, the large-scale structure, and the propagation of gravitational waves.§ RECONSTRUCTION METHODWe shall now provide a derivation for our reconstructed covariant Horndeski action presented in Sec. <ref>.The general approach to this reconstruction is as follows.We consider the sequence of terms of the unitary gauge action (<ref>) contributing at zeroth, first, and second- order. We contrast those with the different ℒ_i contributions to the covariant Horndeski Lagrangian, Eqs. (<ref>) through (<ref>). For this, we put them into the unitary gauge, which is a well defined procedure that has been dealt with in previous work <cit.>.This will identify the Lagrangians that include the required terms in the unitary gauge action, but those will also give rise to extra terms.Using Eq. (<ref>) it is possible to make these extra terms covariant and subtract them from the Horndeski Lagrangian that one originally started with. By construction, one is left with a covariant action that reduces to the required terms in the unitary gauge action after making that transformation.This procedure is only necessary for ℒ_3 and ℒ_4, where for ℒ_2 the reconstruction is straightforward.As discussed in Sec. <ref>, ℒ_5 does not introduce terms in the unitary gauge additional to the contributions arising from ℒ_2-4 and can thus be omitted.With this procedure we obtain a self consistent and well defined reconstruction of a baseline covariant theory from the unitary gauge action that shares the same cosmological background and linear perturbations around it, and to which variations can be applied to move to another covariant theory that is equivalent at the background and linear perturbation level (Sec. <ref>). For the discussion of reconstructing a covariant action from the terms in S^(2), we introduce the notation S^(2)_i with i=1,2,3 referring to S^(2) with all EFT parameters set to zero apart from M_2^4, M̅^3_1, and M̅^2_2, respectively. In Sec. <ref>, we discuss the quadratic contribution to Eq. (<ref>) arising from the zeroth and first-order EFT action (<ref>). The derivation of the first cubic contribution to Eq. (<ref>) from second-order perturbations in the EFT action is discussed in Sec. <ref>. Finally, the quartic term, Eq. (<ref>), is derived in Sec. <ref>.§.§ Quadratic term ℒ_2To start, consider the unitary gauge action up to first order in the perturbations, S^(0,1)_Ω=1=M_*^2/2∫ d^4x √(-g){R-2Λ(t)-Γ(t)δ g^00} ,where we have set Ω=1 (Ω≠1 will be considered in Sec. <ref>).The corresponding covariant action can be obtained through Eq. (<ref>), which yields S^(0,1)_Ω=1 = ∫ d^4x √(-g) {M_*^2/2R - M_*^2Λ(ϕ) .. -M_*^2/2Γ(ϕ) -Γ(ϕ)/2M_*^2X}. This is simply the action of a quintessence model with a non-canonical kinetic term (see Sec. <ref>).The contribution of the first second-order perturbation in the unitary gauge action (<ref>) is S^(2)_1=∫ d^4x √(-g){1/2M^4_2(t)(δ g^00)^2 }. Putting this into covariant form, one obtains the action S^(2)_1=∫ d^4x √(-g){M_2^4(ϕ)/2+M_2^4(ϕ)/M_*^4X+M_2^4(ϕ)/2M_*^8X^2}. Eq. (<ref>) is the contribution that a k-essence model <cit.> makes to Eq. (<ref>) at second order in X. The covariant or unitary gauge combinations S^(0,1)_Ω=1+S^(2)_1 describe the same cosmological background and linear theory of any function G_2(ϕ,X) in Eq. (<ref>) with G_3=G_5=0 and G_4=M_*^2/2. §.§ Cubic term ℒ_3Next, we consider a non-vanishing M̅_1^3 coefficient, which is the first term to give rise to a contribution to the cubic Lagrangian ℒ_3.It appears in the EFT action as S^(2)_2=∫ d^4x √(-g){ -1/2M̅_1^3(t)δ g^00δ K }. We now reconstruct a covariant action that reduces to Eq. (<ref>) to second-order perturbations in the unitary gauge.For this purpose, it is sufficient to consider the special case of G_3(ϕ,X)=ℓ_3(ϕ)X where ℓ_3 is a smooth function of ϕ only. One could do an alternative derivation by making G_3(ϕ,X) a function of an arbitrary power of X. Although the reconstructed covariant action would be different, the linear theory would be the same. After a few integrations by parts, in the unitary gauge adopting Eq. (<ref>) this term becomes <cit.> M_*^-6ℓ_3(ϕ) Xϕ = [ℓ̇_3(t)-3ℓ_3(t)H ] g^00-ℓ_3(t)δ g^00δ K -3ℓ_3(t)H+3H/4ℓ_3(t)(δ g^00)^2-1/4ℓ̇(t)(δ g^00)^2 . We take all the terms apart from that involving δ g^00δ K to the left-hand side of the equation and use Eq. (<ref>) to write δ g^00 in covariant form. Comparing Eqs. (<ref>) and (<ref>), we also make the identification ℓ_3(t) ≡1/2M̅_1^3(t) M_*^-6 . Hence, the covariant action that follows is given by S^(2)_2 = ∫ d^4x √(-g){9H M̅_1^3/8+M_*^2(M̅_1^3)^'/8 +M̅_1^3/2M_*^6X ϕ. . + [3HM̅_1^3/4M_*^4-(M̅_1^3)^'/4M_*^2]X +[(M̅_1^3)^'/8M_*^6-3HM̅_1^3/8M_*^8]X^2} ,which reduces to Eq. (<ref>) at second order in the unitary gauge. Note that after making the replacement (<ref>), there are also extra factors of M_* appearing from the replacement of the time derivative with a derivative with respect to the scalar field via Ṁ̅̇_1^3=M_*^2(M̅_1^3)^'.§.§ Quartic term ℒ_4Finally, we reconstruct the quartic Lagrangian density ℒ_4. The first contribution arises from the background term Ω(t), S^(0,1)_Λ=Γ=0=M_*^2/2∫ d^4x √(-g){Ω(t)R } ,which, after using equation (<ref>), yields the quartic contribution G_4=M_*^2Ω(ϕ)/2. We now proceed to the reconstruction of a covariant action that reduces to the second order unitary gauge action (<ref>) with all the EFT coefficients set to zero apart from M̅^2_2, S^(2)_3=∫ d^4x √(-g) { -M̅_2^2(t)(δ K^2-δ K^μνδ K_μν) . . +1/2M̅_2^2(t)δ R^(3)δ g^00}. For this purpose, consider the quartic Horndeski Lagrangian (<ref>) and transform it into the unitary gauge. This results in <cit.> ℒ_4= G_4R^(3)+(2g^00M_*^4G_4X-G_4)(K^2-K_μνK^μν) -2M_*^2√(-g^00)G_4ϕK. In order to carry out the reconstruction it is necessary to identify G_4 in terms of the EFT parameters. To do this one has to compare the coefficient of R in the covariant Lagrangian with that of R^(3) in the unitary gauge Lagrangian.To compare each term consistently, we will make use of the Gauss-Codazzi relation R^(3)=R-K_μνK^μν+K^2-2∇_ν(n^ν∇_μn^μ-n^μ∇_μn^ν), which relates R to R^(3). Hence, the contribution to the quartic term is G_4(ϕ,X)=M̅_2^2(ϕ)/2(1+X/M_*^4) , and the covariant Horndeski Lagrangian therefore is ℒ_4= M̅_2^2(ϕ)/2(1+X/M_*^4)R -M̅_2^2(ϕ)/M_*^4[ (ϕ)^2-∇_μ∇_νϕ∇^μ∇^νϕ] Note that we have used that δ R^(3) = R^(3) in flat space.Putting this into the unitary gauge gives ℒ_4= -M̅_2^2(δ K^2-δ K^μνδ K_μν)+1/2M̅_2^2δ R^(3)δ g^00+6M̅_2^2H^2-4M̅_2^2HK-3H^2M̅_2^2δ g^00+2M̅_2^2HKδ g^00-Ṁ̅̇_2^2δ g^00 K+1/2Ṁ̅̇_2^2K (δ g^00)^2 . To obtain a covariant action that yields the second-order unitary gauge action (<ref>), we take the last two lines of Eq. (<ref>) and move it to the left-hand side. Care must be taken in the transformation of the term -4M̅_2^2HK. In making it covariant one first has to do an integration by parts to take the derivative in K=∇_μn^μ onto the other coefficients. Using then the definition of n^μ in Eq. (<ref>) one obtains an expansion in powers of δ g^00 that up to second order goes as -4M̅_2^2HK=d/dt(M̅_2^2H){ 4-2δ g^00-1/2(δ g^00)^2}. One can then make the usual replacement for δ g^00 in Eq. (<ref>) and use the result from Sec. <ref> to transform all the terms involving a δ g^00δ K.This yields the covariant actionS^(2)_3 =∫ d^4x √(-g){[1/2M̅_2^2+M̅_2^2/M_*^4X]R -M̅_2^2/M_*^4[(ϕ)^2-∇^μ∇^νϕ∇_μ∇_νϕ] .-M_*^4(M̅_2^2)^''/4-7M_*^2(M̅_2^2)^'H/4-M_*^2H^'M̅_2^2-9H^2M̅_2^2/2+[(M̅_2^2)^''/2+H(M̅_2^2)^'/2M_*^2+2H^'M̅_2^2/M_*^2]X -[(M̅_2^2)^''/4M_*^4-H(M̅_2^2)^'/4M_*^6-H^'M̅_2^2/M_*^6+3H^2M̅_2^2/2M_*^8]X^2+[2HM̅_2^2/M_*^6-(M̅_2^2)^'/M_*^4]Xϕ.}.After putting action (<ref>) back into the unitary gauge, at second order in the perturbations one obtains action (<ref>).Note that a different reconstruction of δ g^00δ R^(3) that is not contained within the Horndeski action was recently presented in Ref. <cit.>.Combining the actions S^(0,1)_Λ=Γ=0, S^(0,1)_Ω=1, S^(2)_1, S^(2)_2, and S^(2)_3 in Eqs. (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), respectively, we obtain the expressions given for G_i in Eqs. (<ref>) through (<ref>), which thus are constructed to produce the same cosmological background and linear perturbations as the EFT action (<ref>).Note that, as discussed in Sec. <ref>, the quintic term G_5 does not introduce additional EFT functions in S^(0-2) and thus its phenomenology at the background and linear perturbation level can be captured by G_2-4. For simplicity, we have therefore adopted a baseline reconstruction with G_5=0 but allow for variations around this solution in Eq. (<ref>).§ SIMPLE EXAMPLESFor illustration, we provide here a brief discussion of the application of our reconstruction for three simple examples. In Sec. <ref>, we show how a quintessence model can be reconstructed and discuss some subtleties about the canonical form of the scalar field action. We then apply the reconstruction to f(R) gravity, cubic galileon gravity and a quartic model in Secs. <ref>, <ref> and <ref> respectively.§.§ QuintessenceLet us assume a measurement of Ω(t)=1, non-vanishing Λ(t) and Γ(t), and vanishing values for the other EFT functions.Applying this to our reconstructed action defined by Eqs. (<ref>) through (<ref>), one finds the action (<ref>).Note that the kinetic contribution is not in its canonical form.To find the canonical form of the action, we perform the field redefinition ∂χ/∂ϕ=1/M_*√(Γ(ϕ))such that in terms of the new scalar field χ, we obtain S=∫ d^4x √(-g){1/2M_*^2R-V(χ)-1/2(∂χ)^2} , where V(χ)=M_*^2(Λ(χ)+1/2Γ(χ) ) . Including a non-minimal coupling term Ω(t) in front of R further allows one to reconstruct a Brans-Dicke action in a similar way by choosing a suitable Γ(t) associated with the Brans-Dicke function ω(ϕ).§.§ f(R) gravityNext, we assume a measurement of varying Ω(t) and Λ(t) while all other EFT functions vanish.This is the scenario that would be expected for a f(R) model. f(R) gravity can be written as a Brans-Dicke type scalar-tensor theory with a scalar field potential and ω=0 (hence, vanishing Γ). The scalar field in this case can be associated with f_R≡ df(R)/dR, where the potential has a particular dependence on f_R,specified by f(R) and R.While we can therefore follow the same procedure as in Sec. <ref> for the reconstruction, we also consider here a slightly different approach (cf. <cit.>). In this case, instead of identifying the time coordinate with the scalar field, one identifies it with the Ricci scalar, adopting a gauge where its perturbations vanish, δ R=0.Hence, in this case, we directly find ℒ_f(R)=Ω(R)R-2Λ(R) = R + [Ω(R)R-R-2Λ(R)] ≡R + f(R). §.§ Cubic GalileonLet us assume a measurement of M_*^2Γ = 4M_2^4 = 3HM̅_1^3=-λ H,and Ω(t)=exp(-2M_*t) with a positive constant λ and all other EFT functions vanishing.Applying this to our reconstructed action, defined by Eqs. (<ref>) through (<ref>), and setting λ=6M_*^5r_c^2, defining a crossover distance r_c, we obtain ℒ=M_*^2/2e^-2ϕ/M_*R-r_c^2/M_*Xϕ+ℒ_M ,which is the Lagrangian density of a cubic galileon model <cit.>.§.§ Quartic LagrangianTo give a simple example of a reconstruction of a model involving G_4, let us assume that the relation M̅_2^2=λ holds for some constant λ. In addition, assume that the other EFT functions are related to H in the following way M̅_1^3 =-4λ H , M^4_2=-λḢ,Γ+Λ =-12H^2,Γ-Λ=8Ḣ.Using these relations in the reconstructed action in Eqs. (<ref>) to (<ref>) it is found that, upon identifying λ=M_*^2, one recovers the following quartic Horndeski Lagrangian ℒ=(M_*^2/2+1/2M_*^2X )R-1/M_*^2[(ϕ)^2-∇_μ∇_νϕ∇^μ∇^νϕ].§ CONCLUSIONSTackling the enduring puzzles behind the nature of cosmic acceleration and the consolidation of gravity with quantum theory has sparked an increased interest in cosmological modifications of gravity. Consequently, a plethora of new conceivable theories of gravity have been put forward. The wealth of cosmological observations acquired in previous decades has enabled tests of General Relativity to be performed at distance scales vastly different from the Solar System, providing a new laboratory to test these theories.The effective field theory of dark energy and modified gravity in the unitary gauge formalism enables a generalized and efficient examination of a large class of theories. So far, much work has gone into expressing a variety of given covariant theories in terms of the EFT unitary gauge functions. In this paper we have examined the inverse procedure. Starting from a given EFT unitary gauge action, for instance provided by measurement, one can derive a covariant Horndeski Lagrangian that shares the same dynamics of the cosmological background and linear fluctuations around it.As the reconstruction cannot be unique, we have focused on the recovery of a baseline covariant Horndeski action that reproduces the desired equivalent background and linear dynamics. We have furthermore characterized the variations of this action that can be performed to move between the covariant theories degenerate at the background and linear level. For illustration, we have applied our reconstruction method to a few simple example models embedded in the Horndeski action: quintessence, f(R) gravity, a cubic galileon model and a quartic model. A range of more involved reconstructions will be presented in a forthcoming paper. The reconstruction has a number of applications. Of particular interest will be the construction of a covariant realization of the linear shielding mechanism shown to be present in Horndeski theories by analysis of its unitary gauge action <cit.> (also see Ref. <cit.>). This mechanism operates in a large class of theories that can become degenerate with ΛCDM in the expansion history and linear perturbations. However, the degeneracy can be broken by the measurement of the speed of cosmological propagation of gravitational waves <cit.>.With the reconstruction, one can also address the question of how well motivated the frequently adopted parametrizations of the EFT functions in observational studies are <cit.>.Furthermore, the reconstruction will enable one to directly employ measurements of the EFT functions to impose constraints on the covariant Horndeski terms, which will be of particular interest to future surveys such as Euclid <cit.> or the Large Synoptic Survey Telescope (LSST) <cit.>.Finally, the covariant reconstruction disentangles the cosmological dependence of the Horndeski modifications in the EFT functions that is due to the spacetime foliation adopted in the unitary gauge.Hence, a reconstructed action from phenomenological EFT functions can be applied to non-perturbative regimes (see e.g. Ref. <cit.>) or non-cosmological backgrounds and used to connect further observational constraints, for instance, arising from the requirement of screening effects in high-density regimes.This list of applications of our reconstructed Horndeski action is far from exhaustive, motivating further examination of the matter in future work. We thank Yuval Nissan and Miguel Zumalacárregui for useful discussions. This work is supported by the STFC Consolidated Grant for Astronomy and Astrophysics at the University of Edinburgh. J.K. thanks STFC for support through an STFC studentship. L.L. also acknowledges support from a SNSF Advanced Postdoc.Mobility Fellowship (No. 161058). A.N.T. thanks the Royal Society for support from a Wolfson Research Merit Award. Please contact the authors for access to research materials. JHEP | http://arxiv.org/abs/1705.09290v2 | {
"authors": [
"Joe Kennedy",
"Lucas Lombriser",
"Andy Taylor"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170525143308",
"title": "Reconstructing Horndeski models from the effective field theory of dark energy"
} |
Improving the Functional Control of Aged Ferroelectrics using Insights from Atomistic Modelling D. M. Duffy December 30, 2023 ===============================================================================================§ INTRODUCTION Biotechnological products are obtained by treating cells as little factories that transform substrates into products of interest. There are three major modes of cell culture: batch, fed-batch and continuous. In batch, cells are grown with a fixed initial pool of nutrients until they starve, while in fed-batch the pool of nutrients is re-supplied at discrete time intervals. Cell cultures in the continuous mode are carried out with a constant flow carrying fresh medium replacing culture fluid, cells, unused nutrients and secreted metabolites, usually maintaining a constant culture volume. While at present most biotechnology industrial facilities adopt batch or fed-batch processes, the advantages of continuous processing have been vigorously defended in the literature <cit.>, and currently some predict its widespread adoption in the near future <cit.>.A classical example of continuous cell culture is the chemostat, invented in 1950 independently by Aaron Novick and Leo Szilard <cit.> (who also coined the term chemostat) and by Jacques Monod <cit.>. In this system, microorganisms reside inside a vessel of constant volume, while sterile media, containing nutrients essential for cell growth, is delivered at a constant rate. Culture medium containing cells, remanent substrates and products secreted by the cells are removed at the same rate, maintaining a constant culture volume. The main dynamical variable in this system is the dilution rate (D), which is the rate at which culture fluid is replaced divided by the culture volume. In a well-stirred tank any entity (molecule or cell) has a probability per unit time D of leaving the vessel. In industrial settings, higher cell densities are achieved by attaching a cell retention device to the chemostat, but allowing a bleeding rate to remove cell debris <cit.>. Effectively only a fraction 0 ≤ϕ≤ 1 of cells are carried away by the output flow D. This variation of the continuous mode is known as perfusion culture.By definition, a continuous cell culture ideally reaches a steady state when the macroscopic properties of the tank (cell density and metabolite concentrations) attain stationary values. Industrial applications place demands on the steady state, usually: high-cell density, minimum waste byproduct accumulation, and efficient nutrient use. However, identical external conditions (dilution rate, media formulation) may lead to distinct steady states with different metabolic properties (a phenomenon known in the literature as multi-stability or multiplicity of steady states) <cit.>. Therefore, for the industry, it becomes fundamental to know in advance, given the cell of interest and the substrates to be used, which are the possible steady states of the system and how to reach them. Moreover, to satisfy production demands, it may be advantageous to extend the duration of a desired steady state indefinitely <cit.>, implying that their stability properties are also of great interest.Fortunately, in the last few years it has been possible to exploit an increasingly available amount of information about cellular metabolism at the stoichiometric level to build genome-scale metabolic networks<cit.>. These networks have been modeled by different approaches <cit.> but Flux Balance Analysis (FBA) has been particularly successful predicting cell metabolism in the growth phase <cit.>. FBA starts assuming a quasi-steady state of intra-cellular metabolite concentrations, which is easily translated into a linear system of balance equations to be satisfied by reaction fluxes. This system of equations is under-determined and a biologically motivated metabolic objective, such as biomass synthesis, is usually optimized to determine the complete distribution of fluxes through the solution of a Linear Programming problem <cit.>. This approach was first used to characterize the metabolism of bacterial growth <cit.>, butlater hasbeen applied also to eukaryotic cells <cit.>. Alternatively, given a set of under-determined linear equations, one can estimate the space of feasible solutions of the system and average values of the reaction fluxes <cit.>.To consider the temporal evolution of a culture, FBA may be applied to successive points in time, coupling cell metabolism to the dynamics of extra-cellular concentrations. This is the approach of Dynamic Flux Balance Analysis (DFBA) <cit.> and has been applied prominently either to the modeling of batch/fed-batch cultures or to transient responses in continuous cultures, being particularly successful in predicting metabolic transitions in E. Coli and yeast <cit.>. However, to the best of our knowledge, the steady states ofcontinuous cell cultures have not been investigated before. First, because DBFA for genome-scale metabolic networks may be a computational demanding task, particularly when the interest is to understand long-time behavior. Second, because it assumes knowledge of kinetic parameters describing metabolic exchanges between the cell and culture medium, that are usually unknown in realistic networks. Moreover, although the importance of toxic byproduct accumulation has been appreciated for decades <cit.>, its impact on steady states of continuous cultures has been studied mostly in simple metabolic models involving few substrates <cit.>, while it has been completely overlooked in DFBA of large metabolic networks. Lactate and ammonia are the most notable examples in this regard and have been widely studied in experiments in batch and continuous cultures <cit.>.Our goal in this work is to introduce a detailed characterization of the steady states of cell cultures in continuous mode, considering the impact of toxic byproduct accumulation on the culture, and employing a minimum number of essential kinetic parameters. To achieve this and inspired by the success of DFBA in other settings we couple macroscopic variables of the bioreactor (metabolite concentrations, cell density) to intracellular metabolism. However, we explain how to proceed directly to the determination of steady states, bypassing the necessity of solving the dynamical equations of the problem. This spares us from long simulation times and provides an informative overview of the dynamic landscape of the system. The approach, presented here for a toy model and for a genome-scale metabolic network of CHO-K1, but easily extensible to other systems, supports the idea that multi-stability, i.e., the coexistence of multiple steady states under identical external conditions, arises as a consequence of toxic byproduct accumulation in the culture. We find and characterize specific transitions, defined by simultaneous changes in the effective cell growth rate and metabolic states of the cell, and find a wide qualitative agreement with experimental results in the literature. Our analysis implies that batch cultures, typically used as benchmarks of cell-lines and culture media, are unable to characterize the landscape of metabolic transitions exhibited by perfusion systems. On the other hand, our results suggest a general scaling law that translates between the steady states of a chemostat and any perfusion system. Therefore, we predict that the chemostat is an ideal experimental model of high-cell density perfusion cultures, enabling a faithful characterization of the performance of a cell-line and media formulation truly valid in perfusion systems. § MATERIALS AND METHODS§.§ Dynamical model of the perfusion system We study an homogeneous population of cells growing inside a well-mixed bioreactor <cit.>, where fresh medium continuously replaces culture fluid (<ref>). The fundamental dynamical equations describing this system are:d X/dt = (μ - ϕ D) X d s_i/dt = -u_i X - (s_i - c_i)Dwhere X denotes the density of cells in the bioreactor (units: gDW / L), μ the effective cell growth rate (units: 1 / hr), u_i the specific uptake of metabolite i (units: mmol/gDW/hr), and s_i the concentration of metabolite i in the culture (units: mM). The external parameters controlling the culture are the medium concentration of metabolite i, c_i (units: mM), the dilution rate, D (units: 1 / day), and the bleeding coefficient, ϕ (unitless), which in perfusion systems characterizes the fraction of cells that escape from the culture through a cell-retention device <cit.> or a bleeding rate. For convenience of notation, in what follows an underlined symbol like s will denote the vector with components {s_i}.Equation <ref> describes the dynamics of the cell density as a balance between cell growth and dilution, while <ref> describes the dynamics of metabolite concentrations in the culture as a balance between cell consumption (or excretion if u_i < 0) and dilution. One must notice that at variance with the standard formulation of DFBA, the last terms in the right-hand side of both equations enable the existence of non-trivial steady states (with non-zero cell density) which are impossible in batch. These are the steady states that are relevant for industrial applications adopting the perfusion model and that we study in what follows.Still, we require a functional connection between variables describing the macroscopic state of the tank (X, s) and the average behavior of cells (u, μ). We start assuming that metabolites inside the cell attain quasi-steady state concentrations <cit.>, so that fluxes of intra-cellular metabolic reactions balance at each metabolite. If N_ik denotes the stoichiometric coefficient of metabolite i in reaction k (N_ik > 0 if metabolite i is produced in the reaction, N_ik<0 if it is consumed), and r_k is the flux of reaction k, then the metabolic network produces a net output flux of metabolite i at a rate ∑_k N_ikr_k, where N_ik = 0 if metabolite i does not participate in reaction k. This output flux must balance the cellular demands for metabolite i. In particular we consider a constant maintenance demand at rate e_i which is independent of growth <cit.>, as well as the requirements of each metabolite for the synthesis of biomass components. If y_i units of metabolite i are needed per unit of biomass produced <cit.>, and biomass is synthesized at a rate z, we obtain the following overall balance equation for each metabolite i:∑_k N_ik r_k + u_i = e_i + y_i z,∀ iIt is also well known that a cell has a limited enzymatic budget <cit.>. The synthesis of new enzymes, needed to catalyze many intracellular reactions, consumes limited resources, including amino acids, energy, cytosolic <cit.> or membrane space <cit.> (for enzymes located on membranes), ribosomes <cit.>, all of which can be modeled as generic enzyme costs <cit.>. We split reversible reaction fluxes into negative and positive parts, r_k = r_k^+ - r_k^-, with r_k^±≥ 0, and quantify the total cost of a flux distribution in the simplest (approximate) linear form <cit.>:α = ∑_k (α_k^+ r_k^+ + α_k^- r_k^-) ≤ Cwhere α_k^+, α_k^- are constant flux costs. The limited budget of the cell to support enzymatic reactions is modeled as a constrain α≤ C, where C is a constant maximum cost. Thermodynamics places additional reversibility constrains on the flux directions of some intra-cellular reactions <cit.>, which can be written as:lb_k ≤ r_k ≤ub_k,lb_k,ub_k ∈{-∞, 0, ∞}.Additionally, some uptakes u_i are limited by the availability of nutrients in the culture. We distinguish two regimes. If the cell density is low, nutrients will be in excess and uptakes are only bounded by the intrinsic kinetics of cellular transporters. In this case u_i ≤ V_i, where V_i is a constant maximum uptake rate determined by molecular details of the transport process. These will be the only kinetic parameters introduced in the model. When the cell density increases and the concentrations of limiting substrates reach very low levels, a new regime appears where cells compete for resources. In this regime the natural condition s_i ≥ 0 together with the mass balance equation (<ref> in steady state) imply that u_i ≤ c_i D/X. In summary,-L_i ≤ u_i ≤min{V_i, c_i D/X}where L_i = 0 for metabolites that cannot be secreted, and L_i = ∞ otherwise. Thus, an important approximation in our model is that low concentrations of limiting nutrients are replaced by an exact zero. The ratio D/X in <ref> establishes the desired coupling between internal metabolism and external variables of the bioreactor.Next, we reason that, although cellular clones in biotechnology are artificially chosen according to various productivity-related criteria <cit.>, the growth rate is typically under an implicit selective pressure. We will consider then that theflux distribution of metabolic reactions inside the cell maximizes the rate of biomass synthesis, z, subject to all the constrains enumerated above. Note that to carry out this optimization it is enough to solve a linear programming problem, for which efficient algorithms are available <cit.>. This formulation is closely related to Flux Balance Analysis (FBA) <cit.>, but some of the constrains imposed here might be unfamiliar. In particular, <ref> has been used before to explain switches between high-yield and high-rate metabolic modes under the name FBA with molecular crowding (FBAwMC) <cit.>, while the right-hand side of <ref> is a novel constrain introduced in this work to model continuous cell cultures. If multiple metabolic flux distributions are consistent with a maximal biomass synthesis rate <cit.>, the one with minimum cost α (cf. <ref>) is selected <cit.>. Summarizing, from the complete solution of the linear program we obtain the optimal z, and the metabolic fluxes u feeding the synthesis of biomass.Finally, the net growth rate of cells μ (see <ref>) is essentially determined by the cellular capacity to synthesize biomass (rate z), but it may also be affected by environmental toxicity. In the examples presented below we considered that: μ = z - σ(s) or μ = z × K(s), corresponding to two different mechanisms explored in the literature <cit.>. In the first case σ(s) is easily interpreted as the death rate of the cell, while 0 ≤ 1 - K(s) ≤ 1 represents a fraction of biomass that must be expended on non-growth related activities, for example, due to increased maintenance demands on account of environmental toxicity (but see also Refs.<cit.> and in particular B. Ben Yahia et al. <cit.> for a recent review of the subject). Both σ(s) and K(s) depend on the concentrations of toxic metabolites in the culture, such as lactate and ammonia. §.§ Metabolic networksToy modelTo gain insight into the kind of solutions expected, we examined first a simple metabolic network that admits an analytic solution. It is based on the simplified network studied by A. Vazquez et al. to explain the Warburg effect <cit.>, and serves as a minimal model of metabolic transitions in the cell <cit.>.A diagram of the network is shown in <ref>. There are four metabolites: a primary nutrient S, an energetic currency E, an intermediate P, and a waste product W. Only S and W can be exchanged with the extracellular medium, and their concentrations in the tank will be denoted by s and w, respectively. The cell can consume S from the medium at a rate u ≥ 0. The nutrient is first processed into P, generating N_F units of E per unit S processed. The intermediate can have two destinies: it can be excreted in the form of W (rate -v ≤ 0), or it can be further oxidized (rate r ≥ 0), generating N_R units of E per unit P. These two pathways are reminiscent of fermentation and respiration. We assume that N_F≪ N_R, which is consistent with the universally lower energy yield of fermentation versus respiration. Therefore, a maximization of energy output implies that the respiration mode is preferred. However, the enzymatic costs required to enable respiration are very high compared to fermentation. Therefore in <ref> only the costs of respiration are significant <cit.>, which implies that this flux is bounded:r ≤ r_maxMetabolic overflow occurs when the nutrient uptake is higher than the respiratory bound r_max. The remaining S must then be exported as waste, W. A balance constraint (cf. <ref>) at the intermediate metabolite P requires that:u + v - r = 0where stoichiometric coefficients are set to 1 for simplicity. Another balance constraint at the internal energetic currency metabolite, E, leads to:N_F u + N_R r - e - y z = 0where e denotes an energetic maintenance demand. The currency E is a direct precursor of biomass, at a yield y. Finally, the waste byproduct W is considered toxic, inducing a death rate proportional to its concentration:μ = z-σ = z-τ w The parameters were set as follows. The stoichiometric coefficients N_F = 2, N_R = 38 are the characteristic ATP yields of glycolysis and respiration, respectively <cit.>. Maintenance demand is modeled as a constant drain of ATP at a rate e = 1.0625 mmol/gDW/h, typical of mammalian cells <cit.>. The maximum respiratory capacity is computed as r_max = F_thr×Vol×DW = 0.45mmol/g/h, where F_thr = 0.9 mM/min is a glucose uptake threshold (per cytoplasmic volume) beyond which mammalian cells secrete lactate <cit.>, Vol = 3 pL and DW = 0.9 ng are the volume <cit.> and dry weight, respectively, of mammalian (HeLa) cells, the later estimated from the dry mass fraction (≈ 30%, <cit.>) and total weight (=3 ng <cit.>) of one HeLa cell. The concentration of substrate in the medium was set c = 15 mM, which is a typical glucose concentration in mammalian cell culture media (for example, RPMI-1640 <cit.>). Next, V = 0.5 mmol/gDW/h, also measured for HeLa cells <cit.> (the measured flux is per protein weight, so we multiplied by 0.5 to obtain a flux per cell dry weight, since roughly half of a generic cell dry weight is protein <cit.>). The parameter y = 348 mmol/gDW was adjusted so that the maximum growth rate was ≈ 1day^-1, which is within the range of duplication rates in mammalian cells <cit.>. Finally, the toxicity of waste was set as τ = 0.0022 h^-1mM^-1, obtained from linearizing the death rate dependence on lactate in a mammalian cell culture reported by S. Dhir et al. <cit.>. Genome-scale metabolic network of CHO-K1Exploiting the increasingly available information about cellular metabolism at the stoichiometric level <cit.>, a metabolic network can be reconstructed containing the biochemical reactions occurring inside a cell of interest. These reconstructions typically contain data about stoichiometric coefficients (N_ik, cf. <ref>), thermodynamic bounds (lb_k, ub_k, L_i, cf. <ref>), and a biomass synthesis pseudo-reaction (y_i, cf. <ref>) <cit.>. Motivated by the fact that most therapeutic proteins requiring complex post-transnational modifications in the biotechnological industry are produced in Chinese hamster ovary (CHO) cell lines <cit.>, we analyzed the steady states of a genome scale model of the CHO-K1 line <cit.>. Based on the latest consensus reconstruction of CHO metabolism available at the time of writing, containing 1766 genes and 6663 reactions, a cell-line-specific model for CHO-K1 was built by Hefzi et al. <cit.>, comprising 4723 reactions (including exchanges) and 2773 metabolites (with cellular compartmentalization). It accounts for biomass synthesis through a virtual reaction that contains the moles of each metabolite required to synthesize one gram of biomass. The network recapitulates experimental growth rates and cell-line-specific amino acid auxotrophies.In order to enforce <ref>, we complemented this network with a set of reaction costs. Following T. Shlomi et. al <cit.>, we assigned costs as follows: α_k^± = MW_k^± / k_cat,k^±, where MW_k^± and k_cat,k^± are the molecular weight and catalytic rate of the enzyme catalyzing reaction k in the given direction. The parameters MW_k^±, k_cat,k^± were gathered by T. Shlomi et. al from public repositories of enzymatic data. Missing values are set to the median of available values. An estimate of the enzyme mass fraction C = 0.078 mg/mgDW was obtained for mammalian cells by the same authors. A constant maintenance energetic demand (cf. term e_i in <ref>) was added in the form of an ATP hydrolysis drain at a flux rate 2.24868 mmol / gDW / h <cit.> (the reported value is for mouse LS cells, which we converted by accounting for the dry weight of CHO cells <cit.>). The maximum uptake rate of glucose was set at V_glc = 0.5 mmol/gDW/h, from previous models of cultured CHO cells fitted to experimental data <cit.> (which also closely matches the values obtained from kinetic measurements on other mammalian cell lines <cit.>). However, kinetic parameters needed to estimate V_i for most metabolites are not known at present. Based on data of CHO cell cultures in our facility (not shown), as well as data in the literature <cit.>, we estimated that the uptake rates of amino acids is typically one order of magnitude slower than the uptake rate of glucose, accordingly we set V_i = V_glc / 10 for amino acids. Other metabolites have an unbounded uptake (V_i = ∞). In the simulations we used Iscove's modified Dulbecco's medium (IMDM), and set infinite concentrations for water, protons and oxygen.The two toxic byproducts most commonly studied in mammalian cell cultures are ammonia and lactate. Their toxicity is primarily attributed to their effects in osmolarity and pH <cit.>. It has been suggested that the accumulated toxicity may result in increased maintenance demands <cit.> and in reduced biomass yields <cit.>. Parameters describing these effects quantitatively vary over an order of magnitude <cit.> depending on culture conditions and cell-line. In our model we incorporate these effects through the factor K and for the sake of specificity in this example we use:K = (1 + s_nh4 / K_nh4)^-1(1 + s_lac / K_lac)^-1with K_nh4 = 1.05 mM, K_lac = 8 mM <cit.>, and set μ = K× z. §.§ Additional details Numerical simulations were carried out in Julia <cit.>. Linear programs were solved with Gurobi <cit.>. The CHO-K1 metabolic network <cit.> was read and setup with all relevant parameters using a script written in Python with the COBRApy package <cit.>. All scripts (which also include parameter values) are freely available in a public Github repository <cit.>. § RESULTS AND DISCUSSION§.§ General properties of steady states In this section we present the general procedure to determine the steady states of <ref> and discuss some general results of our model that are independent of the specificities of the metabolic networks of interest. The first step is to set the time-dependence in <ref> to zero,d X/dt = (μ - ϕ D) X = 0d s_i/dt = -u_i X - (s_i - c_i)D = 0Note that <ref> depends on X and D only through the ratio 1 / ξ = D / X (known in the literature as cell-specific perfusion rate, or CSPR <cit.>), such that ξ is the number of cells sustained in the culture per unit of medium supplied per unit time (the units of ξ are cells × time / volume). If we recall that in our cellular model, u_i is constrained by a term that also depends on X and D only through ξ (cf. <ref>), it immediately follows that the values of the uptakes and metabolite concentrations in steady state must be functions of ξ, which we denote by u_i^*(ξ) and s_i^*(ξ) respectively. To compute u_i^*(ξ), solve the linear program of maximizing the biomass synthesis rate (z) subject to <ref>, but replacing <ref> with:-L_i ≤ u_i ≤min{V_i, c_i / ξ}.The resulting optimal value of z will be denoted by z^*(ξ). Moreover, once u_i^*(ξ) is known, the stationary metabolite concentrations in the culture follow from <ref>:s_i^*(ξ) = c_i - u_i^*(ξ) ξThen, given z^*(ξ), the effective growth rate in steady state can also be given as a function of ξ, μ^*(ξ), by evaluating K or σ using the concentrations s_i^*(ξ) from <ref>. Next, <ref> implies that the dilution rate at which a steady state occurs must also be a function of ξ, that we denote by D^*(ξ). Combining this with the relation ξ = X/D, we obtain the steady state cell density, X^*(ξ), as well:D^*(ξ) = μ^*(ξ) / ϕ,X^*(ξ) = ξμ^*(ξ) / ϕ.Note that while <ref> are satisfied by any D ≥ 0 when X = 0, the value D_max = D^*(0) given by <ref> is actually the washout dilution rate, i.e., the minimum dilution rate that washes the culture of cells. Clearly all steady states with non-zero cell density are required to satisfy D^*(ξ) < D_max.From <ref> it is evident that the function μ^*(ξ) encodes all the information needed to get the values of X in steady state at different dilution rates and for any value of the bleeding coefficient ϕ. On the other hand, if multiple values of ξ are consistent with the same dilution rate (i.e. if the function D^*(ξ) is not one-to-one), the system is multi-stable (i.e., multiple steady states coexist under identical external conditions). A necessary condition multi-stability is that μ^*(ξ) is not monotonously decreasing. Since the biomass production rate z^*(ξ) is a non-increasing function of ξ (proved in the Appendix), a change in the monotonicity of μ^*(ξ) must be a consequence of toxic byproduct accumulation, modeled through the terms K and σ.A noteworthy consequence of <ref> is that a plot displaying the parametric curve (ϕ D^*(ξ), ϕ X^*(ξ)) as a function of ξ is invariant to changes in ϕ. This means that for a given cell line and medium formulation, this curve can be obtained from measurements in a chemostat (which corresponds to ϕ = 1), and the result will also apply to any perfusion system with an arbitrary value of ϕ. Moreover, since s_i^*(ξ) and u_i^*(ξ) are independent of ϕ, cellular metabolism in steady states is equivalent in the chemostat and any perfusion system (with an arbitrary value of ϕ), provided that the values of ξ = X/D in both systems match.Finally, we mention that generally a threshold value ξ_m exists, such that a steady state with ξ = X/D is feasible only if ξ≤ξ_m. When ξ > ξ_m, some of the constrains in <ref> cannot be met. In degenerate scenarioswe could have ξ_m = ∞ (e.g., this could be the case if the maintenance demand in <ref> is neglected) or ξ_m≤ 0 (e.g., if the medium is so poor that the maintenance demand cannot be met even with a vanishingly small cell density). The parameter ξ_m arising in this way in our model, coincides with the definition of medium depth given in the literature <cit.>, and it quantifies for a given medium composition the maximum cell density attainable per unit of medium supplied per unit time. StabilityTo determine the stability of steady states we compute the Jacobian eigenvalues of the system <ref>. If the real parts are all negative the state is stable, but if at least one eigenvalue has a positive real part, the state is unstable <cit.>. The critical case where all eigenvalues have non-negative real parts but at least one of them has a zero real part is dealt with using the Center Manifold Theorem <cit.>. The Appendix contains a detailed mathematical treatment. Briefly, an steady state is unstable if μ^*(ξ) is increasing in a neighborhood, and stable otherwise. We stated above that steady states of a given cell line in continuous culture, using a fixed medium formulation, can be given as functions of ξ. The condition for stability stated here is also uniquely determined by ξ and, in particular, it is independent of ϕ. Therefore, a steady state in a chemostat (with ϕ = 1) is stable if and only if the same steady state in perfusion (with a matching value of ξ, but arbitrary ϕ) is also stable. Since our results are qualitatively invariant to changes in ϕ, we set ϕ = 1 in what follows. §.§ Insight from the toy model We first consider the small metabolic network depicted in <ref>. In this example, maximization of growth sets the nutrient uptake (u) and respiratory flux (r) at the maximum rates allowed by their respective upper bounds, <ref>. Employing <ref> to determine the waste secretion rate (v) from u,r, we obtain:u = min{ V, c / ξ},r = min{u, r_max},v = r - u.Thus the toy model admits simple analytical expressions giving the rates of metabolic fluxes in steady states as functions of ξ. A minimum nutrient uptake rate u_m is required to sustain the maintenance demand e. Since most cell types are able to grow under certain conditions without waste secretion, we make the biologically reasonable assumption that u_m≤ r_max (which is satisfied by the parameters chosen in Materials and Methods). It then follows that u_m = e / (N_F + N_R). There are three critical thresholds in ξ that correspond to important qualitative changes in the culture:ξ_m = c / u_m, ξ_sec = c / r_max, ξ_0 = c / V.These transitions can be interpreted in the following way: 1) if ξ≥ξ_m the growth rate is zero because the maintenance demand cannot be met; 2) if ξ_sec≤ξ≤ξ_m, cells grow without secreting waste; 3) if ξ≤ξ_sec, there is waste secretion; 4) for ξ≥ξ_0 cells are competing for the substrate and growth is limited by nutrient availability (cf. discussion before <ref>); 5) finally, if ξ≤ξ_0 there is nutrient excess and cells are growing at the maximum rate allowed by intrinsic kinetic limitations. We emphasize that the threshold ξ_sec carries a special metabolic significance, because it controls the switch between two qualitatively distinct metabolic modes: if ξ≤ξ_sec, respiration is saturated and the intermediate P overflows in the form of secreted waste, with a lower energy yield; on the other hand, if ξ≥ξ_sec, the cell relies entirely on respiration to generate energy, with a higher yield (cf. <ref>).The medium carries a concentration c of primary nutrient and zero waste content. Under these assumptions, <ref> has the following analytical solution for the steady state values of the metabolite concentrations, s^*(ξ), w^*(ξ):s^*(ξ)= c - min{V ξ, c}w^*(ξ)= max{ 0, c - s^*(ξ) - r_maxξ}Note that s^*(ξ) is a decreasing function of ξ, while w^*(ξ) has at most a single maximum. <ref> can be used to define μ^*(ξ). Then D^*(ξ), X^*(ξ) are given by <ref>.<ref> shows plots of μ^*(ξ), X^*(ξ), s^*(ξ) and w^*(ξ) for thismodel. Parameter values are given in Materials and Methods. As ξ ranges from ξ = 0 to ξ = ξ_m, stable and unstable steady states are drawn in continuous and discontinuous line, respectively. The system is stable in two regimes: ξ≲ 1 × 10^6cells·day/mL, with high toxicity, low biomass yield and low cell density; and ξ≥ξ_sec, with no toxicity, high biomass yield and high cell density that decays as ξ increases. The later states rely solely on respiration for energy generation (<ref>a), while the former states exhibit overflow metabolism (<ref>b). Waste concentration initially increases with ξ until a maximum value is reached. Then w decays during the unstable phase, all the way to zero at ξ_sec, where waste secretion stops and the system becomes stable again. Intuitively, unstable states become stressed due to high levels of toxicity, which also makes the system very sensitive to perturbations. The typical behavior of nutrients and waste products (in particular glucose and lactate, respectively) in continuous cell cultures, as observed in experiments <cit.>, is that as ξ increases, nutrient concentration in the culture decreases while waste initially accumulates <cit.> but eventually phases out as cells switch towards higher-yield metabolic pathways <cit.>. This behavior is qualitatively reproduced by s and w in our toy model.The function μ^*(ξ) is not monotonically decreasing. As explained above, this implies a coexistence of multiple steady states under identical external conditions. This is readily apparent in a bifurcation diagram of the steady values of X versus D, as shown in <ref>a. In a range of dilution rates (0.25 ≲ D ≲ 0.7, units: day^-1), the system exhibits three steady states, one of which is unstable (discontinuous line in the figure), while the other two are stable (continuous line in the figure). Thus a stable state of high-cell density coexists with another of low cell density, over a range of dilution rates. Cellular metabolism in the former state is respiratory (<ref>a), whereas cells in the later state exhibit an overflow metabolism (<ref>b). The unstable state is an intermediate transition state lying between these two extremes.Multi-stability of continuous cultures has been repeatedly observed in experiments <cit.>. In our model it is a direct consequence of toxicity induced by the accumulation of waste <cit.>. Small variations in thedilution rate near D ≈ 0.25day^-1 or D ≈ 0.7day^-1 result in discontinuous transitions where the cell density rises or drops abruptly, respectively. These jumps can be traced around an hysteresis loop, drawn in orange arrows in <ref>a. More generally one may also expect that the system jumps from one state to the other due to random fluctuations. In particular, note that the basin of attraction of the high-cell density state decreases with D (since the discontinuous line of unstable states eventually intersects with the high cell density states). Therefore, our toy model exhibits a plausible mechanism through which increasing dilution rates translate into high cell density states with diminishing resilience to perturbations. The role of toxicity becomes evident if we consider ideal cells resistant to waste accumulation (setting τ = 0). The resulting plot of X vs. D in this case (<ref>b) reveals a single stable steady state for each value of the dilution rate. There is a discontinuous transition at the washout dilution rate (D_max), where the cell density suddenly drops to zero. Away from this value, the system is resilient to perturbations since there are no multiple steady states between which jumps can occur.Multi-stability implies that system dynamics are non-trivial, in the sense that different trajectories might lead to different steady states. Therefore it is important for industrial applications to understand how the system is driven to one or another state. We numerically solved <ref> by performing the FBA optimization at each instant of time, in a manner analogous to DFBA <cit.>. With the parameter values given in Materials and Methods, we simulated the response of the system to three different profiles of the dilution in time. First, in <ref>a, a constant dilution rate of D = 0.6day^-1 is used. Two possible stable steady states are consistent with this dilution rate, attaining different cell densities (cf. <ref>a). Starting from a very low initial cell density, the system responds by settling at the steady state of lowest cell-density. As can be appreciated in the bottom row of <ref>a, this state is characterized by an accumulation of toxic waste that prevents further cell growth. A smarter manipulation of the dilution rate makes the high-cell density state accessible. This is demonstrated in <ref>b, where D starts from a lower value (D = 0.2day^-1), and is gradually increased until the final value (D = 0.6 day^-1) is reached. The final cell density resulting from this smooth increase of the dilution rate is five-fold larger than the one obtained with a constant dilution rate. This state is also characterized by very low levels of waste accumulation (cf. last row of <ref>b). We stress that external conditions in the final steady state (dilution rate and medium formulation) are the same in both cases. Finally, <ref>c shows how the dilution rate can be manipulated to switch from one steady state to another. Starting from the final state of the simulation in <ref>a, the dilution rate is first decreased to a low value (D = 0.2 day^-1), and then it is pushed back up to the starting value (D = 0.6 day^-1). The system responds by switching from the state with low cell-density to the state with high-cell density. These simulations nicely reproduce the qualitative features of the experiment performed by B. Follstad et al.<cit.>, where a continuous cell culture under the same steady external conditions (dilution rate and medium) switches between different steady states by transient manipulations of the dilution rate. The response of the cell density to transient manipulations of the dilution rate best illustrated in the X,D plane (cf. first row of <ref>). Then it becomes obvious that the dilution rate must be pushed down to ≈ 0.2 day^-1, otherwise the system will not leave the low cell density state. §.§ Analysis of the CHO-K1 cell-line using a genome-scale metabolic network We determined the steady states of a continuous cell culture of the CHO-K1 line. Cellular metabolism was modeled using the reconstruction given by Hefzi et al. <cit.>, the most complete available at the time of writing. In the simulations we used Iscove's modified Dulbecco's growth media (IMDM), which is typically employed in mammalian systems. Similar to what we found in the toy model (cf. <ref>), and in qualitative agreement with experimental observations <cit.>, cells exhibited several metabolic transitions between distinct flux modes as ξ was varied. However, in contrast to the the toy model, the CHO-K1 genome-scale metabolic network displays a rich multitude of transitions, as expected from its greater complexity. Because of their importance in the performance of the culture, we focused on metabolic changes that have an impact on macroscopic properties of the bioreactor, i.e., those that affect metabolite exchanges with the extracellular media (u_i). Although many classifications are possible, we organized our discussion by focusing on five qualitatively different modes based on the secretion of lactate and formate. <ref> shows cartoon diagrams of these phases in order of increasing ξ. On the top of each diagram we annotate the nutrients that became limiting for growth during a phase. Blue arrows indicate consumption and red arrows secretion. We focused on metabolites that changed their role between phases. In particular, NH_4 was secreted in all phases and therefore was omitted from the figure to reduce clutter. A more detailed representation of our results is given in <ref>, which shows metabolite concentrations (s_i) and uptakes (u_i) in steady states as functions of ξ for a sub-set of selected metabolites. Red lines in these plots indicate the transitions depicted in <ref>.For small values of ξ we found that glucose and almost all the amino acids available in the media were consumed, but none of them reached limiting concentrations. We call this the nutrient excess phase, where substrate uptake is limited only by intrinsic kinetic properties of cellular transporters. Remarkably lactate was not secreted at this stage, since pyruvate was converted instead to alanine <cit.> (although a small fraction of pyruvate was secreted as well <cit.>). The cell also produced succinate <cit.> and formate, the later being an overflow product of one-carbon metabolism of serine and glycine <cit.>.As ξ continues to increase, the first metabolite that becomes limiting is serine. This marks the end of the nutrient excess phase, coinciding also with the onset of lactate secretion. At this point pyruvate is no longer secreted into the culture. Remarkably, aspartate switches from being a secreted byproduct in the first phase <cit.>, to consumption. Even more striking is that the specific uptake rate of aspartate and proline quickly increase until both reach limiting concentrations. A third phase is entered when succinate and formate production ceases, coinciding with a limitation of glycine. Histidine consumption rises steeply until it too reaches limiting concentrations. Other nutrients that limit growth include tyrosine, tryptophan, arginine, lysine and phenylalanine. This phase is also characterized by secretion of acetaldehyde. Remarkably, formate secretion is resumed in a later phase, where glucose, glutamine and asparagine also become limiting.Finally, as ξ approaches the maximum value ξ_m, lactate and alanine secretion cease. This ideal state attains the highest possible biomass yield per unit of medium supplied per unit time. Note that the increase of ξ has brought an overall qualitative switch to a state of metabolic efficiency where the number of secreted byproducts has dropped significantly, compared to the nutrient excess phase. Notably, the cell-specific ammonia secretion was sustained even in the states of highest biomass yield, indicating a nitrogen imbalance. This behavior has been seen qualitatively in some experiments. For example, using a CHO-derived cell line <cit.>, secretion of ammonia was sustained even after a transition to an efficient metabolic phenotype with low lactate secretion and high cellular yields. However this observation seems to be cell-line dependent, and in another experiment with an hybridoma, ammonia accumulation decreased with increasing ξ <cit.>.All of the secreted metabolites predicted by our model have been verified in experiments in mammalian cell cultures <cit.>, with the exception of acetaldehyde. For this metabolite our search in the literature did not reveal any experimental evidence refuting or validating the presence of this byproduct in mammalian cell culture.The performance of cell-lines and media are typically evaluated by measurements performed in batch experiments <cit.>. Measurements performed in the exponential phase of batch only reveal the behavior of continuous cultures at very low ξ, in conditions of nutrient excess. The existence of a rich multitude of qualitatively distinct metabolic behaviors at higher values of ξ is missed by these experiments and therefore the assessment should not be extrapolated to high-density perfusion systems. As our analysis reveals, several nutrients may switch from basal rates of consumption to growth limiting at later values of ξ, while others go from secreted byproducts to consumption <cit.>. These examples indicate that nutrients could be in excess in a batch experiment but need not be so in the ideal regime of high-cell density perfusion cultures, at high ξ. Our model suggests that a better characterization of a cell-line and media formulation can be obtained in a chemostat, since the full spectrum of values of ξ can be explored in this device and it faithfully reproduces all the metabolic transitions found in perfusion.The effects of toxic byproduct accumulation are explored in <ref>. Although the model can easily accomodate any number of toxic compounds, we considered We considered the toxic effects of the most commonly studied metabolites in this regard: lactate and ammonia, although the model can easily accommodate the effects of additional toxic compounds if necessary. In <ref>a we plot the effective growth rate, μ as function of ξ. Stable states are drawn in continuous line, unstable states are dashed and the red dots indicate the metabolic transitions depicted in <ref>. Note that μ^*(ξ) is not monotonous. In particular, metabolic transitions resulting in lactate and ammonia secretion peaks produce a sink in the curve μ^*(ξ). On the other hand, metabolic transitions associated to the secretion of other non-toxic byproducts do not imply changes in the monotonicity of μ^*(ξ).The non-monotonicity of μ^*(ξ) results in multiple stable states coexisting at the same dilution rate, as evident in the bifurcation diagram <ref>b. This resonates with the results obtained in the simpler model considered above, and is also consistent with many experimental observations of bi-stability in the literature <cit.>. The regime with high-cell density corresponds to a higher value of ξ and exhibits a lower accumulation of toxic byproducts (lactate and ammonia). Metabolism in this regime is also more efficient, with less byproducts secreted (cf. <ref>). On the other hand, low cell density states are wasteful, with high levels of environmental toxicity preventing further cell growth. Again, bi-stability implies the existence of an hysteresis loop (orange arrows in the figure), where the system may suffer abrupt transitions between high and low cell densities. § CONCLUDING REMARKS In this work we have presented a model of cellular metabolism in continuous cell culture. Although similar in spirit to DFBA, our dynamic equations include terms accounting for the continuous medium exchange that enables steady states in this system. We presented a simple method to compute the steady states of the culture as a function of the ratio between cell density and dilution rate (ξ = X / D), scalable to metabolic networks of arbitrary complexity. In the literature 1/ξ is known as the cell-specific perfusion rate (CSPR), introduced by S. Ozturk <cit.> who already made the empirical observation that control of the CSPR can be used to maintain a constant cell environment independent of cell growth <cit.>. Our model theoretically supports this idea and leads to a stronger conclusion: that for a given cell line and medium formulation, the steady state values of the macroscopic variables of the bioreactor are all unequivocally determined by ξ. Therefore, ξ is an ideal control parameter to fix a desired steady state in a continuous cell culture.The model is consistent with multi-stability, a phenomenon repeatedly observed in experiments in continuous cell cultures where multiple steady states coexist under identical external conditions. Moreover, our model accounts for metabolic switches between flux modes, experimentally observed in continuous cell culture in response to variations in the dilution rate <cit.>. These transitions affect the consumption or secretion of metabolites and the set of nutrients limiting growth. As a consequence, the metabolic landscape of steady states in perfusion cell cultures is complex and cannot be reproduced in batch cultures. This has the practical implication that assesments of medium quality and cell line performance carried out in batch <cit.> should not be extrapolated to perfusion, since they might be missleading in this setting.However, our analysis reveals a simple scaling law between steady states in the chemostat and any perfusion system. The landscape of metabolic transitions in the later system can be faithfully reproduced in the chemostat. Thus, for a fixed cell-line and medium formulation, the diagram displaying the values of ϕ X versus ϕ D in steady state is invariant across perfusion systems with any bleeding ratio (ϕ), cf. <ref>, while metabolism is equivalent if the ratio ξ = X / D is the same. The practical consequence is that the chemostat is an ideal experimental model where cell-lines and medium formulations can be benchmarked for their performance in high-cell density industrial continuous cultures. Further, the model predicts that multi-stability is a consequence of negative feedback on cell growth due to accumulation of toxic byproducts in the culture. The qualitative complexity of the ϕ X versus ϕ D diagram depends only on the behavior toxic metabolites. Moreover, multi-stability implies that the system is sensitive to initial conditions and transient manipulations of external parameters. In practice, the dilution rate must be manipulated carefully to bring the system to a desired state. Thus, starting from a seed of low cell density, sharp increases of the dilution may land the system on a steady state of high toxicity and low biomass. On the other hand, slowly increasing the dilution rate will surely lead towards high-cell density states.The conclusions stated above rely on the validity of our assumptions. In particular, we have considered a homogeneous cell population in a well-mixed bioreactor. Both assumptions are behind many models published in the field and provide reasonable fits to experimental data <cit.>. Mechanical stirring of the culture typically achieves a well-mixed solution, but care must be taken to prevent mechanical damage to the cells <cit.> (but see Ref <cit.>). Moreover, that the cell population can be treated attending only to its average properties is justified by the large number of cells in a typical culture (∼ 10^6 – 10^8 cells / mL), although in some settings cell-to-cell heterogeneity might become relevant <cit.>. Next, to develop a specific model of cellular metabolism, we adopted a flux-balance approach <cit.>, where cells are assumed to optimize their metabolism towards growth rate maximization. Although this framework is well supported in the literature <cit.>, it is worth noting that we did not consider the kinetics of intracellular metabolites or additional regulatory mechanisms that may also control metabolic fluxes. Additionally, the quantitative predictions of the model rely on the accuracy of parameters found in the literature and databases. Among these, the flux cost coefficients (α_i, <ref>) are not available for many enzymes. If too many of these parameters are absent, calculations from FBA might be degenerate <cit.>. Another important omission from the present model is that we did not consider explicitly the exchange between the culture and a gaseous phase. In particular, this includes oxygen exchange. Therefore our approach is only valid if this exchange does not become limiting to cellular growth. Despite these limitations, we have shown that the model predictions are in qualitative agreement with experimental data. More importantly, the conclusions stated above are independent of the values of model parameters.§ ACKNOWLEDGEMENTS This project has received funding from the European Union's Horizon 2020 research and innovation programme MSCA-RISE-2016 under grant agreement No. 734439 INFERNET. The authors warmly thank Tamy Boggiano and Ernesto Chico for many helpful discussions and for reviewing this manuscript.§ AUTHOR CONTRIBUTIONS All authors contributed equally to model design; all authors wrote the manuscript; J.F.C.D. performed simulations; all authors analyzed results.§ COMPETING FINANCIAL INTERESTS The authors declare no competing financial interests. § SUPPLEMENTARY MATERIALS§.§ Stability of fixed points The growth rate μ depends on the biomass synthesis rate, z, and on the concentrations of toxic metabolites in the culture. Since for a fixed dilution rate, z depends only on X (cf. <ref>), it follows that we can write the growth rate as a function of X and s, thus μ = μ(X, s).In particular note that the dynamics of non-toxic metabolites can be decoupled from the rest of the system (cf. <ref>). It is enough to determine the stability of a reduced system, where only X and the concentrations s_i of metabolites i that are toxic intervene. In the trivial case where there are no toxic metabolites all fixed points are stable because μ is a non-increasing function of X. We assume that ∂μ / ∂ s_i < 0 for all toxic metabolites i.Let us begin by defining the velocities of change of X and s_i as the right-hand sides of <ref> and <ref>, respectively,F(X, s) = (μ - ϕ D) X G_i(X, s) = -u_i X - (s_i - c_i) DA fixed point X̂, ŝ satisfies F(X̂, ŝ) = 0 and G_i(X̂, ŝ) = 0. To determine its stability, we evaluate the Jacobian (𝐉) of <ref> at X̂, ŝ:𝐉=([ ∂ F/∂ X ∂ F/∂ s_i; ∂ G_j/∂ X ∂ G_j/∂ s_i ])=([ ∂μ/∂ XX̂-∂μ/∂ s_1X̂-∂μ/∂ s_2X̂⋯-∂μ/∂ s_mX̂; -u_1(X̂)-u_1^'(X̂)X̂ -D0⋯0; -u_2(X̂)-u_2^'(X̂)X̂0 -D⋯0;⋮⋮⋮⋱⋮; -u_m(X̂)-u_m^'(X̂)X̂00⋯ -D ])where u_i'(X̂) are the derivatives of u_i(X) with respect to X, evaluated at the fixed point. To evaluate ∂μ / ∂ X, recall that μ = K× z - σ, where K and σ depend only on s, while z is a function of X (at a fixed dilution rate, cf. <ref>). Therefore, it is enough to evaluate z'(X). Computation of z'(X̂) and u_i'(X̂) Before continuing, let us make a short digression into Linear Programming <cit.>. To determine a basic solution of FBA, it is enough to specify: (i) the indexes of fluxes that are away from their lower and upper bounds, and (ii) for the remaining fluxes, whether they are equal to their lower or upper bound. This information is called the basis <cit.>. The full solution can be reconstructed from knowledge of the basis by solving the linear equality constrains. Since the basis is a discrete object, it will remain constant as ξ varies continuously, except for discrete `critical' values of ξ where the basis changes. When the basis remains constant, u_i^*(ξ) and z^*(ξ) have the following forms:u_i^*(ξ) = α_i + β_i / ξ,z^*(ξ) = α + β / ξ,where α_i,β_i,α,β are constant as long as the basis remains fixed. <ref> is simply the generic affine dependency on the upper bounds of the uptakes (cf. <ref>). Since z^*(ξ) is a non-increasing function of ξ, β≥ 0. Using <ref>, it follows that u_i(X̂) = α_i + β_i D / X̂ and z(X̂) = α + β D / X̂. Therefore:û_i + u_i'(X̂) X̂ = α_i,z'(X̂) X̂ = -β / ξ.To obtain α,β,α_i,β_i, we exploit the fact that we will be computing μ^*(ξ) = z(X^*(ξ)), u_i^*(ξ) = u_i(X^*(ξ)) and X^*(ξ) over a sequence of contiguous values of ξ. If ξ_1, ξ_2 are sufficiently nearby:4 α_i = u_i^*(ξ_1) ξ_1 - u_i^*(ξ_2) ξ_2/ξ_1 - ξ_2,β_i= -u_i^*(ξ_1) - u_i(ξ_2)/ξ_1 - ξ_2ξ_1 ξ_2, α= z^*(ξ_1) ξ_1 - z^*(ξ_2) ξ_2/ξ_1 - ξ_2,β= -z^*(ξ_1) - z^*(ξ_2)/ξ_1 - ξ_2ξ_1 ξ_2.The singular case X̂ = 0 has β = β_i = 0, assuming that for very low cell densities growth is not limited by substrate availability (i.e., that the medium is rich; cf. discussion before <ref>).From z'(X̂) we compute ∂μ / ∂ X = K × z'(X̂).Stability of the linearized system The system is stable if the real parts of all the eigenvalues of 𝐉 are negative, and is unstable if at least one eigenvalue has a positive real part <cit.>. The eigenvalues of 𝐉 are:λ_± = 1/2( μ'(X̂) X̂ - D ±√((D + μ'(X̂) X̂)^2 + 4X̂ω)), λ_d = -Dwhere μ'(X̂) denotes the derivative ∂μ / ∂ X evaluated at X̂, andω = -∑_i ∂μ/∂ s_i( û_i + u_i'(X̂) X̂) =- ∑_i ∂μ/∂ s_iα_i.λ_d is a degenerate eigenvalue of order m - 1 (where m is the number of metabolites) and is always negative (we assume that D > 0). The couple λ_± forms a complex conjugate pair with negative real part if (D + μ'(X̂) X̂)^2 + 4X̂ω < 0, which implies ω < 0 < X̂. In this case the system is stable. If (D + μ'(X̂) X̂)^2 + 4X̂ω≥ 0 all the eigenvalues are real and all are negative except possibly λ_+. After some algebra, we find that λ_+ < 0 (the system is stable) or λ_+ > 0 (the system is unstable) according to whether -μ'(X̂) X̂ > ξω or -μ'(X̂) X̂ < ξω, respectively. Since ω < 0 < X̂ implies -μ'(X̂) X̂ > ξω (because μ'(X̂) ≤ 0), the condition -μ'(X̂) X̂ > ξω is sufficient for stability, while -μ'(X̂) X̂ < ξω is sufficient for instability, even if λ_± turn out to be complex.The critical case λ_+ = 0 occurs whenever -μ'(X̂) X̂ = ξω. In this case the stability of the system cannot be resolved by analysis of the linearized system alone, and we must recur to the Center Manifold Theorem <cit.>. As will be shown below, in this case the system is stable. Therefore, the fixed point is stable if -μ'(X̂) X̂≥ξω and unstable if -μ'(X̂) X̂ < ξω. Since μ'(X̂) X̂ and ω are both independent of ϕ (by <ref>), this condition does not depend on ϕ. Then, whether a fixed point is stable or not can be given as a function of ξ only, as asserted in the main text.The condition for stability can be further simplified by noting that μ'(X̂) X̂ / ξ + ω is the derivative of μ^*(ξ) with respect to ξ. Therefore, the system is stable if an only if μ^*(ξ) is non-increasing in a neighborhood. Center manifold stability for the critical case (λ_+ = 0)If -μ'(X̂) X̂ = ξω all eigenvalues are real and negative except λ_+ = 0. In this case the linearized system cannot be used to determine the stability of the fixed point, because the effect of small perturbations along the direction of the eigenvector corresponding to λ_+ (the so-called center manifold) is not captured by the linearized system. Since only one eigenvalue has a zero real part, the Center Manifold Theorem <cit.> can be used to find a reduced one-dimensional system where the stability can be determined. For simplicity we will only consider the case where μ'(X̂) = α_i = 0. The eigenvectors of 𝐉 then are:p_1 = [[ 1; 0; ⋮; 0 ]], p_2 = [[ -ξ∂μ/∂ s_1;1;0;⋮;0 ]], p_3 = [[ -ξ∂μ/∂ s_2;0;1;⋮;0 ]], …, p_m+1 = [[ -ξ∂μ/∂ s_m;0;0;⋮;1 ]]where p_1 corresponds to λ_+, p_2 to λ_-, and the rest to λ_d. These eigenvectors are assembled into a similarity matrix 𝐌 (as columns), which serves to diagonalize 𝐉:𝐌^-1𝐉𝐌=[[ λ_+ 0 0 ⋯ 0; 0 λ_- 0 ⋯ 0; 0 0 λ_d ⋯ 0; ⋮ ⋮ ⋮ ⋱ ⋮; 0 0 0 ⋯ λ_d ]]Introduce new variables x,z_1,…,z_m through the relation:[[ X; s; ]] = [[ X̂;ŝ ]] + 𝐌[[ x; z ]]To find the reduced system, we set z = 0, which implies X = X̂ + x and s = ŝ. Then, differentiating x with respect to time:d x/d t = (μ - ϕ D)(X̂ + x)The system is stable if and only if <ref> is stable at x = 0. We show now that the right-hand side of <ref> is a decreasing function of x, which implies stability. Since s_i = ŝ_i is fixed, K,σ are constant. From <ref> we know that z = α + β D / (X̂ + x) with constant α,β for sufficiently small x. Since X̂ is a fixed point, it follows that K(α + β D / X̂) = σ + ϕ D. Therefore the right-hand side of <ref> is:K(α + βD/X̂ + x - α - βD/X̂) (X̂ + x) = - K β x / ξwhich is decreasing in x. This argument breaks down if β = 0, which occurs only in conditions of nutrient excess, where ξ is low enough that there is no nutrient competition between the cells. In this case β_i = 0 also for all i, implying that û_i = α_i is piece-wise constant in this regime. If toxic metabolites are being secreted, ω > 0 implying -μ'(X̂) X̂ = 0 < ξω, which falls under the umbrella of the non-critical linear stability analysis discussed above. If toxic metabolites are not being secreted, ω = 0. But in the later case X is uncoupled from the rest of the variables, and the system is trivially stable because μ = α is constant. | http://arxiv.org/abs/1705.09708v1 | {
"authors": [
"Jorge Fernandez-de-Cossio-Diaz",
"Kalet León",
"Roberto Mulet"
],
"categories": [
"q-bio.MN",
"q-bio.CB",
"q-bio.PE",
"q-bio.SC"
],
"primary_category": "q-bio.MN",
"published": "20170526203634",
"title": "Characterizing steady states of genome-scale metabolic networks in continuous cell cultures"
} |
squishlist ∙WebSci'17June 25-28, 2017, Troy, NY, USA.This is a longer version of a 2-page poster publised at WebSci'17. Please cite the WebSci poster, not the arxiv version. printacmref=falseAalto University HelsinkiFinland [email protected] University of Haifa Haifa [email protected] Qatar Computing Research Institute [email protected] Celebrity and fandom have been studied extensively in real life. However, with more and more celebrities using social media, the dynamics of interaction between celebrities and fans has changed. Using data from a set of 57,000 fans for the top followed celebrities on Twitter, we define a wide range of features based on their Twitter activity. Using factor analysis we find the most important factors that underlie fan behavior.Using these factors, we conduct analysis on (i) understanding fan behavior by gender & age, and (ii) para-social breakup behavior. We find that (i) fandom is a social phenomenon, (ii) female fans are often more devoted and younger fans are more active & social, and (iii) the most devoted fans are more likely to be involved in a para-social breakup.Our findings confirm existing research on para-social interactions.Given the scale of our study and dependence on non-reactive data, our paper opens new avenues for research in para-social interactions.Characterizing Fan Behavior to Study Para Social Breakups Ingmar Weber December 30, 2023 =========================================================§ INTRODUCTIONSince the advent of electronic media, para-social relationships (PSR) or para-social interactions (PSI) have been a widespread feature of advanced societies <cit.>.Horton and Wohl define PSR as a one-way relationship that is imagined as a mutual relationship. They use the phrase `intimacy at a distance' to describe this phenomenon. It was used to explain the power of persuasion that certain celebrity personae have. Following early studies of PSR based on self-report surveys, the use of Computer Mediated Communications has made the study of such phenomena through non-reactive measures possible <cit.>.Traditionally, the celebrity-fan relationship was very lop-sided, with one way interaction between the celebrity and the fan (typically through the television). With the advent of social media,PSR have become more reciprocal and `social', or at least intensified the illusion of sociability by creating seemingly more personal and frequent communication between a celebrity and their fans. Given this new era of PSI on social media, many studies have looked at how celebrity behavior has been shaped on social media <cit.> and to a certain extent, how fandom has changed <cit.>. Eyal et al. <cit.> introduce the idea of a para-social breakup, the end of a PSR. They find that a para-social breakup is like a regular romantic relationship breakup but not as intense. This para-social breakup (PSB) is manifested on social media by an act of unfollowing the celebrity. Though breakups on social media have been studied in the past <cit.>, para-social breakups have special characteristics and understanding them at a large scale would be of great interest, with applications in brand loyalty, advertising and societal good, in general.Most studies on PSR and PSB to date focus on self-report surveys, where fans report on their feelings, attitudes & behaviors and are done on a small scale. We tackle such problems to build on top of these studies in this paper, and present a first large scale analysis of fan behavior on Twitter and how different fans indulge in PSR and PSB on Twitter. The research questions we tackle in this study are: (i) Can a systematic pattern be found in fan behaviors based on their interactions on Twitter? (ii) Can such a typology reliably predict and help in understanding para social breakups with celebrities? We collect data of 15 celebrities from popular culture and 57,000 of their fans on Twitter. We construct an exhaustive list of features from fan activity and fan-celebrity interaction on Twitter. We then use factor analysis to automatically group these features into three main factors, that explain most of the fan interactions on twitter, namely, devotedness, sociability, and activity. Using these factors as a building ground, we measure the effects of these factors on para-social breakup behavior of the fans (identified using fans unfollowing celebrities on Twitter).We make some interesting and novel observations.(i) Using inferred gender and age groups of the fans, we find that women are more devoted and younger fans tend to be more social & active. (ii) We find a correlation between how devoted a fan is and how social they are,which suggests that PSR on social media is a way to connect not only with the celebrity but with other fans and that this is an important component of fandom. Fans find other like-minded people of Twitter to connect with. This is in line with findings from <cit.>.(iii) We find that the more involved a fan is in a PSR, the more likely that they will end the PSR. We propose a potential explanation for this finding using a cost/benefit analysis. To the best of our knowledge, this is the first study to characterise fan behavior on Twitter andpara-social breakups at such a large scale. Our findings are valuable because of our dependence on non-reactive data.Typical studies in this field are mostly based on surveys, limited in scale and have self-reporting biases.§ RELATED WORK Celebrity on social media: A para-social relationship <cit.> (PSR) describes imagined relationships that people have with a media persona. Before the advent of social media, PSR was mostly (i) one way, and (ii) through television. In recent years, the emergence of social media provided opportunities for increased interaction between celebrities and followers that did not previously exist. Such changes have the potential to make PSR more reciprocal and `social', or at least to intensify the illusion of sociability by creating seemingly more personal and frequent communication <cit.>.Celebrity interactions with fans by posting information about their daily routines and replying directly to fans have become increasingly common these days. Social media is being used as a way of self promotion for the celebrities, which change the way PSR are established and operate <cit.>.Understanding celebrity behavior and fan reactions on social media has an impact on building good consumer relationships with brands <cit.>and loyalty to the social networks <cit.>.Fandom on social media: Though most analysis of celebrities on social media to date have focused on how the celebrities place themselves, there are a few studies on the role fans' in interacting with celebs. <cit.> study fans of NBA teams on Facebook and find four key motives for fans to engage with their favorite teams: `passion, hope, esteem and camaraderie'. Kim et al.<cit.> study the role of celebrity self disclosure on Twitter and the reaction of fans. They report that increased celebrity self-disclosure leads to `enhanced fans' feeling of social presence, thereby positively affecting para-social interaction with celebrities'.Para-social breakups: <cit.> introduce the idea of a para-social breakup, the end of a PSR. They find that a para-social breakup is like a regular romantic relationship breakup but not as intense. Breakups on social media have been studied in the past <cit.>. We define a para-social breakup on Twitter as an act of unfollowing the celebrity and study the impact of fan behavior on a breakup.Work has been done on reasons for unfollowing others on Twitter <cit.>. Kwak et al. conclude that most users unfollow `those who left many tweets within a short time, created tweets about uninteresting topics, or tweeted about the mundane details of their lives.'.In our case of a para-social breakup, we find that most unfollowing happens asan act of defiance against the celeb and despair of not receiving the personal attention that fans perceive they deserve. § DATASET We started with a set of the 15 most followed celebrities from popular culture on Twitter (in May 2015), from <http://followerwonk.com/bio>. The selected celebrities were:@justinbieber, @katyperry, @taylorswift13, @ladygaga, @rihanna, @jtimberlake,@theellenshow, @kimkardashian,@cristiano, @britneyspears, @jlo, @shakira, @selenagomez, @arianagrande, and @ddlovato.Note that we did not include politician @barackobama, as we assume political fandom to have very different characteristics.Each of these celebrities has tens of millions of Twitter followers (fans). Since it is not feasible to analyze all their followers, we sampled a subset of fans for each celebrity. We sampled from three types of fans, defined based on their level of interaction with the celebrity. (1) Involved – This is the set of fans who regularly interact with the celebrity in an intimate manner. To get this set, we first obtained all users replying to the tweets from celebrities and obtained users who replied to tweets from celebrities at least on five tweets with messages containing `i love you' (or similar variations, like, ILY).[This is common for many celebrities on Twitter. As an example, see any tweet from Justin Bieber <https://twitter.com/justinbieber/status/809785232186478592>] (2) Casual – This is a set of fans who interacted with (replied /mentioned/retweeted) the celebrity at least once in the previous year (May 2014 – May 2015). (3) Random – Random sample of followers of the celebrity, sampled randomly from the millions of followers the celebrity has.These fans need not have interacted with the celebrity beyond the act of following.The choice for these three sets was made in order to obtain the complete spectrum of fans, from the highly invested fans, who engage with the celebrity all the time, to people who just follow the celebrity out of curiosity.We first randomly sampled approximately 2,000 users from each set for each celebrity. We then applied simple heuristics to remove bots and inactive accounts (at least 10 tweets/followers/friends, been on Twitter for at least one year, etc.). We then up and down sampled each group for each celeb to be approximately of the same size. This gave us 57,609 fans in total.For these fans, we extracted a set of features based on a range of user interactions on Twitter.In an attempt to be as encompassing as possible, we did not apply strict theoretical considerations in identifying these features. Rather, we included any feature which may be at all relevant to fandom behavior and that we could quantify. We then allowed the factor analysis procedure (see next section) to group these as a way to reflect the underlying structure of the variables. (i) feat1 - Number of tweets, (ii) feat2 - Number of friends, (iii) feat3 - Number of followers, (iv) feat4 - Does the profile description contain an @mention of the celebrity?, (v) feat5 - Days since joining Twitter, (vi) feat6 - Fraction of tweets containing retweets, (vii) feat7 - Fraction of tweets containing mentions, (viii) feat8 - Fraction of tweets containing replies, (ix) feat9 - Fraction of tweets containing images, (x) feat10 - Fraction of tweets containing urls, (xi) feat11 - Fraction of tweets containing mentions of a celeb, (xii) feat12 - Fraction of tweets containing retweets of a celeb, (xiii) feat13 - Is the fan followed back by the celeb?, (xiv) feat14 - Fraction of followers followed by the celeb, (xv) feat15 - Fraction of friends followed by the celeb, (xvi) feat16 - Fraction of friends who also follow the same celeb, (xvii) feat17 - Fraction of followers who also follow the same celeb.For all the users, we used Face++ Api[<http://www.faceplusplus.com/>] on their profile pictures to obtain estimates of their age and gender.Face++ uses computer vision and data mining techniques applied to a large database of celebrities to generate estimates of age and sex of individuals from their pictures. We were able to obtain confident age and gender predictions for around 50% of the users (27,889 users).§ IDENTIFYING FACTORS OF FANDOMSince having 17 features makes it hard to interpret the underlying dynamics of the features clearly, we applied factor analysis to find a smaller number of factors that capture the correlations between the features.Factor analysis is a simple statistical method <cit.> to first find correlations among variables in order to then find a lower number of unobserved variables called factors.All the feature variables were first z-normalized (to have zero mean and standard deviation one). We create a matrix where the rows are the fans and columns are the 17 features (of size 57,609 × 17). We then computed pairwise correlations between the 17 features,and ran a factor analysis on this matrix. Using a cut off for the eigenvalue of 1.25, we obtained three factors that explain about 57% of the variance (as proposed in <cit.>). The three main factors we inferred are: * Factor1: {feat4, feat11, feat12}. Looking at the features that load well with this factor, we define this factor to indicate the Devotedness of a fan. * Factor2: {feat13, feat14, feat15}. These set of features seem to indicate the Sociability of fandom. * Factor3: {feat1, feat2, feat3, feat5}, which indicates the Activity of the fan on Twitter. Note that we were able to identify these feature groups (factors) in a completely unsupervised manner.Quantifying fandom: Using the three factors obtained above, we are now able to get a concise representation of the features that encode fandom on Twitter. For each fan, we compute three scores, Devotedness = median(feat4, feat11, feat12) Sociability = median(feat13, feat14, feat15) Activity = median(feat1, feat2, feat3, feat5)Note that since the features are z-normalised, the absolute values of these scores do not give much information, but relative trends, comparing different scores do.Validation:To validate our factor analysis we tested whether the three factors could reconstruct the fan types of our sample that were discerned based on other criteria, such as frequency of tweets, intimacy of interactions with the celebrity, etc. The results are shown in Figure <ref>. The figure indicates that involved fans have a high devotedness, activity and sociability compared to the other groups, which is in line with the way we defined the groups.This is non obvious as the type of attributes weconsider to obtain the factors was defined in a generic way, without taking into account the fan type.Next, we computed correlations between the various factors for all the fans. We find that devotedness and sociability are highly correlated (Pearson's R 0.65, p<1e-6). We do not find meaningful correlations between devotedness/sociability and activity.Differences in gender and age group: Using data from the 27,000 fans for whom we could get gender and age data, we tried to see if there are any visible differences between genders in terms of the three factors. We compared the mean values of the factors for the two groups (men and women) and there are significant differences. Using a Mann-Whitney U test, we find that women have a higher devotedness and sociability, while men have higher activity (p<0.0001).Next, we bucketed the age estimates into 4 groups - 0–14, 15–25, 26–35 and 36+. We computed the mean devotedness, sociability and activity values for all fans in each age group. We find that younger fans (from the 0–14 and 15–25 buckets) are more devoted, social and active (statistical significance tested using a Kruskals-Wallis test,p<1e-6). Detailed results are omitted due to lack of space. § PARA-SOCIAL BREAKUPSIn this section, we try to use the fan behavior characteristics we obtained to understand para-social breakups.We assume that a para-social breakup on Twitter manifests itself as an act of unfollowing the celebrity. We tracked all the 57k fans for a period of 26 weeks (between 21 May 2015 – 21 Nov 2015), and got data on whether they still follow the celebrity every week. At the end of the data collection period, we recorded 2,369 fans unfollowing a celebrity during this period.We also estimate when the fans started following the celebrity using <cit.>.As a first step, we tried to predict if a user would unfollow or not, just based on the three factors as features. Since unfollowing is a rare event, we first over sample from the rare group using SMOTE <cit.>. Then, using a random forest classifier, and 10-fold cross-validation, we were able to predict if a user will unfollow or not with a 66% accuracy. This may not look like a great accuracy, but it is still much better than random. Given that we are only using three features (the three factors), the factors seem to capture some signal in the unfollowing behavior.Next, we model unfollowing behavior using Survival analysis. Survival analysis is a statistical tool for analyzing the expected time to an event (unfollowing, in this case). It can be used to answer questions such as: whether a group of fans is more likely to unfollow than others.We set up survival analysis as follows: (i) Event: The act of unfollowing a celeb, (ii) Survival time/event time: Months since following the celebrity to unfollow. e.g. if a fan follows a celeb in Jan 2015 and unfollows in Oct 2015, this variable is 10. We have data going back to the last 4 years (48 months). (iii) Censoring event: All users who haven't unfollowed yet. For example, if a user followed the celebrity in Jan 2015 and is still following, we censor the user after 23 months (Jan 2015 to Dec 2016) (iv) Survival function: Probability of unfollowing after x months. Given this set up, our first task is to see if there is any difference in para-social breakups between the three fan types (involved, casual and random). Figure <ref> shows the survival probability for the various fan types using a Kaplan Meier non-parametric analysis.Compared across groups, involved users have a statistically significantly higherprobability of unfollowing. We can also see that the chance of unfollowing for random users is almost non-existent.This finding might be counter intuitive, given we expect involved fans to feel “closer” to the celebrity. We can interpret this behavior in terms of a cost/benefit analysis. The more a fan engages the celeb, the more likely it is for the relationship to end. The more involved a fan is the more the emotional cost they invest in the relationship and thus the more likely they are to end it when it does not reward them.These high investors tend to be young and female whereas as older fans are less invested and less prone to disappointment and breakup.We manually looked at the profile descriptions of users who unfollowed, before and after unfollowing, and found interesting examples.(i) Before: `@arianagrande followed and faved (30.11.14)', After: `dreams dont work unless you do. good vibesss!!' (ii) Before: `thank you justin my life', After:`find your purpose' (iii) Before: `queen faved x2. pls follow 2, ma queen', After: `im loving the pain, i never wana live without it'.These examples show cases of despair and a sense of defiance after not getting the personal attention they desired.Next, we compared the unfollowing behavior with respect to the factor scores obtained for each fan. Since these values are continuous (not categorical), we applied a Cox proportional hazard regression analysis to obtain the coefficients for the three factors.The coefficients for sociability and activity were statistically significant (p < 0.001 ), with values, 0.421 and 0.418 respectively. Using the exponent of the coefficient (e^coeff), we can conclude that users that have higher sociability and activity, will have a lower unfollowing time, i.e. they will unfollow earlier.This could mean that when fandom is not wholly directed at the celebrity but rather also has a communal component, it is more stable. Going back to the cost/benefit perspective we offered above, we can say that the socialbility of fandom adds to the benefits fans receive from their PSRs so that even if a celebrity disappoints on Twitter or in real life, the community of fans provides benefits to its members such that they are less likely to break off this relationship.§ CONCLUSIONSIn this paper, we presented a large scale analysis of fan behavior on Twitter.Based on fan-celeb interaction and fan activity, we first obtained the important factors that underlie fan behavior.Using these factors, we analysed (i) fan behavior by gender & age, and (ii) para-social breakup behavior.Our results show that (i) fandom is a social phenomenon, (ii) female fans are often more devoted and younger fans are more active & social, and (iii) the most devoted fans are more likely to be involved in a para-social breakup.In summary, our analyses point to the complexity that social media use adds to fandom behavior and the various ways that fans develop PSRs on Twitter. As celebrities - and now presidents -continue to use social media as a central avenue to cultivate their following, engage their fans and influence society, it is crucial that more studies are conducted to understand the ways such relationships develop and are maintained.§ ACKNOWLEDGEMENTS.This work has been partly supported by the Academy of Finland project “Nestor” (286211) and the EC H2020 RIA project “SoBigData” (654024). ACM-Reference-Format | http://arxiv.org/abs/1705.09087v1 | {
"authors": [
"Kiran Garimella",
"Jonathan Cohen",
"Ingmar Weber"
],
"categories": [
"cs.SI"
],
"primary_category": "cs.SI",
"published": "20170525082031",
"title": "Characterizing Fan Behavior to Study Para Social Breakups"
} |
firstpage–lastpage Kato-Milne Cohomology and Polynomial Forms Kelly McKinnie October 2, 2017 ========================================== ∂̣Łαα_coldα_hotṀ_∗Ω_∗Ω̇Ω_KṀ_inṀ_dṀ_critṀ_outṀĖṖν̇M_⊙F_optL_inL_coolR_inr_inr_LCr_outr_in,maxr_cor_eL_discL_xL_dL_optL_xF_optM_dN_Hδ E_burstδ E_xB_∗υ_ffβ_bB_eB_pB_zB_|phiB_At_intt_diff_̊mr_mr_AB_Ar_Sr_pT_pδ M_in_cT_effυ_ffT_irrF_irrT_crit0minP_0,minA_Vα_hotα_coldτ_c∝ω_∗υ_ffυ_υ_rυ_Kυ_escυ_outυ_ϕυ_diffυ_r,eυυ_Bτ_Bh_Ah_ec_sc_s,eh_inρ^'ρ_dρ_sρ_d^'ρ_eρ_outAlfvén 18SGR 0418+572942AXP 0142+61Çalışkan “” g s^-1erg s^-12cm^-2 The optical excess in the spectra of dim isolated neutron stars (XDINs) is a significant fraction of their rotational energy loss-rate. This is strikingly different from the situation in isolated radio pulsars. We investigate this problem in the framework of the fallback disc model. The optical spectra can be powered by magnetic stresses on the innermost disc matter, as the energy dissipated isemitted as blackbody radiation mainly from the inner rim of the disc. In the fallback disc model, XDINs are the sources evolving in the propeller phase with similar torque mechanisms. In this this model, the ratio of the total magnetic work that heats up the inner disc matter is expected to be similar for different XDINs. Optical luminosities that are calculated consistently with the the optical spectra and the theoretical constraints on the inner disc radii givevery similar ratios of the optical luminosity to the rotational energy loss rate for all these sources. These ratios indicate that a significant fraction of the magnetic torque heats up the disc matter while the remainingfraction expels disc matter from the system. For XDINs, the contribution of heating by X-ray irradiation to the optical luminosity is negligible in comparison with the magnetic heating. The correlation we expect between the optical luminosities and the rotational energy loss-rates of XDINs can be a property of the systems with low X-ray luminosities, in particular those in the propeller phase.pulsars: individual– accretion – accretion discs § INTRODUCTION Dim isolated neutron stars (XDINs) form one of the young neutron star populations with several distinguishing properties (see e.g. Haberl 2007 for a review). Among other single neutron star systems, they have relatively low X-ray luminosities (∼ 10^31 - 10^32) that hinder their detection at large distances. All seven confirmedXDINs lie within a distance of 500 pc, and have cooling ages less than 10^6 yr. This indicates that they are rather abundant in the Milky Way with birth rates comparable to those of radio pulsars (Popov, Turolla & Possenti2006).All seven XDIN sources were detected in the opticalband with luminosities higher than the extrapolations of their X-ray spectra, while none of them have confirmed detection in the radio band. Another distinguishing property ofXDINs is their period clustering in the same range (P ∼ 3 - 12 s ) as that of anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs) (for a review of AXPs and SGRs see Mereghetti 2008).The source of X-ray luminosity of XDINs is likely to be the intrinsic coolingof the neutron star with effective temperatures ∼ 50 - 100 keV (Haberl 2007; Turolla 2009; Kaplan et al. 2011). What is the torque mechanism slowing down these systems? A neutron star evolving in vacuum loses rotational energy and angular momentum through magnetic dipole radiation. In this case, the dipole field strength on the pole of the star, B_0, can be estimated from the observed period, P, and period derivative, , using the dipole torque formula which gives B_0 ≃ 6.2 × 10^19 (P )^1/2. For XDINs, B_0 values inferred from the vacuum dipole torque assumption are in the10^13 - 10^14 G range, close to those of AXP and SGRs (B_0 ≳ 10^14 G).In the presence of a fallback disc around an XDIN, disc torques dominate the dipole torque.In the fallback disc model (Chatterjee, Hernquist & Narayan 2000; Alpar 2001),estimated B_0 values can be about one or two orders of magnitude lower than those inferred from the magnetic dipole torque formula. Neutron stars with fallback discs can reach the individual source properties of XDINs and AXP/SGRs with B_0 ≲ 10^12 GandB_0 ≳ 10^12 Grespectively (Ertan et al. 2014; Benli & Ertan 2016).In this work, we focus on the optical emission properties of XDINs (Kaplan et al. 2011 and references therein) in the fallback disc model. We will use the results of earlier work on the long-term evolution of individual XDINs (Ertan et al. 2014) and on the inner disc radius of neutron stars in the propeller phase (Ertan 2017). These results indicate that XDINs are evolving in the propeller phase under the effect of disc torques.The source of the X-ray luminosity is the intrinsic cooling of theneutron star. With the current mass-flow rates of XDINs (≲ 10^9) expected from the long-term evolution model, viscous dissipationin the inner disc cannot produce the optical luminosity of these sources. The X-ray irradiation of the upper and lower surfaces of the disc is not sufficient either. We think of two alternative sources that can power the optical emission of XDINs, namely the X-ray irradiation of the inner rim of the disc and the magnetic stresses that heat up the matter in the disc-field interaction region. For both heating mechanisms, we expect that the surface of the inner rim of the disc is the main site of optical/UV emission. Depending on the current X-ray luminosity, disc mass-flow rate and and the rotational energy loss-rate, one of these mechanisms could dominate the other. Our results exclude X-ray irradiation as a possible source with enough power for the optical excess. We find that the magnetic stresses acting on the inner disc, which is likely to produce a strong relation between the rotational power and the heating rate, could be the source of the optical luminosity of XDINs.In Section 2, we estimate the upper and lower limits on the total optical fluxes of XDINs from the available optical data. In Section 3, we investigate the inner disc conditions of XDINs and show that the narrow disc-field interaction boundary heated by the magnetic stresses can produce the observed optical properties of the six XDINs with confirmed period and period derivatives. We discuss our results in Section 4 and summarise our conclusions in Section 5. § OPTICAL SPECTRA AND LUMINOSITIES OF XDINS The model X-ray spectra seen in Figs. 1–6 were obtained by Kaplan et al. (2011) from the blackbody model fits to the unabsorbed X-ray data of XDINs with effective temperatures given in Table 1. This emission is very likely to be produced by the intrinsic cooling of the neutron stars. All these sources have detections in the optical/UV bands (Kaplan et al. 2011, and references therein). It is seen that the optical data of the sources remain above the extrapolation of the X-ray spectrum to low energies. This is known as the optical excess of XDINs. With the current properties of XDINs indicated by the long-term evolution model with fallback discs, heating by viscous dissipationacrossthe disc is negligible compared to the X-ray irradiation flux, , provided by the cooling luminosity of the neutron star. At the low X-ray luminosities of XDINs,is not sufficient to produce the optical excess from the upper and lower surfaces of the disc. For a neutron star with mass M = 1.4 and radius R = 1× 10^6 cm, the irraditation flux can be written as ≃ 1.2 C / (π r^2), where the irradiation efficiency C(h/r)includes effects of irradiation geometry and the albedo of the disc surfaces (Fukue 1992). Pressure scale-height hr^9/8, and h/rr^1/8 has a weak r dependence.During the accretion phase the value of C is estimated to be≳ 10^-4 for XDINs for a large range of accretion rates, like in low-mass X-ray binaries (Dubus et al. 1999). After termination of the accretion phase of XDINs, the mass-flow rate of the disc, , decreases rapidly to very low levels (≲ 10^9), which also decreases the h/r ratios to less than about 10^-3 and the irradiation efficiency C to a few 10^-5 forall XDINs. All seven XDINs have upper limits in the infrared/SPITZER bands (Posselt et al. 2014). We obtain the illustrative model infrared spectra (ν < 3× 10^14 Hz) given in Figs. 1–6 with C = 3× 10^-5 and cos i = 0.5 where i is the inclination angle between the normal of the disc and the line of sight of the observer. The inner disc radii taken to obtain the IR spectra consistently match the inner disc radii that canproduce the optical spectra (Section 3).As seen in Figs. 1–6, the contribution of the emission from the irradiated upper/lower disc surfaces to the optical luminosity is too weakto account for the observed optical spectra. The detections of two sources, namely RX J0806.4-4123 and RX J2143.0+0654,in the Herschel PACS red band should be taken with some caution, because whether these detections are from the sources themselves or from their bright neighbours is not clear yet, due to the so-called “confusing neighbours" (Posselt et al. 2014).In this work, we propose that the optical flux is emitted mainly from the inner rim of the disc in the form of a blackbody spectrum. To test this idea and the alternative sources for powering the optical luminosity, we first estimate the total observed optical fluxand the effective temperature, , of the emitting region from the spectra of the sources.The two model curves (dashed and dashed-dotted) seen in Figs. 1–6 correspond to the minimum and themaximumallowed by the error bars of the data points respectively. Thecorresponding to a particularis found by integrating thespectrum produced by this . For each XDIN source,andvalues are given in Table 1. In Section 3, we use these results to estimate the optical luminosities of thesources in consistence with their inner disc properties.§ WHAT POWERS THE OPTICAL EMISSION OF XDINS? For isotropic X-ray emission from the star the fraction of the X-ray luminosity, ,that is absorbed by the disc's inner rim is estimated to be ∼ / whereis the half thickness of the disc at the inner radius of the disc, . The / ratio depends very weakly on both the disc mass-flow rate, , and . We expect the ratio of the optical luminosity, , toto be similar for different XDINs, if the inner disc is heated mainly by the X-ray irradiation. The second alternative source to power the optical emission is the interaction between the inner disc and the dipole field of the star.Part of the work done by the magnetic stresses, at the expense of the rotational energy of the star, could be converted into heat in the disc.Results of recent analytical and numerical work indicate that the inner disc-field interaction takes place in a narrow boundary (Lovelace, Romanova & Bisnovatyi-Kogan 1995; Hayashi, Shibata & Matsumoto 1996; Goodson, Winglee & Boehm 1997; Miller & Stone 1997; Lovelace, Romanova & Bisnovatyi-Kogan 1999; Uzdensky, Königl & Litwin 2002; Uzdensky 2004; Ustyugova et al. 2006). This means that, like X-ray irradiation, the magnetic stresses also heat up the innermost region of the disc, and the resultant radiation is likely to be emitted mostly from the surface of theinner rim of the disc.Independently of thedetails of the heating inside the boundary, if this mechanism, rather than the X-ray irradiation, is responsible for the optical emission, the ratio ofto the rotational energy loss-rate, , is expected to be similar for different XDINs that slow down with similar propeller torques. The rotationalpower = I Ω || of XDINs, where I is the moment of inertia, Ω is the rotational frequency of the neutron star, andis the spin-down rate, is given in Table 2, along with the periods, period derivatives, estimated distances, and the X-ray luminosities of the sources. To estimateusingobtained from the spectral fits, we need the inner disc radius , and the area, A, of the inner rim of the disc. §.§ The inner disc radius in the propeller phase The XDINs are expected to be in the propeller phase of their evolution in interaction with a fallback disc (Ertan et al. 2014).Where is the inner disc radius in the propeller phase? At the co-rotation radius = (GM)^1/3Ω^-2/3, where G is the gravitational constant, and M is the mass of the neutron star, the field lines and the disc matter move with the same angular velocity. Since the escape speed = √(2) r whereis the local Keplerian angular velocity in the disc,the disc mattercan be accelerated to aboveat radii r where (r) / Ω≥√(2). Since r^-3/2, the field lines co-rotating with the star have speeds greater than (r) at radii greater than r_1 = 2^1/3 = 1.26 .That is, the inner disc radiusshould be greater than r_1 = 1.26 in a steady propeller phase. Recently, from simple analytical calculations, Ertan (2017) estimated the maximum possible inner disc radius in the propeller phase. Theoretical and numerical work on the disc-field interaction show that the field lines interact with the inner disc in a narrow boundary (see e.g. Lovelace et al. 1995). In this interaction region, the field lines cannot slip through the disc in the azimuthal direction due to the long magnetic diffusion time-scale that is similar to the viscous time-scale(Fromang & Stone 2009), and is still orders of magnitude smaller than the time-scale of interaction between the field lines and the inner disc matter, ≃ |Ω - |^-1. The field lines inflate and open up within , expel the matter along the open field lines, and subsequently reconnect on the dynamical time-scale completing the cycle (Lovelace et al. 1999; Ustyugova et al. 2006). Outside the boundary, the disc and the field lines are expected to be decoupled. The outflow speed of the matter depends on the dipole field strength and the mass density of the disc matter. At a given radius, the maximum amount of angular momentum that can be injected into the matter is limited by the interaction time-scale, unlike in models assuming that the closed field lines could remain threaded across a large boundary region. For a given mass-flow rate of the disc and dipole field strength, the minimum angular momentum transfer required to accelerate the matter to above the escape speed can be sustained up to a certain critical radius. This is the maximum inner disc radius at which a steady propeller mechanism can be built up, and is given by≃ 3.8 × 10^-2 _-1^2/7 (ω - 1)^-4/7 (/)_-2^4/7 (Ertan 2017) where _-1 =/ 0.1 is the kinematic viscosity parameter (Shakura & Sunyaev 1973), (/)_-2 = (/)/ 10^-2, ω =/ ≥√(2) is the fastness parameter, andis the conventionalradius≃ (G M)^-1/7 μ^4/7 ^-2/7 (Lamb, Pethick & Pines 1973; Davidson & Ostriker 1973) whereis the spherical accretion rate and μ = (B_0 /2)R^3 is the magnetic dipole moment of the star. Equation (<ref>) is valid only in the propeller phase, that is, when ω > √(2). Because of sharp radial dependence of magnetic stresses we expect that the actual inner disc radius is near the radius given by equation (<ref>).The / ratio should be calculated for an unperturbed geometrically thin disc. For a standard thin disc, h/r ^3/20 r^1/8 has a very weak dependence on both r and the disc mass-flow rate (see e.g. Frank, King & Raine 2002). In equation (<ref>), the h/r ratio at the inner disc radius should also be calculated for an unperturbed standard disc (Ertan 2017).For disc accretion,is usually assumed to be close to the radius where the viscous stresses in a standard disc is balancedby the magnetic stresses, and with different assumptions,is usually inferred to be close towithin a factor of ∼ 2 (e.g. Frank et al. 2002). It was shown that a steady propeller mechanism cannot be sustained at a radius close to(Ertan 2017). From equation (<ref>), it can be calculated thatshould be at least ∼ 15 times smaller than , in the propeller phase. Whenis sufficiently greater thansuch that ω -1 ∼ω, equation (<ref>) gives ^7/13 ^-2/13. This indicates that once a system enters the propeller phase,remains close toeven with very low . For an order of magnitude decrease inbelow the accretion-propeller transition level, the inner disc radius increases only by a factor of 1.4. In the propeller phase,is the radius where all the inflowing matter, with the rate , is thrown out of the system with speeds greater than the escape speed. The critical accretion rate for the accretion-propeller transition can be estimated by setting ω = √(2) in equation (<ref>) which gives ≃ 4 × 10^11 g s^-1 _-1 P^-7/3 μ_30^2 (/)_-3^2 (Ertan 2017) where μ_30 is the magnetic dipole moment of the star in units of 10^30 G cm^3, P is the spin period of the star, and ( / )_-3 is the h/r ratio at the inner disc radius in units of 10^-3, a typical value for XDINs. Above this critical accretion rate, the system is expected to be in the accretion phase. The accretion-propeller transition condition estimated from equation (<ref>) is consistent with the observed properties of the transitional milli-second pulsars which show transitions between the accretion powered X-ray pulsar state and the radio pulsar state (Papitto et al. 2015, Archibald et al. 2015).From our earlier work on the long-term evolution of XDINs, we have estimated μ_30 to be in the 0.2 - 0.7 range, and the current mass-flow rates≲ 10^9 .For all XDINs,∼ 10^9 - 10^10, and the actual disc accretion rates, ,are below the critical rate, thus the XDINs should be in the propeller phase currently. Using thevalues estimated from the long-term evolution model for each source, the maximum inner disc radii from equation (<ref>) are found to be less than around 2.8. These results constrainof these sources into a narrow range (1.26 << 2.8).The/ values are weakly dependent on , and we find /≲ 10^-3 for all these sources with ∼. These results allow us to constrain and the area A = 2π 2 of the rim of the inner disc for a given XDIN source. We will estimate the optical luminosities using these well constrainedand / values and the effective temperatures obtained from the spectral fits.§.§ Estimation of the optical luminosity To calculate the optical luminosity, , we need only the total area of the inner rim of the disc, A, from the theoretical calculations, and the effective temperature from the observed spectrum. We define a geometrical factor, η, that represents the fraction of the total area of the inner rim of the disc, A, that is visible and projected onto a plane perpendicular to the line of sight, A^' = η A. Since we can see less than half of the total area A, η is estimated to be less than about sin i / π where i is the angle between the normal of the disc and the line of sight. The total optical luminosity can be written as = 4 π d^2 η^-1 = σ^4 A where d is the distance to the source, σ is the Stefan-Boltzmann constant andis the effective temperature of the disc's inner rim. In equation (<ref>), the inner rim area A of a source is well restricted independently of the spectrum as discussed in Section 3.1.is the observed flux obtained from the spectrum with this particular . In the spectral fits, error bars of data points limitsinto ranges with different widths for different sources (Table 1). For a given source, these restrictions put upper and lower limits on , and thus on d^2 / η value. The maximumcorresponds either to < or to a lower luminosity calculated by usingwith the maximum allowed values ofand . The minimumis obtained with= 1. 26 and the minimumpermitted by the spectral fits.The results of our earlier work on the long-term evolution of these sources indicate that they all slow down in the propeller phase with similar torque mechanisms (Ertan et al. 2014). In this model, we expect the fraction of the rotational energy-loss rate of the star heating up the inner disc matter to besimilar for these systems. To test whether the observed optical luminosities of XDINs could be produced by this heating mechanism, we calculate the allowedranges of the six XDINs, and compare their / ratios.For the illustrative results given in Table 3, it is seen that the / ratios of the six XDINs are found to be similar (0.45 - 0.90). The last term in equation (<ref>) does not depend on η or d.is calculated using the allowed values of A from the theoretical model, andindependently from the spectrum. For a given source, calculatedcorresponds to a certain value of d^2 / η (see equation (<ref>)). We have estimated the η values for the 6 XDINs using the distances employed earlier in their long-term evolution model. In Table 3, it is seen that these η values are distributed in a plausible range (0.01 ≲η≲ 0.07). If the distance of a source is modified, the same / ratio can be obtained by modifying the η as well. The maximum value of η could be ∼ 0.3 due to the restrictions by the viewing geometry. This does not impose a significant limitation on distances, and our results for / ratios do not change within a factor≳ 2 of the distances given in Table 2.In this picture, a self-consistent explanation of the optical excess of XDINs requires similar / ratios for all these sources, and our results show thatthis is possible only with high / values close to unity (see Table 3), which indicates thata significant fraction of the work done by the magnetic torques is converted into thermal energy in the boundary layer. The resultant blackbody emission from the surface of the inner rim of the disc is in good agreement with the optical data. Note that optical luminosities of XDINs, even assuming isotropic emission, are too high to havea magnetospheric origin (see Kaplan et al. 2011 for a detailed discussion on the alternative sources of the optical excess for the neutron stars without a fallback disc).The / ratiosare also given in Table 2.For RX J0420.0-5022, the ratio is 2.8, which cannot be attained by the X-ray irradiation, since onlya small fraction, /, ofcan illuminate the inner rim of the disc. For the other sources, the / ratios are greater than 10^-2. These ratios would be possible if / > 10^-2 which is not likely to be attained with the temperatures that can be sustained with the X-ray irradiation flux illuminating the inner discs of XDINs. More importantly, thevalue that can be produced by X-ray irradiation, at the inner rim of the disc (∼ 1 eV) remains much below the minimumallowed by the optical spectra for all these sources (given in Table 1). In sum, the X-ray irradiation cannot produce the optical luminosity of XDINs consistently with their opticalspectra. The contribution of X-ray irradiation to the heating of the inner rim of the disc seems to be negligible for XDINs. For other systems with relatively high X-ray luminosities, the X-ray irradiation could dominate the magnetic heating. We estimate that the inner rim of the disc is heated dominantly by the magnetic stresses for the systems with / ≳/.§ CONCLUSIONSThe optical luminosities, , of the six XDINs are orders of magnitude greater than the total optical luminosity that can be produced by the viscous dissipation in their discs. We have estimated and comparedwith possible sources of power from the neutron star. First, we have tested the effect of X-ray irradiation of the disc by the cooling luminosity of the neutron star. For one XDIN, namelyRX J0420.0-5022, > which directly eliminates X-ray irradiation as an alternative source of optical excess of this particular source. For the other XDINs, we have found that that the X-ray irradiation of the upper/lower surfaces and the inner rim of the disc cannot produce the optical luminosities consistently with the effective temperatures indicated by the optical spectra. Next, we have considered the rotational energy loss-rateof the neutron star, = I Ω ||, which is determined mainly by interaction of the inner disc with magnetosphere of the star in the fallback disc model.The rotational poweris a measured quantity, and the thermal radiation powered by the heating effect of the magnetic stresses on the disc matter is expected to be emitted mostly from the inner rim of the disc. This interaction mechanism is likely to be similar in different XDIN sources evolving in the propeller phase producing similar /ratios if the source of theirare indeed powered by the magnetic torques. We have obtained our results with this basic assumption without addressing the details of the magnetic torque and heating mechanisms. We have estimated the ranges ofvalues allowed by the observed spectra, and the constraints on the inner disc radii and thickness placed by the earlier work on the evolution of XDINs and on the inner disc radii in the propeller phase. We have found that thevalues compatible with these constraints match a similar fraction offor all the six XDINs (Table 3).Note that the optical luminosities of XDINs cannot be produced by magnetospheric emission as in isolated neutron stars without fallback discs. Typical / ratio is less than 10^-6 for radio pulsars (Zavlin & Pavlov 2004) which is several orders of magnitude lower than those in XDINs.The rotational powers of XDINs are lower than their X-ray luminosities. Nevertheless, our results indicate that a large fraction of their rotational power heats up the inner disc matter, while a small fraction of the X-ray luminosity (∼/) can illuminate the inner rim of the disc.We have estimated that the inner discs of the systems with/>/ are heated mainly by the magnetic stresses. All the known XDINs show this property. To sum up, the optical excess of XDINs can be explained by the emission from the inner rims of their fallback discs due to heating during the propeller process powered by magnetic stresses at the expense of the star's rotational energy. 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G., 2004, ApJ, 616, 452 | http://arxiv.org/abs/1705.09547v1 | {
"authors": [
"Unal Ertan",
"Sirin Caliskan",
"M. Ali Alpar"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170526120315",
"title": "Optical excess of dim isolated neutron stars"
} |
Department of Mathematics, University of Michigan, Ann Arbor, Michigan [email protected] Bayraktar is supported by the National Science Foundation under grant DMS-1613170 and the Susan M. Smith Professorship. Alexander Munk is supported by a Rackham Predoctoral Fellowship.We gratefully acknowledge E. Jerome Benveniste, Charles-Albert LeHalle, Sebastian Jaimungal as well as the participants in thede Finetti Risk Seminar (jointly organized by Bocconi University and the University of Milan),LUISS's Mathematical Economics and Finance Seminar, University of Michigan's Financial/Actuarial Mathematics Seminar, and the Fields Institute's Quantitative Finance Seminar, for their valuable suggestions.Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 [email protected][2010]Primary 91G80; Secondary 60H30, 60H10, 34M35Oft-cited causes of mini-flash crashes include human errors,endogenous feedback loops, the nature of modern liquidity provision. We develop a mathematical model whichcaptures aspects of these explanations. Empirical features of recent mini-flashcrashes are presentin our framework.For example,there are periods whenno such events will occur.If they do, even just before their onset,market participants may not know withcertaintythat a disruption will unfold.Our mini-flash crashes can materialize in both low and high trading volume environments and may be accompanied bya partial synchronization in order submission.Instead of adopting a classically-inspired equilibrium approach, we borrowideas from theoptimal execution literature. Each of our agents begins withbeliefs about howhis own trades impact prices and how prices would move in his absence.They, along withother market participants,then submit orders which are executed at a common venue.Naturally, this leads us to explicitly distinguish betweenhow prices actuallyevolve and our agents' opinions.In particular, every agent'sbeliefs will be expressly incorrect. Mini-Flash Crashes, Model Risk, and Optimal Execution Alexander Munk=====================================================§ OVERVIEW Amidst the violent market disruption on May 6, 2010, the infamous Flash Crash,“Over 20,000 trades across more than 300 securities were executed at prices more than 60%away from their values just moments before.Moreover, many of these trades were executed at prices of a penny or less, or as high as $100,000, before prices of those securities returned to their `pre-crash' levels” (<cit.>).Today, this particular event remains so memorable due to its remarkable scale.In fact, lesser versions of the Flash Crash, or mini-flash crashes, happen quite often.Anecdotal evidence suggests that there may be over a dozen every day (<cit.>).A rigorous empirical analysis uncovered “18,520 crashes and spikes with durations less than 1,500 ms” in stock prices from 2006 through 2011 (<cit.>).The exhaustive documentation on Nanex LLC's “NxResearch” site offers further corroborationas well (<cit.>). Why do such phenomena occur? Several answers have been proposed.Roughly, most point tohuman errors,endogenous feedback loops, the nature of modern liquidity provision.These ideas can be viewed as different ways to rationalize how an extreme (local or global) dislocation in supply and demand can arise in modern markets.Our contribution is the development of a modelwhich captures this. Our model also appears to exhibit features ofhistorical mini-flash crashes.For instance, there areperiods in which extremeprice moves will not manifest. If they do, accompanying trade volumes can be high or low. Some market participants may partially synchronize their trading during a mini-flash crash. Our agents may not know thata mini-flash crash is about to begineven just before its onset. Our results seem to be aligned with intuitive expectations as well.For example, our mini-flash crashes can begin if some of our agents are toouncertain about their initial beliefs, inaccurate in their understanding of price dynamics, slow to update their models and objectives, or willing to take on risk. We construct our model beginning witha finite population of agents trading in a single risky asset,each of whom must decide how to act based upon his own preferences, beliefs, and observations. Our specifications are drawn fromideas in the price impact and optimal execution literature and are given in Sections <ref> - <ref>. We imagine that our agents' orders are submitted to a single venue, where they are executed together with trades from other (unmodeled) market participants.This naturally compels us to make an explicit distinction betweenhow the risky asset price actually evolves and our agents' beliefs about its future evolution (see Section <ref>). Since we view our agents assimultaneously solving their own optimal executionproblems, we avoid certain strongassumptions that would have beenimplicitly needed, if we had useda classical equilibrium-based approach. We use a similar framework to <cit.> in that our players' interaction with the rest of the world (in addition to each other) is given by a price impact model.An additional consequence is thatwe precisely describe the errors in ouragents' beliefs. Potentially, each agent could be wrong both about how his trades affect prices andhow prices would move in his absence.To the best of our knowledge,this general setup appears to be a new paradigm for modelingheterogeneous agent systems in the contexts of optimal execution and mini-flash crashes.We are ready to begin presentingour work in detail.We highlight key background material and our paper's contributionsin relation to it in Section <ref>. Our agents and their beliefs are described in Section <ref>.We describe the correct dynamics of the risky asset's price in Section <ref>. General analysis of the dynamical system when our agents act as prescribed by Section <ref>but prices actually move as in Section <ref>are given in Section <ref>. Using the material in Section <ref>, we obtain explicit characterization of mini-flash crashes when uncertain agents are semi-symmetric in<ref>. Numerical examples illustrating our main results are given in Section <ref>.Our longer proofs are contained in Appendices <ref> - <ref>.§ BACKGROUND & CONTRIBUTIONS In this section, we clarify our contributions and explain how they fit into the current literature.We already mentioned that existing theorieson the causes of mini-flash crashes could beviewed as falling into one of five categories (see Section <ref>).Here are further details.*Human errors (and, relatedly, improper risk management) are among themost commonly cited causes of mini-flash crashes (<cit.>, <cit.>, <cit.>).The SEC claims that the majority of mini-flash crashes originate from such sources, in fact (<cit.>).When we read aboutfat finger trades, rogue algorithms, or glitches in the media, typically human errors are indirectly responsible. For example, due to a bug in thesystems at the Tokyo Stock Exchange and a typo in a trade submitted by Mizuho Securities,the share price of the recruitment agency J-Com fell in minutes from 672,000 to 572,000 on December 8, 2005 (<cit.>). *Mini-flash crashes may be caused bythe rapid, endogenous formation of positive feedback loops (<cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>).As Johnson et al. put it, “Crowds of agents frequently converge on the same strategy and hence simultaneously flood the market with the same type of order, thereby generating the frequent extreme price-change events” (<cit.>). A separate empirical study on the Flash Crash of May 6, 2010, specifically,determined that at its peak, “95% of the trading was due to endogenous triggering effects” (<cit.>).*The nature of liquidity provision in modern markets is thought tocause some mini-flash crashes (<cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, <cit.>). Today, the majority of liquidity is provided by participants that are free from formal market-making obligations (<cit.>). In particular, they can instantly vanish, effectively taking oneor both sides of the order book at some venue with them.A mini-flash crash can arise either directly as bid-ask spreads blow out or indirectly when a market order (of any size) tears through a nearly empty collection of limit orders.Such a phenomenon has been called fleeting liquidityand may have contributed to the occurrence of38% of mini-flashcrashes from 2006 - 2011 (<cit.>).This proposed explanation is deeply intertwinedwith a crucial empirical observation: Mini-flash crashes occur in bothhigh and low trading volume regimes. For instance, the trading volume during the 30s mini-flash crashof “WisdomTree LargeCap” Growth Fundon November 27, 2012 was nearly eight times the average daily trading volumefor this security (<cit.>).The empirical study by Florescu et al.offers extensive evidence that mini-flash crashesoften occur during low trading volume periods as well. Aspects of (<ref>,(<ref>,and (<ref> are reflected in our work.For example, the human error theory arises in each of the following ways:*Every agent believes thata mini-flash crash is a null event.On the contrary, there are cases in whichone will occur almost surely (see Theorems <ref> and<ref>).*Every agent thinks that his trades affect prices through specifictemporary and permanent price impact coefficients (see Section <ref>).His estimates for these parameters might bewrong (see Section <ref>).*Every agent's trades may also indirectly impact pricesby inducing others to make differentdecisions than they would otherwise(see Section <ref> and Section <ref>).This potential effect is not modeled by our agents (see Section <ref>).More generally, even if we have a single agent in our setup trading with other unspecified market participants,the parameters in his fundamental valuemodel might be inaccurate (see Section <ref> and Section <ref>). *No agent revises the general class of his beliefs,admissible strategies,or objectives on our time horizon (see Sections <ref> - Section <ref>). In some cases, a mini-flash crash will not occur if this period is fairly short but will if it is too long (see Lemmas <ref> and <ref>).*Every agent is averse tohis position's apparent volatility risks(see Section <ref>).In some cases, there will be nomini-flash crash when our agents are sufficientlyaverse to these risks; otherwise, there will be one (see Lemmas <ref> and <ref>).*Every agent has the opportunity toupdate the drift parameter inhis price model based upon his observations (see Section <ref>).In some cases, a mini-flash crash will unfoldbecause our agents are too easily persuadedto revise their priors(see Lemmas <ref> and <ref>).*Every agent has a model for how pricesare affected by the temporary impactof trades (see Section <ref>).In some cases, there will be a mini-flashcrash if our agents sufficientlyunderestimate the role of aggregate temporary impact.No such disturbance will occur otherwise.Our agents may be more prone to inducemini-flash crashes in this waywhen there are many of them (see Lemmas <ref> and <ref>). Notice that some of our agents' human errors directlycause mini-flash crashes, though not all (see Proposition <ref>). We highlight this observation in Figures <ref> - <ref>.Implicitly, the occasional absence of mini-flash crashes also agrees with (<ref>. Despite the regularity of these disruptionson a market-wide basis,individual securities may rarely experience such an event.Similarly, traders' models and strategies do roughly achieve their intended goals much of the time,which we observe as well (see Proposition <ref>).Several key ideas from theendogenous feedback loop theoryare present in our paper.For example, if a mini-flash crash does occur,it almost surely does so because of“endogenous triggering effects.” Specifically, our mini-flash crashes arise when some of our agentsbuy or sell at faster and faster rates, which they only dobecause they started trading more rapidly in the first place (see Section <ref> and Lemma <ref>).As predicted by this theory, some of our agents also “converge on the same strategy” during mini-flash crashes: In certain cases, the agents driving these events all buy or sell together with the same (exploding) growth rate (see Theorems <ref> and <ref>).Figures <ref>,<ref>,and <ref> graphically illustrate this partial synchronization. We do not explicitly model liquidity providers in our framework,as we view our agents as submitting market orders to a single venue (see Section <ref>).We still view our paper as reflecting (<ref>,at least in some sense, since our mini-flash crashes can be accompanied by both high and low trading volumes (see Corollary <ref> andTheorems <ref> and <ref>).Visualizations of this point are providedin Figures<ref>,<ref>, <ref>,<ref>, <ref>, and <ref>.§ AGENTS AND THE EXECUTION PRICE In this section we describe ouragents and their individualoptimal execution problems. We end this section by introducing the “true" execution price.§.§ Agents' ModelsOur agents trade continuously by optimally selecting a trading rate from a particular class of admissible strategies.To motivate our specifications of their choices and objectives, we first define their models and beliefs.All trades submitted at time t are executed immediately at the price S_t^exc.At each time t, every agent observes the correct value of S_t^exc.No agent knows the true dynamics of the stochastic process S^exc, though. In our framework, Agent j ∈{1,⋯, N}'s models the the risky asset without his own trading by S_j,t^unf = S_j,0 + β_j t +W_j,t,t ∈[ 0, T ],where W_j is a Wiener process and β_j is a normally distributed with mean μ_jand variance ν_j^2 which is independent of the Wiener process. (In what follows we will write down P_j for the probability measure of agent j.) This drift term represents the price pressure that Agent j believes will arise due to the trades of(other) institutional investors.Agent j approximates the average behavior of uninformed or noise traders using the Brownian term.[ Almgren & Lorenz provide further details regarding the interpretation of (<ref>) (<cit.>).A possible extension of our work could replace (<ref>) with one of the more recent models considered in the literature on optimal trading problems with a learning aspect (<cit.>, <cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>). ] If ν_j^2 > 0, then we call Agent j an uncertain agent. If ν_j^2 = 0, then we call Agent j a certain agent. Regardless of whether he is certain or uncertain in this sense,we will soon see that Agent j can always be viewed as certain about many things,e.g, he will not change theform of his models, objectives, or admissible strategies on [ 0 , T ]. From thisperspective, one might partially connect our work on mini-flash crashesto explanations of longer term financial bubbles based on overconfident investors (<cit.>).Intuitively, Agent j's selection of (<ref>) makes the most sense when N is large and T is short.Notice that Agent j makes no attempt to precisely estimate the number of other market participants,nor their individual goals or beliefs.That he believes he cannot improve the predictive accuracy of (<ref>) by doing so appears to suggest that the population of traders is of sufficient size. Together with the fact that real drifts and volatilities are non-constant,(<ref>) only seems even potentially plausible over short periods.Let 𝒜_j be the space of ℱ_j,t^unf-adapted processes θ_j of trading speeds such that θ_j,·( ω) is continuous on [ 0 , T ] for P_j-almost surely, E^P_j[ ∫_0^Tθ_j,t ^2 dt ] < ∞,and x_j + ∫_0^T θ_j,tdt = 0 P_j-a.s.For any θ_j∈𝒜_j, we denote by X_j,t^θ_j = x_j + ∫_0^tθ_j,sds,the agent's inventory. The agent models the execution price as S_j,θ_j^exc, which is given byS_j,θ_j,t^exc = S_j,t^unf + η_j,per∫_0^tθ_j,sds + 1/2η_j,temθ_j,t,t ∈[ 0, T ].Agent j has chosen the deterministic positive constants η_j,per and η_j,tem in (<ref>) prior to time t =0. Agent j could be viewed astaking into account his own effects on the execution pricevia an Almgren-Chriss reduced-form model (<cit.>, <cit.>, <cit.>).η_j,per would denote Agent j's estimate for his permanent price impact parameter,while he would approximate his temporary price impact parameter with η_j,tem.§.§ Each Agent's Optimization Problem Agent j's objective is to maximize the following objective function: E^P_j[ - ∫_0^T θ_j,tS_j,θ_j,t^excdt-κ_j/2∫_0^T( X^θ_j_j,t)^2 dt ].This frequently used criteria balanceshis realized trading revenue and risks associated with delayed liquidation.Agent j selects the deterministic risk aversion parameter κ_j > 0 based upon his appetite. Let us denote τ_j( t ) ≜√(κ_j/η_j,tem)( T - t ) ,t ∈[ 0 , T ].(<ref>) has a unique optimizerθ_j^⋆∈𝒜_j almost surely. When ω∈Ω_j is chosen such that W_j,·( ω) is continuous on [ 0 , T ],X^θ^⋆_j_j ( ω) satisfies the linear ODE θ^⋆_j,t( ω)= - √(κ_j/η_j,tem)( τ_j( t ) ) X^θ^⋆_j_j,t( ω) +tanh( τ_j(t ) /2 ) [ μ_j+ ν_j^2 ( S_j,t^unf( ω) - S_j,0) ]/√(η_j,temκ_j)( 1 + ν_j^2 t ),t ∈( 0, T ) X^θ^⋆_j_j,0( ω)= x_j .See Appendix <ref>. The first term in (<ref>) arises from our constraint thatAgent j must liquidate by the terminal time (see (<ref>)). In fact, the weighting factor - √(κ_j/η_j,tem)( τ_j( t ) )tends to - ∞ as t ↑ T.Intuitively, the reason that Agent j believes that X^θ^⋆_j_j,t and θ_j,t^⋆ remain finite as t ↑ T is that X^θ^⋆_j_j,t tends very rapidly to zero.Agent j thinks that he learns about β_j's realized value over time, which is captured by the second termin (<ref>)since E^P_j[ β_j | ℱ_j,t^unf] = μ_j+ ν_j^2 ( S_j,t^unf - S_j,0) /1 + ν_j^2 t P_j-a.s.(<cit.>).The factor tanh( τ_j(t ) /2 )/√(η_j,temκ_j)is bounded by 1 / √(η_j,temκ_j) and tends to zero as t ↑ T.The second term may either dampen or amplify the effects of the first.Agent j believes that the weighting factorsreflect that his need to liquidate musteventually overwhelm his desire to profit by trading in the directionof the risky asset's drift. An immediate observation from Lemma <ref> is the following observation for certain agents, which we record as a corollary for ease of referencing.If ν_j^2 = 0, then X^θ^⋆_j_jdoes not depend on S^unf_j. In particular, it isdeterministic and satisfies the linear ODE θ^⋆_j,t = - √(κ_j/η_j,tem)( τ_j( t ) ) X^θ^⋆_j_j,t +μ_jtanh( τ_j(t ) /2 ) /√(η_j,temκ_j),t ∈( 0, T ) X^θ^⋆_j_j,0 = x_j .§.§ Execution Price We now specify how S^exc actually evolves.While each agent observes thesame realized path of this process, in general, no agent knows the correct dynamics.[ There is a single trivial case where this is not true.If N = 1, β̃ = β, ν^2_1 = 0,η̃_1,tem = η_1,tem, and η̃_1,per = η_1,per,our lone agent's model would be exactly right. ]An agent's trading decisionsare entirely determined byhis beliefs, preferences, and observations of a single realized path of S^exc. Let ( Ω̃, ℱ̃ , {ℱ̃_t}_0 ≤ t ≤ T , P̃) be a filtered probability space satisfying the usual conditions. The space is equipped with an ℱ̃_t-Wiener process under P̃, which we denote by W̃. We also have the following deterministic real constants:β̃,S_0, η̃_1,per, …, η̃_N,per,andη̃_1,tem, …, η̃_N,tem .β̃ can be arbitrary; however, the remaining constants are strictly positive. The true execution price S^exc under P̃is the ℱ̃_t-adapted process S_t^exc = S_0 +β̃t + ∑_i = 1^N η̃_i,per( X^θ_i^⋆_i,t -x_i) + 1/2∑_i = 1^N η̃_i,temθ_i,t^⋆ + W̃_t, t ∈[ 0, T ]. Equation (<ref>) can be viewed as a multi-agent extension of the Almgren-Chriss model (<cit.>, <cit.>, <cit.>). Models of this form, particularly when the η̃_j,tem's (η̃_j,per's) are all identical,have been applied in the context of predatory trading (<cit.>). On the other hand, <cit.> consider a mean-field game model where the interactions between the players are through the price as it is here.Although both of these papers address latency and learning in their models misspecification of agents's models is an important element in our framework. Moreover, our agents do not observe each other or know each other's parameters. In fact, in our finite player set-up they do not even know the number of players N.In what follows we will say that a mini-flash crash occurs, if the S_t^exc tends to either +∞ or -∞ on our time horizon. The parameters η̃_j,per and η̃_j,tem are the correctvalues of Agent j's permanent and temporary price impact parameters, respectively.We allow these quantities to have arbitrary relationships to Agent j's corresponding estimates η_j,per and η_j,tem.For instance, Agent j might underestimate his permanent impact (η_j,per < η̃_j,per) but perfectly estimate his temporary impact (η_j,tem = η̃_j,tem).Similarly, Agent j's prior β_j for the correct drift β̃ may be accurate or severely mistaken. Comparing our descriptions of S_j,θ_j^exc in (<ref>) and S^exc in (<ref>), we see that Agent j proxies each term in (<ref>) as follows:η_j,per( X^θ_j^⋆_j,t -x_j) ⟷η̃_j,per( X^θ_j^⋆_j,t -x_j)1/2η_j,temθ_j,t^⋆ ⟷1/2η̃_j,temθ_j,t^⋆S_j,0 + β_j t +W_j,t ⟷S_0 +β̃t + ∑_i ≠ jη̃_i,per( X^θ_i^⋆_i,t -x_i)+ 1/2∑_i ≠ jη̃_i,temθ_i,t^⋆ + W̃_t.§ ANALYSIS OF THE DYNAMICAL SYSTEMWhen our agents implement the strategies that they believe are optimal (see Lemma <ref>) but S^exc has the dynamics in (<ref>), what happens?The goal of Section <ref> is to offer some general answers to this question.To simplify our presentation, we begin by introducing and analyzing additional notation (see Definition <ref> and Lemma <ref>).We find that our agents' inventories and trading rates evolve according to a particular ODE system with stochastic coefficients (see Lemma <ref>).Under certain conditions, the system can have a singular point(see Lemma <ref>).For convenience, we study what unfoldswhen this singular point is of the first kind (see Proposition <ref>).We also examine the case in which there is no singular point (Proposition <ref>). We will have an even mix of deterministicand stochastic maps. In what follows, we always explicitly indicate ω-dependence to distinguish between the two. Our equations are solved pathwise, so we do not encounter probabilisticconcerns.We will fix ω∈Ω̃such that W̃_·( ω) has a continuous path.Define the maps[ Φ_i : [ 0 , T ] ⟶ ℝ; A : [ 0 , T ] ⟶M_K ( ℝ); B : [ 0 , T ) ⟶M_K ( ℝ); C ( ·, ω) : [ 0 , T ] ⟶ ℝ^K; ]by Φ_i ( t )≜tanh( τ_i(t )/ 2 )ν_i^2/√(η_i,temκ_i)( 1 + ν_i^2 t )A_ik( t)≜{[ 1 - 1/2( η̃_i,tem - η_i,tem) Φ_i ( t )ifi = k;-1/2η̃_k,temΦ_i ( t ) ifi ≠k;].B_ik( t)≜{[ ( η̃_i,per- η_i,per)Φ_i ( t )- √(κ_i/η_i,tem)( τ_i( t ) ) ifi = k; η̃_k,perΦ_i ( t )ifi ≠k; ].C_i( t , ω)≜Φ_i ( t ) [ μ_i/ν_i^2 +( S_0 - S_i,0)+β̃t - ∑_ k ≤ K k ≠ i η̃_k,per x_k- x_i( η̃_i,per- η_i,per). . + ∑_ k > K η̃_k ,per( X^θ_k^⋆_k,t-x_k)+ 1/2∑_k > Kη̃_k,temθ_k,t^⋆+ W̃_t ( ω) ] .Here, i ∈{ 1, …, K } are the uncertain agents, whose behavior we are set out to characterize. The behavior of the certain agents are already described. They are not influenced by the execution price but they do have an influence on it. Observe that we can now write the dynamics in(<ref>) as θ^⋆_j,t( ω)= - √(κ_j/η_j,tem)( τ_j( t ) ) X^θ^⋆_j_j,t( ω) + Φ_j ( t ) [ μ_j/ν_j^2+ ( S_j,t^unf( ω) - S_j,0) ]when Agent j is uncertain.When A has a root on [ 0, T ], we let t_e denote the smallest one (see Lemma <ref>). Suppose that A has a root on [ 0, T ].If t_e > 0, then S^exc( ω), the X^θ_j^⋆_j( ω)'s andthe θ^⋆_j( ω)'s are all uniquely defined and continuous on [ 0, t_e ).Moreover,the uncertain agents' strategies are characterized byA (t )θ_t^u,⋆( ω)=B ( t ) X^u,θ^⋆_t(ω)+ C ( t, ω) ,t ∈( 0, t_e ) X^u,θ^⋆_0(ω)= x^u,whereθ_t^u,⋆( ω) denotes the first K-entries of θ_t^⋆( ω). When A does not have a root on [ 0, T ], the same statements hold after replacing t_e with T.See Appendix <ref>.Lemma <ref> does not addressthe behavior of our uncertain agents' inventories and trading rates as t ↑ t_e or t ↑ T.The difficulties are that A is non-invertible at t_e,while B's entries explode at T (see Lemma <ref>). The approach for resolving these issues is well-established (see Chapter 6 of <cit.>).We sketch the key points when A has a root on [ 0, T ] and t_e > 0. Analyzingthe effects of B's explosion at T is similar (see Proposition <ref>). We begin by considering the homogeneous equation corresponding to (<ref>):A (t )Ẋ_t^u ( ω)=B ( t ) X_t^u (ω) ,t ∈( 0, t_e ) X_0^u (ω)= x^u.We change notation to emphasize that (<ref>) no longerdescribes the uncertain agents' optimal strategies.We next write (<ref>) in a more convenient form. Suppose that A has a root on [ 0, T ] and t_e > 0. Near t_e, the solution of (<ref>) satisfies (t - t_e )^m+1Ẋ_t^u( ω)=D ( t ) X_t^u(ω) .In (<ref>), m is a nonnegative integer such that the multiplicity of the zero ofA at t_e is ( m + 1 ). D is a particular analytic map for which D ( t_e ) has rank 0 or 1 (see (<ref>)). See Appendix <ref>.Let us denote the unique non-zero eigenvalue of D ( t_e ) in the above lemma by λ.Suppose that A has a root on [ 0, T ], t_e > 0,and m = 0.If λ∉ℤ, then for somesmall ρ > 0, X^u,θ^⋆_t( ω)=P ( t ) [ ∑_j = 1^K-1( y_j ( ω) - ∫_t_e - ρ^t F_j ( s, ω)/| s - t_e | ds)v_j.+. | t - t_e |^λ( y_K ( ω)-∫_t_e - ρ^tF_K ( s, ω) /| s - t_e |^1+ λds) v_K]for t ∈( t_e - ρ, t_e ).Here,* {v_1, …, v_K} is an eigenbasis for D ( t_e ) (v_K corresponds to λ);* P is a (non-singular-)matrix-valued analytic function on [ t_e - ρ, t_e ] such that P ( t_e ) = I_K(see (<ref>)); * { y_1 ( ω) , … , y_K ( ω) } are constants (see (<ref>));* and { F_1 ( · , ω), … , F_K ( · , ω) } are continuousreal-valued functions on [ t_e - ρ , t_e ] (see (<ref>)). We get θ^u,⋆( ω) and S^exc( ω) on ( t_e - ρ, t_e ) by differentiating (<ref>) and by substituting X^θ^⋆( ω) and θ^⋆( ω) into (<ref>), respectively. See Appendix <ref>.Suppose that A does not have a root on [ 0, T ].Then S^exc( ω), the X^θ_j^⋆_j( ω)'s andthe θ^⋆_j( ω)'s are all uniquely defined and continuous on [ 0, T ].Moreover, lim_t ↑ T X^θ^⋆_t( ω) = 0.Each agent believes that his terminal inventory will be zero almost surely (see (<ref>)).Proposition <ref> specifies conditionsunder which the agents are effectively correct in this regard. See Appendix <ref>.§ EXPLICIT CHARACTERIZATIONS OF FLASH CRASHES FOR SEMI-SYMMETRIC UNCERTAIN AGENTSIn this section we thoroughly analyze a broad but tractable class of scenarios.This will enable us to both theoretically and numerically investigate theoccurrence of mini-flash crashes.We specify that our uncertain agents' parameters areidentical, except for theirinitial inventories x_j, means of their initial drift priors μ_j,and their initial estimates for the fundamental price S_j,0.Such agents are nearly symmetric, so we call them semi-symmetric.We say that our uncertain agents are semi-symmetricwhen there are positive constants η̃_tem, η_tem, η̃_per, η_per, ν^2,andκsuch that for each i ∈{ 1, …, K }[ η̃_i,tem = η̃_tem, η_i,tem = η_tem, η̃_per = η̃_i,per,; η_i,per = η_per,ν_i^2 = ν^2 ,κ_i = κ . ] Since τ_j's and the Φ_j's are the same for j ≤ K(see Definitions <ref>, <ref>, and <ref>).We denote these functions by τ and Φ, respectively.Definition <ref> implies that the diagonal entries of A are identical,as are the off-diagonal entries.The same is true for B (see Definition <ref>).Such a simplification considerably reduces the difficulties in computing A, λ,and an eigenbasis for D ( t_e ) (see (<ref>) and Lemma <ref>).The x_j's, μ_j's, and S_j,0's only enter in C,which also has a nice structure (see (<ref>)).§.§ ResultsSuppose that the uncertain agents are semi-symmetric.Then A has a root on [ 0 , T ] and t_e > 0 if and only ifν^2( K η̃_tem - η_tem)tanh( T/2√(κ/η_tem)) /√(η_temκ)> 2.In this case, the zero of A ( ·) at t_e is of multiplicity 1. By varying our parameters in (<ref>) one at a time we can obtain the following interpretations discussed in Section <ref>: * (<ref>) holds when ν^2 is high.Since ν^2 is the variance of the uncertain agents' drift priors,we are led to (<ref> in Section <ref>. * (<ref>) holds when ( K η̃_tem - η_tem) is high.A given uncertain agent believes that his own temporary impact parameter is η_tem,while the actual collective temporary impact parameter induced by the uncertain agents isK η̃_tem.Then ( K η̃_tem - η_tem) is large whenever each uncertain agent severely underestimates his own temporary impact orthere are many uncertain agents, giving (<ref> in Section <ref>. * (<ref>) holds when T is high.Since [ 0 , T ] is our time horizon, we get(<ref> in Section <ref>. Note that T must be small enough for ouragents' modeling rationale to hold (see Section <ref>);however, T need not be too large here, as the value of tanh reaches 95% of its supremumon [ 0 , ∞) for argumentsgreater than 1.8. * (<ref>) holds when κ is low. We conclude(<ref> in Section <ref>,as κ measures our uncertain agents' aversion to volatility risks (see Section <ref>).Observe that both the numerator and the denominator of the LHS in (<ref>)roughly look like√(κ) for small κ; however, when κ is large,the whole LHS looks like 1 / √(κ) since tanh is bounded by 1 on [ 0 , ∞). See Appendix <ref>.As observed in (<ref>), when A has a root on [0, T ], we haveΦ( t_e ) = 2/ K η̃_tem - η_tem .No agent would think to compute t_e since they all believe that a mini-flash crash is a null event; however, (<ref>) makes it especially clear that they could not do so anyway.Suppose that the uncertain agents are semi-symmetric and (<ref>) holds.Thenλ = 2 [ √(κ/η_tem)( τ( t_e ) ) -2 ( K η̃_per - η_per/ K η̃_tem - η_tem) ]/( K η̃_tem - η_tem) Φ̇( t_e )and the corresponding eigenvector is v_K = [ 1, …, 1 ]^⊤.By slightly perturbing η̃_per and/or η_per, if necessary,we can ensure that λ∉ℤ.In this case, D ( t_e ) is diagonalizable and the remaining vectors in aneigenbasis for D ( t_e ) (all with the eigenvalue zero) are v_1 = [ -1, 1, 0, …, 0 ]^⊤ , …,v_K-1 = [ -1, 0, …, 0, 1 ]^⊤. With the exceptions of η̃_per and η_per, all parameters in (<ref>) determine whether or not A has a root on [ 0, T ](see Lemma <ref>).They also fix the value of t_e (see Remark <ref>).Hence, to interpret (<ref>), we only consider the rolesof η̃_per and η_per.These parameters enter (<ref>) via K η̃_per - η_per/ K η̃_tem - η_tem .Intuitively, (<ref>) can be viewed as the ratio of two terms: The numerator measures how far a given uncertain agent's estimate of his own permanent impact is from the uncertain agents' actual collective permanent impact.The denominator, which must be positive due to Lemma <ref>, is the corresponding measure for the temporary impact. One might call (<ref>) a mistake ratio. Since Φ̇( t_e ) < 0 by Lemma <ref>, λ is positive only when(<ref>) is high enough. We have λ < 0 when the uncertain agents' total permanent impact and a single uncertain agent's' estimate of his ownpermanent impact are too close or when his estimate exceeds the cumulative permanent impact. More precisely, {λ > 0 }{1/2√(κ/η_tem)( τ( t_e ) ) ( K η̃_tem - η_tem)< K η̃_per - η_per}{λ < 0 }{1/2√(κ/η_tem)( τ( t_e ) ) ( K η̃_tem - η_tem)> K η̃_per - η_per} .Whether a mini-flash crash is accompanied by high or low trading volumesis effectively determined by which inequality in (<ref>)holds (seeTheorems <ref> and <ref> and Sections <ref>, <ref>, and <ref>). See Appendix <ref>.Suppose that the uncertain agents are semi-symmetric and (<ref>) holds.Assume that λ∉ℤ and λ < 0 (see Lemma <ref>).Let ρ, y_K ( ω), and F_K ( · , ω) be defined as in Proposition <ref>.Then { y_K ( ω) >lim_t ↑ t_e∫_ t_e - ρ^tF_K ( s, ω)/| s - t_e |^1+λds}{lim_t ↑ t_e X^u, θ^⋆_t( ω)=lim_t ↑ t_eθ^u,⋆_t( ω)= [ + ∞, …, + ∞]^⊤ , lim_t ↑ t_e S^exc_t( ω) = + ∞}and{ y_K ( ω) < lim_t ↑ t_e∫_ t_e - ρ^tF_K ( s, ω)/| s - t_e |^1+λds}{lim_t ↑ t_e X^u, θ^⋆_t( ω)=lim_t ↑ t_eθ^u,⋆_t( ω)= [ - ∞, …, - ∞]^⊤ , lim_t ↑ t_e S^exc_t( ω) = - ∞}. Moreover,* The integral limits in (<ref>) and (<ref>) exist and are finite.* Either (<ref>) or (<ref>) holds P̃-a.s. * At t_e - ρ,the events (<ref>) and (<ref>) both have positive P̃-probability; however, the P̃-probability of one event tends to 1 (while the other tends to 0) if we let ρ↓ 0. Since P ( t_e ) = I_K (see Proposition <ref>),(<ref>) and Lemma <ref> imply that y_K ( ω) will belarge and positive when the uncertain agents hold significant, similar long positions.y_K ( ω) will be of high magnitude but negative, if the uncertain agentscarry substantial, similarly-sized short positions.Hence, a spike in S^exc( ω) is more likely when the uncertain agentsare synchronized aggressive buyers, while the odds of a collapse improve when they are synchronized heavy sellers. These effects play the deciding role as t ↑ t_e, as the integral limits in (<ref>) and (<ref>) are finite. Still, due to how we can decompose F_K in our case (see (<ref>)),large fluctuations in the fundamental pricecan make the mini-flash crash's direction unclear until just before t_e (see Figure <ref>). See Appendix <ref>. Suppose that the uncertain agents are semi-symmetric and(<ref>) holds.Assume that λ∉ℤ and λ > 0 (see Lemma <ref>).Then P̃-a.s.,lim_t ↑ t_e X^θ^⋆_t( ω)exists in ℝ^N.If any coordinates of θ^u,⋆_t( ω) explode, thenS_t^exc( ω) and all coordinates of θ^u,⋆_t( ω)explode in the same direction. For instance, when λ > 1, {lim_t ↑ t_e[| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)- W̃_t ( ω) /| s - t_e |^1+ λds] = + ∞}{lim_t ↑ t_eθ^u,⋆_t( ω)= [ + ∞, …, + ∞]^⊤ , lim_t ↑ t_e S^exc_t( ω) = + ∞} and{lim_t ↑ t_e[| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)- W̃_t ( ω) /| s - t_e |^1+ λds] = - ∞}{lim_t ↑ t_eθ^u,⋆_t( ω)= [ - ∞, …, - ∞]^⊤ , lim_t ↑ t_e S^exc_t( ω) = - ∞}. Moreover, * Either (<ref>) or (<ref>) holds P̃-a.s. * At t_e - ρ,the events (<ref>) and (<ref>) both have positive P̃-probability; however, the P̃-probability of one event tends to 1 (while the other tends to 0) if we let ρ↓ 0. We make no rigorous statement regarding the λ∈( 0 , 1 ) case.Most of Theorem <ref>'s proofwould still be valid (see Appendix <ref>);however, the final estimates are especially convenientwhen λ > 1(see (<ref>) - (<ref>)).The over-arching purpose of Theorem <ref> is only to illustrate that mini-flash crashes can occur in low trading volume environments (see Section <ref>).Nevertheless, we suspect that mini-flash crashes might unfold when λ∈( 0 , 1 ), e.g.,see Section <ref> and (<ref>) - (<ref>). See Appendix <ref>. § NUMERICAL ILLUSTRATIONS §.§ Example 1: No mini-flash crashOur mini-flash crashes do not always occur (see Lemmas <ref> and <ref>).In Section <ref>,we illustrate this by numericallysimulating a scenario in whichA has no root on [ 0 , T ].By Lemma <ref> and (<ref>),we know that A is non-vanishing on [ 0 , T ] if and only if ( K η̃_tem - η_tem) Φ( 0 ) < 2.One selection of parameters for which (<ref>) is satisfied is[N = 3,K = 2,T = 1,S_0 = 100,; β̃ = 1, η̃_tem = 1, η_tem = 0.75, η̃_per = 1,;η_per = 1,ν^2 = 2, κ = 5 ,x_1 = 2,; x_2 = -2, μ_1 = 15,μ_2 = -10,S_1,0 = 100,;S_2,0 = 100, η̃_3,tem = 1,η_3,tem = 1, η̃_3,per = 1,; μ_3 = -3,ν_3^2 = 2,κ_3 = 5, x_3 = 2 .; ]In fact, the LHS of (<ref>) then equals 1.1095.Observe that there is no need to specify η_3,per and S_3,0 as they are irrelevant(seeCorollary <ref>,Definition <ref>, and Lemma <ref>). Again, our purposes are only illustrative here,and we leave the reproduction of a specific practically meaningful scenario for a future work. Since K = 2 and N = 3, we have two uncertain agents and one certain agent inthe coming plots.We label the corresponding curves with U1, U2, and C1.For example, the label U1 will signify a quantity for Agent 1, the first uncertain agent.In Figures <ref> and<ref>,we plotinventories and trading rates.The execution price isdepicted in Figure <ref>.The diagrams exhibit all of the important qualities thatwe expect based upon our theoretical results.Here are a few key features:* All agents liquidate their positions by the terminal time T(see (<ref>) and Figure <ref>).* S^exc( ω), the X^θ_j^⋆_j( ω)'s andthe θ^⋆_j( ω)'s are all continuous on [ 0, T ](see Proposition <ref> and Figures <ref> - <ref>). * The uncertain agents' trading rates appear toexhibit a Brownian component (see Lemma <ref> and Figure <ref>). * The certain agent's trading rate appears to be smooth on [ 0 , T ](see Corollary <ref> and Figure <ref>). * The agents need noteither strictly buy or strictly sell throughout [ 0, T ](see Figure <ref>). * Even so, the agents may decideto strictly buy or strictly sell throughout [ 0, T ](see Figure <ref>).* The uncertain agents' trading rates do not appear to synchronize(see Figure <ref>). §.§ Example 2: A mini-flash crash with low trading volume Our mini-flash crashes can be accompanied by low trading volumes(see Theorem <ref>).In Section <ref>, we visualize thisby studying a concrete scenario in whichA has a root on [ 0 , T ];t_e > 0;the zero of A at t_e is of multiplicity 1;λ∉ℤ;and λ > 0. The behavior of the X^θ_j^⋆_j( ω)'s is then characterized byCorollary <ref> and Theorem <ref>.Theorem <ref> would rigorouslydescribe S^exc_t ( ω) and the θ^⋆_j,t( ω)'s as t ↑ t_e, if λ > 1.To improve the quality of our plots,we consider a situation where λ∈( 0 ,1 ) instead (see Remark <ref>). By Lemmas <ref> and <ref>,we must select parameters such that (<ref>) is satisfied and λ = 2 [ √(κ/η_tem)( τ( t_e ) ) -2 ( K η̃_per - η_per/ K η̃_tem - η_tem) ]/( K η̃_tem - η_tem) Φ̇( t_e )is a positive non-integer.We can keep most of our choices in (<ref>) the same and only make a few revisions:[η̃_tem = 0.5, η_tem = 0.2,η̃_per = 0.8,; η_per = 0.025, ν^2 = 3,κ = 1 . ]As in Section <ref>, we do not seek to replicate a particular historical situation.We immediately get (<ref>), as its LHS is 4.3302.Using Remark <ref> and (<ref>), we can show that t_e = 0.2691 andλ = 0.5939 . Again, we have two uncertain agents and one certain agent.We retain the { U1, U2, C1}- labeling system from Section <ref>.The inventories, trading rates, and execution price are plotted in Figures <ref> - <ref>.To aid our illustration,we truncate the time domainsin Figures <ref> - <ref> to[ 0, 0.75 ( t_e - 10^-6) ] and[ 0, t_e - 10^-6]for the left and right plots, respectively.The qualities that we expect based upon Theorem <ref>,and Remark <ref> are all present.We offered some applicablecomments in Section <ref>,so we only add a few new observations here.* All agents' inventories approach a finite limit as t ↑ t_e (see Theorem <ref> and Figure <ref>).* The execution price and the uncertain agents' trading rates explode as t ↑ t_e (seeTheorem <ref>,Remark <ref> andFigures <ref> - <ref>). * The uncertain agents' tradingrates synchronize as t ↑ t_e(seeTheorem <ref>,Remark <ref>, and Figure <ref>). * That an explosion in S^exc( ω) will occur as well as its direction becomes increasingly obvious as t ↑ t_e; however, it is not necessarily clear at first(see Theorem <ref>,Remark <ref>,and Figure <ref>).§.§ Example 3: A mini-flash crash with high trading volume Our mini-flash crashes can also be accompanied by hightrading volumes (see Theorem <ref>).We illustrate this in Section <ref> by simulating a case in whichA has a root on [ 0 , T ];t_e > 0;the zero of A at t_e is of multiplicity 1;λ∉ℤ;and λ < 0. The behaviors ofS^exc( ω), theX^θ_j^⋆_j( ω)'s, and the θ^⋆_j( ω)'sare then described by Corollary <ref> andTheorem <ref>.We especially wish to emphasize the stochastic explosion direction and do this in two ways.First, we choose the same deterministicparameters tocreate Figures <ref> - <ref>.The difference is that one realization of W̃_· is used in Figures <ref> - <ref>,while another is used in Figures <ref> - <ref>. We denote the corresponding ω's by ω_up and ω_dn,since thereare spikes and crashes in the former and latter plots, respectively. Second, Figures <ref> - <ref> themselves suggest that the explosion direction is random.This is particularly true inFigures <ref> - <ref>,since we initially notice thatthe price rapidly rises asthe uncertain agents' buying rates synchronize.Only moments before the mini-flash crash do we see the price collapsing and the uncertain agents' aggressively selling together. Now, we need to choose parameters such that (<ref>) is satisfied and λ = 2 [ √(κ/η_tem)( τ( t_e ) ) -2 ( K η̃_per - η_per/ K η̃_tem - η_tem) ]/( K η̃_tem - η_tem) Φ̇( t_e )is a negative non-integer due to Lemmas <ref> and <ref>.Compared to Section <ref>, we setη̃_per = 0.5, η_per = 0.5and keep every other parameter the same.As in Sections <ref> - <ref>,we do not have in mind a special historical example here.Since we have only changed η̃_per and η_per, thevalues of( K η̃_tem - η_tem) Φ( 0 ) and t_edo not differ from Section <ref>; however, λ is now negative:( K η̃_tem - η_tem) Φ( 0 ) = 4.3302,t_e = 0.2691,andλ = -0.4531. The numbers of uncertain and certain agentsare still two and one, respectively. We also retain the { U1, U2, C1 }-labeling system from Sections <ref> - <ref>.Figures <ref> and <ref> depict the agents' inventories.We plot the agents' trading rates in Figures <ref> and<ref>.The execution price appears in Figures <ref> and <ref>.To help with our visualization, the time domains in theleft plots in Figures <ref> - <ref> andFigures <ref> - <ref> are truncated to[ 0 , 0.94( t_e - 10^-6) ] and[ 0 , 0.75( t_e - 10^-6) ],respectively. Our observations regardingFigures <ref> - <ref> are in agreementwith Corollary <ref> and Theorem <ref>.We have already made note of many important aspects in Sections <ref> - <ref> and only remark upon the new details. * The execution price, as well as the uncertain agents' inventories and trading rates, all explode inthe same direction as t ↑ t_e(see Theorem <ref> and Figures <ref> - <ref>).* The explosions take place at the deterministic time t_e (see Theorem <ref> and Figures <ref> - <ref>). * The explosion direction depends on ω∈Ω̃(see Theorem <ref> and Figures <ref> - <ref>). * Theexplosion direction cannot be known with complete certainty before t_e(see Theorem <ref> and Figures <ref> - <ref>).* The explosion rates in the price and uncertain agents' trading rates in Section <ref> are slower than in Section <ref> (seeFigures <ref> - <ref>,Figures <ref> - <ref>,and Figures <ref> - <ref>).We did not explicitly state this previously; however, this is to be expectedsince trading rates are integrable in Section <ref> but not in Section <ref>. § CONCLUSION In this paper we show how mini-flash crashes might occur when agents learn and make their decisionsbased upon misspecified models. We give a necessary and sufficient condition for a mini-flash crash to occur and observe that if the agents * are too uncertain about their prior information* sufficiently underestimate the aggregate temporary impact* have long trading periods* have low risk aversion then mini flash crashes occur.Our numerical section has three main examples illustrating our three main results. In the first example, we see that not all human errors directlycause mini-flash crashes illustrating Proposition <ref>. Despite the regularity of the human errorson a market-wide basis,individual securities may rarely experience such an event.Similarly, traders' models and strategies do roughly achieve their intended goals much of the time,as it is observed in the real markets.The two other examples are about the nature of the mini-flash crashes. If a mini-flash crash does occur,it almost surely does so because of“endogenous triggering effects.” As predicted by Theorems <ref> and <ref>), some of our agents also “converge on the same strategy” during mini-flash crashes: In certain cases, the agents driving these events all buy or sell together with the same (exploding) growth rate. The two theorems and the examples illustrating them are to demonstrate that mini-flash crashes can be accompanied by both high and low trading volumes.§ SECTION <REF> PROOFS §.§ Proof of Lemma <ref>Step step 1 orig:Denote the usual P_j-augmentation of {ℱ_j,t^unf}_0 ≤ t ≤ T by {ℱ̃_j,t^unf}_0 ≤ t ≤ T.Let 𝒜̃_j be the space of ℱ̃_j,t^unf-progressively measurable processes θ_j such that (<ref>) and (<ref>) hold.We, again, define the process X^θ_j_j by (<ref>) forany strategy θ_j∈𝒜̃_̃j̃.Agent j's auxiliary problem is to maximize E^P_j[ - ∫_0^T θ_j,tS_j,θ_j,t^excdt-κ_j/2∫_0^T( X^θ_j_j,t)^2 dt ] over θ_j∈𝒜̃_j.It is not difficult to show thatE^P_j[ β_j | ℱ̃_j,t^unf] = E^P_j[ β_j| ℱ_j,t^unf]P_j-a.s. Step step 4 orig: By (<ref>) and (<ref>),- ∫_0^T θ_j,tS_j,θ_j,t^excdt= - ∫_0^T θ_j,tS_j,t^unfdt - ∫_0^T θ_j,t[ η_j,per(X_j,t^θ_j - x_j) + 1/2η_j,temθ_j,t] dtfor θ_j ∈𝒜̃_j. Section 7.4 of <cit.> and (<ref>) imply thatthe process {W_j,t}_0 ≤ t ≤ T with W_j,t ≜ S_j,t^unf - S_j,0 - ∫_0^t E^P_j[ β_j | ℱ_j,s^unf] dsis an ℱ_j,t^unf-Wiener process under P_jand S_j,t^unf = S_j,0+∫_0^t E^P_j[ β_j | ℱ_j,s^unf] ds +W_j,t .After integrating by parts and recalling (<ref>) and (<ref>), we get E^P_j[ - ∫_0^T θ_j,tS_j,t^unfdt]=E^P_j[-X_j,T^θ_jS_j,T^unf + ∫_0^T X_j,t^θ_jE^P_j[ β_j | ℱ_j,t^unf] dt] + x_j S_j,0.We also have E^P_j[ - ∫_0^T θ_j,t[ η_j,per(X_j,t^θ_j - x_j) + 1/2η_j,temθ_j,t] dt ]= E^P_j[- 1/2η_j,per( X_j,T^θ_j-x_j)^2- 1/2η_j,tem∫_0^T θ_j,t^2 dt]. Now X_j,T^θ_j =0 P_j-a.s. by the definition of 𝒜̃_j in Step <ref>.Since x_j, S_j,0 and S_j,T^unf do not depend on Agent j's choice of θ_j∈𝒜̃_j, (<ref>) implies that θ_j^⋆ maximizes (<ref>) over θ_j ∈𝒜̃_j if and only if it maximizes E^P_j[ ∫_0^T X_j,t^θ_jE^P_j[ β_j | ℱ̃_j,t^unf] dt- 1/2η_j,tem∫_0^T θ_j,t^2 dt -κ_j/2∫_0^T( X^θ_j_j,t)^2 dt] . Clearly, θ_j^⋆ maximizes (<ref>) over θ_j ∈𝒜̃_j if and only if it minimizesE^P_j[ 1/2∫_0^T( X_j,t^θ_j -E^P_j[ β_j | ℱ̃_j,t^unf]/κ_j)^2 dt + η_j,tem/2 κ_j∫_0^Tθ_j,t^2 dt ] . Step step 5 orig:Recall the definition of τ_j(·) from (<ref>) and let[ K_j ( t , s ) ≜√(κ_j/η_j,tem)( sinh( τ_j( s ) )/cosh(τ_j( t ) ) - 1) ,0 ≤ t ≤ s < T;; β̂_j,t≜ E^P_j[ 1/κ_j( 1 - 1/cosh( τ_j(t ) )) . ; ; ·. ∫_t^T E^P_j[ β_j | ℱ̃_j,s^unf] K_j ( t , s ) ds|ℱ̃_j,t^unf] ,t ∈[ 0 , T ). ]We see from Theorem 3.2 of <cit.> that (<ref>) has a unique solution θ_j^⋆∈𝒜̃_j.Moreover, the corresponding optimal inventory process X^θ^⋆_j_j satisfies the linear ODE d X^θ^⋆_j_j,t = √(κ_j/η_j,tem)( τ_j( t ) ) ( β̂_j,t - X^θ^⋆_j_j,t) dt X^θ^⋆_j_j,0 = x_jd P_j⊗ dt-a.s. on Ω_j×[ 0 , T ).Using Fubini's theorem and (<ref>), we get that E^P_j[∫_t^T E^P_j[ β_j | ℱ̃_j,s^unf] K_j ( t , s ) ds|ℱ̃_j,t^unf]=E^P_j[ β_j| ℱ_j,t^unf] ,P_j-a.s.The tanh half-angle formula together with (<ref>) and (<ref>) imply that (<ref>) can be re-written as θ^⋆_j,t = - √(κ_j/η_j,tem)( τ_j( t ) ) X^θ^⋆_j_j,t +tanh( τ_j(t ) /2 ) [ μ_j+ ν_j^2 ( S_j,t^unf - S_j,0) ]/√(η_j,temκ_j)( 1 + ν_j^2 t ),t ∈( 0, T ) X^θ^⋆_j_j,0 = x_j . Step step 7 orig: We know that θ^⋆_j satisfies (<ref>) and (<ref>), as all strategies in 𝒜̃_j have these properties.Now W_j,·( ω) is continuous on [ 0 , T ]for P_j-almost every ω∈Ω_j. When such an ω is chosen,(<ref>) becomes (<ref>).The latter is a first order linear ODE with continuous coefficients, soθ^⋆_j, ·( ω) is continuous on [ 0 , T ) (e.g., by Chapter 1.2 of <cit.>). Since our terminal inventory constraint is deterministic, we observe that lim_t ↑ Tθ^⋆_j,t( ω)exists and is finite from (28) and (29) in the proof of Theorem 3.2 in <cit.>,as well as (<ref>) in Step <ref>.In particular, we can view the paths of θ^⋆_j on [ 0 , T ] asP_j-a.s. continuous.[Alternatively, we could give an argument using singular point theory as in Section <ref>.]We conclude by noting that θ_j^⋆ is also ℱ_j,t^unf-adapted by (28) and (29) in the proof of Theorem 3.2 in <cit.>,(<ref>),and (<ref>) in Step <ref>. § SECTION <REF> PROOFSWe frequently reference various easy properties of the functions in Definition <ref>.We collect these below for convenience. We will leave the proof to the reader.Fix j ∈{ 1, …, K }. We have the following: * Φ_j is a strictly decreasing nonnegative function on [ 0 , T ] with Φ_j( T ) = 0. *The entries of A are analytic on [ 0 , T ] and A ( T ) = I_K. *If A has a root on [ 0, T ], we can find the smallest one which we denote by t_e.In this case, t_e < T and the zero of A at t_e is of finite multiplicity.*The entries of B are analytic on [ 0 , T ) but lim_t ↑ T B_jj( t ) = -∞.* C ( ·, ω)'s entries are continuous on [ 0, T ]. §.§ Proof of Lemma <ref> Let j ∈{ 1, …, K }.At each time t,Agent j observes the correct value of S_t^exc( ω), interprets this value as the realized value of S_j,θ_j^⋆,t^exc( ω), and computes S_j,t^unf( ω).[By abuse of notation, we evaluate S_j,θ_j^⋆,t^exc and S_j,t^unf are evaluated at ω; however, Agent j would evaluate these quantities at some ω_j ∈Ω_j. We adopt similar conventions in the sequel without further comment.]By (<ref>), it follows thatS_t^exc(ω)= S_j,θ_j^⋆,t^exc(ω) = S_j,t^unf(ω) + η_j,per(X_j,t^θ_j^⋆(ω)- x_j) + 1/2η_j,temθ_j,t^⋆(ω).After substituting (<ref>) into (<ref>), we haveS_j,t^unf(ω)- S_j,0 =( S_0 - S_j,0)+β̃t + ∑_i ≤ K i ≠ jη̃_i ,per( X^θ_i^⋆_i,t(ω) -x_i) + ∑_ i > K η̃_i ,per( X^θ_i^⋆_i,t-x_i)+ 1/2∑_i ≤ K i ≠ jη̃_i,temθ_i,t^⋆(ω)+ 1/2∑_i > Kη̃_i,temθ_i,t^⋆ + ( η̃_j,per- η_j,per)(X_j,t^θ_j^⋆(ω) - x_j) + 1/2( η̃_j,tem - η_j,tem)θ_j,t^⋆(ω) + W̃_t(ω). The quantity on the LHS of (<ref>) plays a role in determining Agent j's strategy (see Lemma <ref>). Substituting (<ref>) into (<ref>) and applying the half-angle formula for tanh(·),we getA_jj( t ) θ_j,t^⋆(ω)- ∑_ i ≤ K i ≠ j A_ji( t ) θ_i,t^⋆(ω)= B_jj( t )X^θ_j^⋆_j,t(ω) + ∑_ i ≤ K i ≠ j B_ji( t )X^θ_i^⋆_i,t(ω) + C_j( t , ω) .It follows that the uncertain agents' strategies are characterized by the ODE systemA (t )θ_t^u,⋆( ω)=B ( t ) X^u,θ^⋆_t(ω)+ C ( t, ω) X^u,θ^⋆_0(ω)= x^u. Corollary <ref>, Lemma <ref> and a standard existence and uniqueness theorem (see Sections 1.1 and 3.1 of <cit.>) finish the argument. §.§ Proof of Lemma <ref>As t ↑ t_e, [ { A (t )Ẋ_t^u( ω) =B ( t ) X_t^u(ω) };; {[ A (t ) ] Ẋ_t^u( ω) =[ adjA (t ) ] B ( t ) X_t^u(ω) } . ]Here, adj denotes the usual adjugate operator. We can find a non-negative integer m such that the multiplicity of the zero of A at t_e is( m + 1 ) by Lemma <ref>.Hence, there is a unique non-vanishing analytic function f such that A ( t ) = ( t - t_e )^m+1 f ( t )on a small neighborhood of t_e.Note that f is non-vanishing, as the zeroes of A are isolated and A ( T ) = 1 (see Lemma <ref>).We then define the analytic (see Lemma <ref>) map D byD ( t ) ≜[ adjA (t ) ] B ( t ) / f ( t )and arrive at (<ref>). Since A ( ·) has a root at t_e,the rank of A ( t_e ) is no more than K - 1.We conclude by observing that adj A ( t_e ) has rank 1 when A ( t_e )has rank K - 1; otherwise, adj A ( t_e ) must be the zero matrix.The comments about the rank of D ( t_e ) immediately follow.§.§ Proof of Proposition <ref> D ( t_e ) ≠ 0 since λ≠ 0.Then (<ref>) has a singular point of the first kind at t_e (see our discussion above).λ∉ℤ by hypothesis, so Theorem 6.5 of <cit.> implies that a fundamental solution of (<ref>)on [ t_e - ρ , t_e ) for some ρ > 0 is given byP ( t ) | t - t_e |^D ( t_e ) .In (<ref>),P( ·) is an analytic M_K ( ℝ)-valued function with P ( t_e ) = I_K.Moreover, P ( t ) is invertible for all t ∈[ t_e - ρ , t_e ) and[Any fundamental solution of (<ref>) is invertible everywhere, as are matrix exponentials.]( P ( t ) | t - t_e |^R )^-1= | t - t_e |^- R[ P ( t ) ]^-1 .The solution of (<ref>) satisfies(t - t_e ) θ_t^u,⋆( ω) =D ( t ) X^u,θ^⋆_t(ω)+ adj[ A ( s ) ] C ( s , ω)/ f ( s ).near t_e (argue as in Lemma <ref>).Since P ( t ) | t - t_e |^D ( t_e )ρ^-D ( t_e )[ P ( t_e - ρ)]^-1is also a fundamental solution of (<ref>) on [ t_e - ρ , t_e )[See Theorem 2.5 ofCoddington & Carlson (<cit.>).] and equals I_K at t_e - ρ,we can apply variation of parameters[See Theorem 2.8 of Coddington & Carlson (<cit.>).] to obtainX^u,θ^⋆_t( ω) = P ( t ) | t - t_e |^D ( t_e )[ ρ^-D ( t_e )[ P ( t_e - ρ)]^-1]·[ X^θ^⋆_t_e - ρ( ω) +∫_t_e - ρ^t ( P ( t_e - ρ)ρ^D ( t_e )| s - t_e |^-D ( t_e ) [ P ( s ) ]^-1) ( adj[ A ( s ) ] C ( s , ω)/( s - t_e ) f ( s ))ds] . We can find an eigenbasis {v_1, …, v_K} for D ( t_e ) such that v_K corresponds to λ.We then definethe continuousreal-valued functions { F_1 ( · , ω), … , F_K ( · , ω) } on [ t_e - ρ , t_e ] and the constants{ y_1 ( ω) , … , y_K ( ω) } as certain eigenbasis coordinates:∑_j = 1^KF_j ( s, ω) v_j ≜[ P ( s ) ]^-1adj[ A ( s ) ] C ( s , ω)/ f ( s ) ∑_j = 1^Ky_j ( ω) v_j ≜ρ^- D ( t_e )[ P ( t_e - ρ)]^-1 X^θ^⋆_t_e - ρ( ω) .Taken with (<ref>), these definitions immediately give (<ref>) after recalling that for any matrix Q ∈ M_K ( ℝ) witheigenvalue γ and corresponding eigenvector v,we have| t - t_e |^Q v = | t - t_e |^γ v. §.§ Proof of Proposition <ref> We know that S^exc( ω), the X^θ_j^⋆_j( ω)'s andthe θ^⋆_j( ω)'s are all uniquely defined and continuous on [ 0, T )(see Lemma <ref>).Corollary <ref> implies thatX^θ_j^⋆_j( ω) andθ^⋆_j( ω) are continuous at T for j > K (the certain agents). It also gives uslim_t ↑ T X^θ_j^⋆_j,t( ω) = 0for j > K.It remains to show that lim_t ↑ T X^u,θ^⋆_t( ω) = 0andlim_t ↑ Tθ^u,⋆_t( ω) ∈ℝ^K.As discussed above, one difficulty is that the diagonal entries of B in (<ref>) explode at T (see Lemma <ref>); however, the approach for resolving this issueis similar to that used to analyze solution behavior near t_e. First, we show that (<ref>) (after replacing t_e with T) has a singular point of the first kind at T.Now sinh( τ_j( ·) ) has a zero of multiplicity 1 at T since d sinh( τ_j( t ) ) /dt|_t = T = - √(κ_j/η_j,tem)cosh( τ_j( t ) )|_t = T = - √(κ_j/η_j,tem).Hence, there is a unique non-vanishing analytic function g_j such that sinh( τ_j( t ) ) = (t - T ) g_j ( t ) and g_j ( T ) = - √(κ_j/η_j,tem)on a small neighborhood of T. Near T, it follows that the entries of ( t - T ) B ( t) are given by( t - T ) B_ik( t) = {[ ( t - T ) ( η̃_i,per- η_i,per)Φ_i ( t ); - √(κ_i/η_i,tem)( cosh( τ_i( t ) )/ g_i ( t ) ) ifi = k;; ( t - T ) η̃_k,perΦ_i ( t )ifi ≠k; ].(see Definition <ref>).On this region, the solution of (<ref>) satisfies (t - T ) Ẋ_t^u( ω)=A^-1( t ) ( t - T ) B ( t ) X_t^u(ω) .By (<ref>) and Lemma <ref>,(<ref>) has a singular point of the first kind at T. Second, we find a fundamental solution of (<ref>) near T.We know thatA^-1( T ) = ( t - T ) B ( t)|_t = T = I_Kby (<ref>), (<ref>), and Lemma <ref>.Theorem 6.5 of <cit.> implies that a fundamental solution of (<ref>)on [ T - δ , T ) for some δ > 0 is given byQ ( t ) | t - T |^I_K= Q ( t ) | t - T |.In (<ref>),Q is an analytic M_K ( ℝ)-valued function with Q ( T ) = I_K.Also, Q ( t ) is invertible for all t ∈[ T- δ , T ).[Any fundamental solution of (<ref>) is invertible everywhere.]Finally, we use our fundamental solution to solve (<ref>) and conclude the proof.Notice that tanh( τ_j ( ·) ) also has a zero of multiplicity 1 at T sinced tanh( τ_j( t ) ) /dt|_t = T =- 1/2√(κ_j/η_j,tem)sech^2 ( τ_j( t ) /2 )|_t = T=- 1/2√(κ_j/η_j,tem) .There is a unique non-vanishing analytic function h_j such that tanh( τ_j( t ) / 2) = (t - T ) h_j ( t )on a neighborhood of T.In particular, the entries of C (t , ω) / ( t - T ) near T are given byC_i( t , ω)/( t - T )= (h_i ( t ) ν_i^2/√(η_i,temκ_i)( 1 + ν_i^2 t ))[ μ_i/ν_i^2 +( S_0 - S_i,0)+β̃t - ∑_ k ≤ K k ≠ i η̃_k,per x_k.- x_i( η̃_i,per- η_i,per)+ ∑_ k > K η̃_k ,per( X^θ_k^⋆_k,t-x_k) . + 1/2∑_k > Kη̃_k,temθ_k,t^⋆+ W̃_t ( ω) ] .Since Q ( t )| t - T | δ^-1 Q^-1( T - δ)is also a fundamental solution of (<ref>) on [ T - δ, T )[See Theorem 2.5 ofCoddington & Carlson (<cit.>).] and equals I_K at T- δ,we can apply variation of parameters[See Theorem 2.8 of Coddington & Carlson (<cit.>).] to obtainX^u,θ^⋆_t( ω)= Q ( t ) | t - T | δ^-1Q^-1( T - δ) ·[ X^θ^⋆_T - δ( ω) +∫_T - δ^t ( Q ( T - δ)δ| s - T |^-1Q^-1( s ))A^-1( s ) C ( s , ω)ds] .By (<ref>), (<ref>), and Corollary <ref>, we get (<ref>). § SECTION <REF> PROOFS §.§ Proof of Lemma <ref>First observe that(<ref>) is equivalent to( K η̃_tem - η_tem) Φ( 0 ) > 2,by Definitions <ref> and <ref>. By Definitions <ref> and <ref>,we see that A is now given by A_ik( t)≜{[ 1 - 1/2( η̃_tem - η_tem) Φ( t ) ifi = k;-1/2η̃_temΦ( t )ifi ≠k; ]. . A short calculation shows that A (t )=[ 1 +1/2η_temΦ( t )]^K-1[1 - 1/2( Kη̃_tem - η_tem) Φ( t )] .The first term in (<ref>) is always at least 1. The second term is non-zero at 0 but does have a root on ( 0, T ] if and only if(<ref>) holds.[In fact, t_e is the unique root of A in this case.] Both of these observations come from Lemma <ref>.Now, (<ref>)implies that K η̃_tem > η_tem.Since t_e is a zero of A, we must have that 1 - 1/2( Kη̃_tem - η_tem) Φ( t_e ) = 0.Hence, by Lemma <ref>, d [A ( t ) ] /dt|_t = t_e =- 1/2( Kη̃_tem - η_tem)[ 1 +1/2η_temΦ( t )]^K-1Φ̇( t )|_t = t_e > 0.§.§ Proof of Lemma <ref>By (<ref>), (<ref>), and Lemma <ref>, f ( t_e )= d [A ( t ) ] /dt|_t = t_e= - 1/2( Kη̃_tem - η_tem)[ 1 +1/2η_temΦ( t_e )]^K-1Φ̇( t_e ) .A short calculation shows that adj A( t ) is given by[ adj A ( t ) ]_ik= ( 1 +1/2η_temΦ( t ) )^K-2{[ 1 - 1/2[ ( K - 1 ) η̃_tem - η_tem] Φ( t ) ifi = k; 1/2η̃_temΦ( t )ifi ≠k; ]. . It follows that [ ( adj A ( t ) ] B ( t ) ]_ik = η̃_perΦ( t ) ( 1 +1/2η_temΦ( t ) )^K-1 + ( 1 +1/2η_temΦ( t ) )^K-2( η_perΦ( t )+ √(κ/η_tem)( τ( t ) )) ·{[ 1/2[ ( K - 1 ) η̃_tem - η_tem] Φ( t )- 1ifi = k; ;- 1/2η̃_temΦ( t ) ifi ≠k;]. .One can then check that the only potentially non-zero eigenvalue of D (t_e ) =[ adj A ( t_e ) ] B ( t_e ) /f ( t_e )is given by λ = - 2 [ ( K η̃_per - η_per)Φ( t_e ) -√(κ/η_tem)( τ( t_e ) ) ]/( Kη̃_tem - η_tem) Φ̇( t_e ) with corresponding eigenvector v_K as above.We get (<ref>) from (<ref>) after applying (<ref>). Recall that Φ( t_e ) > 0 and Φ̇( t_e ) < 0 by Lemma <ref>.Since t_e, Φ, and τ do not depend on η̃_per or η_per,we can ensure that λ∉ℤ by perturbing the latter parameters.D ( t_e ) is then diagonalizable as observed in Proposition <ref>,and v_1, … , v_K-1 can be computed using (<ref>). §.§ Proof of Theorems <ref> and <ref> Since our uncertain agents are semi-symmetric,C_i( t , ω)=Φ( t )W̃_t ( ω)+Φ( t ) [ β̃t+ ∑_ k > K η̃_k ,per( X^θ_k^⋆_k,t-x_k)+ 1/2∑_k > Kη̃_k,temθ_k,t^⋆]+ Φ( t ) [ μ_i/ν^2 +( S_0 - S_i,0) - ∑_ k ≤ K k ≠ i η̃_per x_k- x_i( η̃_per- η_per)]for t ≤ t_e by Definition <ref>.For convenience, we introduce the following deterministic function[ The function c is deterministic by Corollary <ref>.] c and the constants c_1, …, c_K:c ( t )≜[ β̃t+ ∑_ k > K η̃_k ,per( X^θ_k^⋆_k,t-x_k)+ 1/2∑_k > Kη̃_k,temθ_k,t^⋆]∑_i = 1^K c_i v_i ≜[ [ μ_1/ν^2 +( S_0 - S_1,0) - ∑_ k ≤ K k ≠ 1 η̃_per x_k- x_1( η̃_per- η_per);⋮; μ_K/ν^2 +( S_0 - S_K,0) - ∑_ k ≤ K k ≠ K η̃_per x_k- x_K( η̃_per- η_per) ]] .Using (<ref>), we get that C ( t , ω)= W̃_t ( ω) Φ( t ) v_K+ c ( t ) Φ( t ) v_K+ Φ( t ) ∑_i = 1^K c_i v_i .By (<ref>), { v_1 , …, v_K } is an eigenbasis for adj[ A ( t ) ].Moreover,( 1 +1/2η_temΦ( t ) )^K-2[1 - 1/2( Kη̃_tem - η_tem) Φ( t )]is the eigenvalue corresponding to each ofv_1 , …, v_K-1, while ( 1 +1/2η_temΦ( t ) )^K-1corresponds to v_K. By (<ref>), it follows that ∑_j = 1^KF_j ( t, ω) v_j =[ P ( t ) ]^-1adj[ A ( t ) ] C ( t , ω)/ f ( t )=W̃_t ( ω)(Φ( t ) ( 1 +1/2η_temΦ( t ) )^K-1/ f ( t ))[ P ( t ) ]^-1v_K+(Φ( t ) ( 1 +1/2η_temΦ( t ) )^K-1( c ( t ) + c_K ) / f ( t ))[ P ( t ) ]^-1v_K+(Φ( t ) ( 1 +1/2η_temΦ( t ) )^K-2[1 - 1/2( Kη̃_tem - η_tem) Φ( t )] / f ( t )) ·∑_i = 1^K-1 c_i [ P ( t ) ]^-1v_i. It follows that we can findanalytic deterministic functions F_j,1 and F_j,2 such that F_j( t , ω) ≜W̃_t ( ω)F_j,1( t ) + F_j,2( t )for each j ∈{ 1, …, K }.[ Note that c is continuously differentiable on [ 0 , t_e ] by Corollary <ref>.]Since P ( t_e ) = I_K (see Proposition <ref>), (<ref>) and Remark <ref> further imply that F_j,1( t_e) = F_j,2( t_e) =⋯ = F_K-1,1( t_e ) = F_K-1,2( t_e ) = 0and F_K,1( t_e) =-Φ^2 ( t_e ) /Φ̇( t_e )> 0and F_K,2( t_e) =-Φ^2 ( t_e ) /Φ̇( t_e ) (c ( t_e ) + c_K) .While F_K,1( t_e) > 0, determining the sign of F_K,2( t_e)is difficult, in general, as it depends upon the sign of c ( t_e ) + c_K(see (<ref>)). We see from (<ref>) and (<ref>)that the expression F_j ( s, ω)/| s - t_e |is bounded near t_e for each j < K and almost every ω∈Ω̃.In particular, the coordinates of both ∑_j = 1^K-1( y_j ( ω) - ∫_t_e - ρ^t F_j ( s, ω)/| s - t_e | ds) P ( t )v_jand its time derivative are bounded near t_e for such ω as well. Since P ( t_e ) = I_K, the v_K-coordinate of P( t ) v_K tends to 1 t ↑ t_e.For j < K, the v_j-coordinate of P( t ) v_K tends to 0 as t ↑ t_e.In each situation, we can also obtain Lipschitz bounds on the convergence.Due to (<ref>)and (<ref>),potential explosions in the coordinates of X^u,θ^⋆_t( ω)are characterized by lim_t ↑ t_e[ | t - t_e |^λ( y_K ( ω)-∫_t_e - ρ^tW̃_s (ω) F_K,1( s) + F_K,2( s) /| s - t_e |^1+ λds)] .Specifically, {| (<ref>)| < +∞} {lim_t ↑ t_eX^u,θ^⋆_t( ω)exists in ℝ^K} {(<ref>)= +∞} {lim_t ↑ t_e X^u,θ^⋆_t( ω)= [ + ∞, …, + ∞]^⊤} {(<ref>)= -∞} {lim_t ↑ t_e X^u,θ^⋆_t( ω)= [ - ∞, …, - ∞]^⊤} .To finish the proof, we separately consider the λ < 0and λ > 0 cases. λ < 0 Case.Assume that λ < 0.It follows that lim_t ↑ t_e∫_ t_e - ρ^t | W̃_s (ω) F_K,1( s) | /| s - t_e |^1+λds< ∞andlim_t ↑ t_e∫_ t_e - ρ^t |F_K,2( s) | /| s - t_e |^1+λds< ∞ .Clearly, lim_t ↑ t_e| t - t_e |^λ= + ∞ ,meaning that { y_K ( ω)- lim_t ↑ t_e∫_ t_e - ρ^tF_K,2( s )/| s - t_e |^1+λds>lim_t ↑ t_e∫_ t_e - ρ^tW̃_s ( ω) F_K,1( s)/| s - t_e |^1+λds}{lim_t ↑ t_e X^u, θ^⋆_t( ω)= [ + ∞, …, + ∞]^⊤}and{ y_K ( ω)- lim_t ↑ t_e∫_ t_e - ρ^tF_K,2( s )/| s - t_e |^1+λds<lim_t ↑ t_e∫_ t_e - ρ^tW̃_s ( ω) F_K,1( s)/| s - t_e |^1+λds}{lim_t ↑ t_e X^u, θ^⋆_t( ω)= [ - ∞, …, - ∞]^⊤} Arguing as in our discussion of (<ref>), we see that the hypotheses in(<ref>) and (<ref>) also imply that {lim_t ↑ t_eθ^u,⋆_t( ω)= [ + ∞, …, + ∞]^⊤}and{lim_t ↑ t_eθ^u,⋆_t( ω)= [ - ∞, …, - ∞]^⊤} ,respectively.[ In particular, the coordinates of θ^u,⋆_t( ω) will asymptotically explode at the rate | t - t_e |^-λ - 1.]Conditional on ℱ̃_t_e - ρ, the RHS of the inequality in(<ref>) (and <ref>)is deterministic. Since F_K,1( t_e ) > 0(see (<ref>)), we finish our proof of Theorem <ref>. λ > 0 Case.Assume that λ > 0.We can find a constant R_0 ( ω) such that | y_K ( ω)-∫_t_e - ρ^tW̃_s (ω) F_K,1( s) + F_K,2( s) /| s - t_e |^1+ λds| ≤R_0 ( ω) /| t - t_e |^λ .Hence, (<ref>) is bounded as t ↑ t_e and lim_t ↑ t_e X^u,θ^⋆_t( ω)exists in ℝ^K by our previous comments. By our discussion surrounding (<ref>),we see that explosions in the coordinates of θ^u,⋆_t( ω)are characterized bylim_t ↑ t_e [ - λ| t - t_e |^λ - 1( y_K ( ω)-∫_t_e - ρ^tW̃_s (ω) F_K,1( s) + F_K,2( s) /| s - t_e |^1+ λds) . . -(W̃_t (ω) F_K,1( t ) + F_K,2( t) /| t - t_e | ) ] .More precisely, {(<ref>)= +∞} {lim_t ↑ t_eθ^u,⋆_t( ω)= [ + ∞, …, + ∞]^⊤} {(<ref>)= -∞} {lim_t ↑ t_eθ^u,⋆_t( ω)= [ - ∞, …, - ∞]^⊤} .Suggestively, we first rewrite the expression in (<ref>) asF_K,2( t)(λ| t - t_e |^λ - 1∫_t_e - ρ^t 1 /| s - t_e |^1+ λds - 1 /| t - t_e | ) +λ| t - t_e |^λ - 1∫_t_e - ρ^t F_K,2( s)- F_K,2( t)/| s - t_e |^1+ λds - λ| t - t_e |^λ - 1 y_K ( ω)+ W̃_t ( ω) F_K,1( t)(λ| t - t_e |^λ - 1∫_t_e - ρ^t 1 /| s - t_e |^1+ λds - 1 /| t - t_e | ) +λ| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)[F_K,1( s)- F_K,1( t)] /| s - t_e |^1+ λds+ λ| t - t_e |^λ - 1F_K,1( t)∫_t_e - ρ^t W̃_s ( ω)- W̃_t ( ω) /| s - t_e |^1+ λdsLet R_1 and R_2 be the deterministic Lipschitz coefficients for F_K,1 and F_K,2.The first two lines of (<ref>) are deterministic,and we can obtain the following bounds:| F_K,2( t)( λ| t - t_e |^λ - 1∫_t_e - ρ^t 1 /| s - t_e |^1+ λds - 1 /| t - t_e | ) | ≤| F_K,2( t ) | | t- t_e |^λ - 1/ρ^λ | λ| t - t_e |^λ - 1∫_t_e - ρ^t F_K,2( s)- F_K,2( t)/| s - t_e |^1+ λds | ≤( λ R_2 /1 - λ)( ρ^1 - λ| t- t_e |^λ - 1 - 1 )In (<ref>), the third line isdeterministic conditional on ℱ̃_t_e - ρ.Lines 4 - 6 of (<ref>) are stochastic conditional on ℱ̃_t_e - ρ.Letting R_3 ( ω) be the maximum of| W̃_t ( ω) | on [ t_e - ρ, t_e ],we notice that| W̃_t ( ω) F_K,1( t)(λ| t - t_e |^λ - 1∫_t_e - ρ^t 1 /| s - t_e |^1+ λds - 1 /| t - t_e | )| ≤ F_K,1( t ) | W̃_t ( ω) | | t- t_e |^λ - 1/ρ^λ | λ| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)[F_K,1( s)- F_K,1( t)] /| s - t_e |^1+ λds | ≤( λ R_1 R_3 ( ω)/1 - λ)( ρ^1 - λ| t- t_e |^λ - 1 - 1 ) .When λ > 1, it follows that we see that (<ref>) has thesame behavior as lim_t ↑ t_e[| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)- W̃_t ( ω) /| s - t_e |^1+ λds](all other terms tend to 0 P̃-a.s.).Using integration by parts,| t - t_e |^λ - 1∫_t_e - ρ^t W̃_s ( ω)- W̃_t ( ω) /| s - t_e |^1+ λds ∼𝒩( 0 ,| t - t_e |^2λ - 2∫_t_e - ρ^t(| s - t_e |^-λ/λ - 1/λρ^λ)^2 ds ) .Asymptotically, the variance in (<ref>) grows like | t - t_e |^-1 as t ↑ t_e, completing the proof of Theorem <ref>.siam | http://arxiv.org/abs/1705.09827v2 | {
"authors": [
"Erhan Bayraktar",
"Alexander Munk"
],
"categories": [
"q-fin.TR",
"math.OC",
"math.PR",
"Primary 91G80, Secondary 60H30, 60H10, 34M35"
],
"primary_category": "q-fin.TR",
"published": "20170527150151",
"title": "Mini-Flash Crashes, Model Risk, and Optimal Execution"
} |
ICEx-UFMG, Avda. Presidente Antônio Carlos 6627, Belo Horizonte-MG, BR31270-901 [email protected] 37D30,57R30 We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics is given by (a continuum of parameters in) the Froeschlé family. These types of coupled systems appear as some induced maps in models for the study of Arnold diffusion. Consequently, we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics. The methods are also suitable for studying cocycles over shift spaces, and do not demand any low dimensionality condition on the fiber.Random products of Standard maps Pablo D. Carrasco December 30, 2023 ================================§ INTRODUCTIONLet f be a 𝒞^2 diffeomorphism of a compact Riemannian manifold M preserving a smooth probability measure μ. A central tool to detectchaotic behavior in the system is by means of its Lyapunov exponents. For p∈ M, v∈ T_pM∖{0}, the Lyapunov exponent of v is χ(p,v)=lim sup_n→+logd_pf^n(v)/n.It is a consequence of Oseledet's Multiplicative Ergodic Theorem <cit.> that the above limit exists for μ almost every p∈ M and everyv∈ T_pM∖{0}. The existence of non zero Lyapunov exponents on a set of positive μ measure guarantees abundance of exponentially diverging orbits, either for the future or the past. Of particular importance is the case when all exponents are different from zero. The system (f,μ) is non-uniformly hyperbolic (NUH) if its Lyapunov exponents are non-zero μ-a.e., in other words for μ almost every p∈ M it holds ∀ v∈ T_pM∖{0},lim_n+logd_pf^n(v)/n≠ 0.The concept of non-uniformly hyperbolicity was introduced by Y. Pesin as a generalization of the classical hyperbolic diffeomorphisms, and allow for much more flexibility than its uniform counterpart. However, notwithstanding the large industry dedicated to their study, NUH systems are still not very well understood, mainly because establishing the existence of non zero exponents requires an asymptotic analysis of almost every orbit. This difficulty is reflected in the very limited available pool of examples of truly (i.e. non Anosov) NUH diffeomorphisms.The obstacle appears already for conservative surface maps, where showing non-uniform hyperbolicity is reduced (since the sum of exponents for a conservative map is zero <cit.>) to establishing positivity of a single exponent. Perhaps the most famous example where non-uniform hyperbolicity is unknown is given by Chirikov-Taylor standard family (cf. <cit.>): for r>0, s_r:𝕋^2=^2/(2πℤ)^2→𝕋^2 is the diffeomorphism given bys_r(x,y)=[2 -1;10 ]·[ x; y ]+r[ sin(x);0 ]. Interacting pairs of these maps were introduced to study Arnold diffusion for systems of coupled oscillators <cit.>, albeit the original research was mainly numerical. Each s_r preserves the Lebesgue measure, and it is a major open problem in smooth ergodic theory to show the existence of non-zero Lyapunov exponents. Such behavior is evidenced numerically (cf. <cit.>), but currently NUH of s_r is still unknown for even a single parameter <cit.>.It is know however that the dynamics ofs_r is very complicated. See for example <cit.> and references therein.In <cit.> P. Berger and the author proposed to study a random version of the standard map. We considered a linear hyperbolic automorphism of A:𝕋^2→𝕋^2 and studied the diffeomorphism[Here [r] denotes the integer part of r.]f_r(x,y,z,w)=(A^[2r]·(x,y),s_r(z,w)+P∘ A^[r]·(x,y)) P(x,y)=(x,0).The coupled map f_r can be seen as a random perturbation of the dynamics of s_r: computing its derivative one verifies that the action df_r|{0}×^2 coincides with ds_r|{0}×^2, so one can think f_r as a family of standard maps driven by the random motion determined by the base dynamics. This point of view (studying random versions of maps) is well established, and related to the study of classical fast-slow systems of differential equations. For considerations on the statistical properties of similar maps the reader can check for example <cit.>.In that article we established the following. There exists r_0>0 such that for r≥ r_0 one can find a 𝒞^2 neighborhood 𝒰_r of f_r and C(r)>0 such that g∈𝒰_r is conservative ⇒for Lebesgue-a.e.(p),∀ v∈ T_p^4∖{0}, |χ(p,v)|>C(r). Thus every such g (in particular f_r) is NUH. The constant C(r) goes to infinity as r.The goal of this paper is to refine some of the ideas and techniques introduced in <cit.> and establish positivity of center exponents for a larger class of examples. In the referred article most of the arguments were subordinate to a specific case, as the main motivation of the authors was to study a concrete center behavior (given by the standard family), whereas here we are more interested in developing general results with ample applicability, particularly without the low dimensionality restriction in the fiber. For unfolding broad methods, we fine-tune the central notions of admissible curves and adapted fields introduced in <cit.>, providing a more abstract definition readily suitable for dealing with more general systems. Our theorems will be stated in large generality, hence it seems appropriate to give an example of their consequences.The reader is referred to the third Section where more general versions of this and other examples are discussed thoroughly, and where a comparison between the present methods and others in the literature is also given.Consider an hyperbolic matrix A∈ Sl(2,ℤ), e≥ 2 a natural number and r>0; define g_r:×^2e→×^2e by the formulag_r(z_0,z_1,⋯, z_e)=(A^[2r](z_0),s_r(z_1)+P∘ A^[r](z_0),⋯, s_r(z_e)+P∘ A^[r](z_0))where z_i=(x_i,y_i)∈. Then g_r is a generalized version of the map f_r of <cit.>, having e coupled standard maps acting on the fiber instead of just one. Note in particular that if V={0}×^2e then dg_r|V=ds_r×⋯× ds_r, the product taken e times.As consequence of the results in this article we get the following. There exists r_0>0 such that for r≥ r_0 one can find a 𝒞^2 neighborhood 𝒰_r of g_r such that g∈𝒰_r is conservative ⇒for Lebesgue-a.e.(p),∀ v∈ T_p^2(e+1)∖{0}, |χ(p,v)|>3/5log r.It is worth to point out that other available methods in the literature seem to be unfitted to study this type of map, and the above result is, at least as known to the author, the only formal proof of the existence of non-zero Lyapunov exponents for this higher dimensional version of coupled standard maps. It is also interesting to remark that the map g_r is partially hyperbolic (see the next section for the definition), hence in passing we provide a concrete example of a partially hyperbolic system with interesting higher dimensional center behavior.A particular instance of coupled standard maps is given by thefamily of diffeomorphisms u_r,τ:𝕋^2×𝕋^2→𝕋^2×𝕋^2,u_r,τ(x,y,z,w)=(2x-y+rsin(x)+τsin(x+z),x,2z-w+rsin(z)+τsin(x+z),z)where 0≤τ<1 above is a small parameter. Similar types of systems appeared in the study of Arnold diffusion; in the specific case of u_r,τ above, an equivalent map was studied by B. P. Wood,A. J. Lichtenberg and M. A. Lieberman in <cit.> where they provided numerical evidence of Arnold diffusion between the stochastic layers, for small τ. See also the Froeschlé family example in the third Section. It is apparent that these systems are at least as complex, if not more, than s_r. From the previous theorem we get. Define the family of maps g_r,τ:𝕋^6→𝕋^6 by g_r,τ(x_1,…,x_6):=(A^[2r](x_1,x_2),u_r,τ(x_3,…,x_6)+φ_r(x_1,x_2))φ_r(x_1,x_2):=(P∘ A^[r](x_1,x_2),P∘ A^[r](x_1,x_2)). where A∈ SL(2,ℤ) is hyperbolic. Then there exists r_0>0 such that for r≥ r_0 one can find τ_0(r)>0such that for 0≤τ≤τ_0(r) the map g_r,τ is 𝒞^2 - robustly NUH, meaning that any volume preserving map which is sufficiently 𝒞^2 - close to g_r,τ is also NUH.§ STATEMENT OF THE MAIN RESULT §.§ Products of conservative systems with some hyperbolicity A difficult and not well understood problem in smooth ergodic theory is establishing the existence of non zero exponents for systems that are hyperbolic on a large but not invariant set. One of the main complications is that in visits to the complement of the hyperbolic set, vectors that were previously expanded canbe sent to one of the contracted directions, thus loosing expansion. This complication is particularly present in the conservative setting, since typically the complement of the hyperbolic set has positive measure, and thus is recurrent. The Chirikov-Taylor standard map family (<ref>) falls into this category; for a higher dimensional example one can simply take the product of several standard maps. As we mentioned before, no parameter such that any of these systems is NUH is currently known.In this article we will be considering a random version of these products. In pursuit of generality and to free ourselves from the specific formula of the map, we will enumerate the properties required on each of the factors for our method to work. Nevertheless, we suggest the reader to keep in mind the specific example given in (<ref>).Notation: - Consider V, W⊂^d non-trivial sub-spaces such that ^d=V⊕ W. The cone of size α>0 centered around V in V⊕ W is the set Δ_α:={(v,w)∈ V⊕ W: w<αv}∪{(0,0)}. For such a set, Δ_α denotes its closure in ^d, and CΔ_α=^d∖Δ_α is the complementary cone of Δ_α. We will mostly consider cones centered around ×{0} in ×{0}⊕{0}×; note that in this particular case such a cone is of the form Δ_α:=·{(1,w)∈^2:|w|<α} whereas its complimentary cone is CΔ_α:=·{(1,w)∈^2:|w|≥α}∪·(0,1). We will need to consider also finite decompositions of CΔ(α) into sub-cones. For concreteness, let CΔ^+_ =·{(1,w)∈^2:w≥α}∪·(0,1) CΔ^-_ =·{(1,w)∈^2: w≤ -α}∪·(0,1). Then CΔ_=CΔ^+_∪CΔ^+_ and CΔ^+_∩CΔ^-_=·(0,1). These notations extend naturally to cone fields on . - If T:(V,·_V)→ (W,·_W) is a linear map between normed vector spaces, we denote its operator norm and conorm by T:=sup{T(v)_W:v_V=1} m(T):=inf{T(v)_W:v_V=1} In case that T is invertible, it holds T^-1=1/m(T). - Let N be a compact manifold and π:E→ N a continuous vector bundle equipped with a Riemannian metric ·. If T:E→ E is a bundle map we will write T :=max_p∈ N{T_p:T_p:E_p→ E_t(p)} where t is the induced map by T on N.If T is an automorphism, it holdsT^-1=1/min_p∈ N{m(T_p)}. - A subset of the form A=[a,b]×⊂ or A=×[a,b]⊂ is called a band of base [a,b], and we say that it has length l(A)=b-a. As a mild abuse of language, we also call a band to the union of finitely many sets as before, provided that the bases are in the same coordinate of .To express quantitatively the required conditions we find convenient to work in the context of parametrized families. Consider a family of conservative diffeomorphisms {S_r:→}_r satisfying the following conditions. i) The existence of a continuous cone field Δ_r={Δ_r(y)=Δ__r,r(y)⊂^2}_y∈ on . Here we are identifying for y∈^2, T_y=^2. ii)The existence of bands 𝒞_r,ℬ_r⊂ whose bases are in the same coordinate of . The set 𝒞_r will be referred as the critical region of S_r. To simplify the exposition and with no loss of generality, we will assume that the bases of these bands are in the first coordinate. We denote by β(r) the smallest expansion for vectors in Δ_r at points outside the critical region, and by ζ(r) the smallest contraction at points in ^2 , i.e.β(r) :=inf{m(d_yS_r|Δ_r(y)):y∉𝒞_r}ζ(r) :=inf{m(d_yS_r):y∈}(=1/dS^-1).Hypotheses on S_r: there exist σ∈ℕ,0<R<2π such that for r large the following conditions are verified. S-1 It holds * the length of the critical region converges to zero as r goes to infinity, l(𝒞_r)0. * Associated to CΔ^+_r,CΔ^+_r in the decomposition of CΔ_r there exist bands ℬ_r^+,ℬ_r^-⊂ℬ_r such that ℬ_r=ℬ_r^+∪ℬ_r^- and furthermore min{l(ℬ_r^+),l(ℬ_r^-)}≥ R. * The size of Δ_r is bounded from below as a function of r, inf_r min{_r(p):p∈}>0. * Vectors in Δ_r at points outside the critical region are uniformly expanded β(r)> 1≥ζ(r) * The maximal contraction of S_r is controlled by the expansion in the cone field Δ_r by the following relation β(r)^6ζ(r)^1/σ>1. S-2 * Invariance of Δ_r: if y∉𝒞_r then v∈Δ_r(y)⇒ d_yS_r(v)∈Δ_r(S_r(y)). * Restitution of the expansion direction: it holds * y∈ℬ_r^+, v∈CΔ_r^+(y) then d_yS_r(v)∈Δ_r(S_r(y)). * y∈ℬ_r^-, v∈CΔ_r^-(y) then d_yS_r(v)∈Δ_r(S_r(y)). It is further assumed that the angle between d_yS_r(v) and the boundary of Δ_r(S_r(y)) is bounded away from zero uniformly in r. * More generally, we could consider a finite decomposition into sub-cones CΔ=CΔ^1∪⋯∪CΔ^k. In this case we require the existence of associated bands ℬ_r^1,⋯,ℬ_r^k satisfying analogous properties as the listed above. * The techniques of this article also permit to consider families {S_r:^d→^d}_r with d>2 satisfying similar conditions. In this case, the bands are replaced by (finite union of) sets of the form^k-1×[a,b]×^d-k, and there are some dimension restrictions that have to be imposed on the cones (if Δ_r is a cone centered around ^k then k≥d/2 in order to satisfy the last part of S-2). Since this complicates the notation, and no compelling example of this situation seems available, the author decided to consider the case d=2 and leave the problem of the general case to the interested reader. See also the comments after the proof of Proposition <ref>. It is worth to emphasize that the conditions postulated on S_r do not guarantee positivity of any of its exponents. Note also that the quantitative requirement in the expansion hypothesis (the last part of S-1) is very weak.We will be considering the productS_r=S_1,r×⋯× S_e,r:^2e→^2e of e factors S_i,r:→ satisfying S-1,S-2 above: as an mild abuse of language we say in this case that {S_r}_r satisfies S-1,S-2.We will identify _i={0}×⋯××⋯×{0}⊂^d (in the i-th position), and likewise ^2_i={0}×⋯×^2×⋯×{0}⊂^2e=T_y^2e, ∀ y∈^2e. In the same way, objects associated to the map S_i,r will be denoted by the corresponding subscript (for example, 𝒞_i,r denotes the critical region of S_i,r). §.§ Partially hyperbolic skew products As we mentioned in the Introduction there are very few available methods to establish non uniform hyperbolicity. It is not surprising then that a fair amount of the literature imposes some extra hypothesis on the map f considered, customarily the existence of some uniformly hyperbolic directions. A case of special interest is when the diffeomorphism f is partially hyperbolic. We recall the definition below. The diffeomorphism f:M→ M is weakly partially hyperbolic (wPH) if there exists a continuous df-invariant splitting TM=E^cs⊕ E^u (i.e. d_pf(E_p^u)=E_f(p)^u,d_pf(E_p^cs)=E_f(p)^cs ∀ p∈ M) and constants C>0,λ>1,0<K<1 such that for every p∈ M, for every unit vectors v∈ E^u_p,w∈ E_p^cs and for all n≥ 0 it holds * d_pf^n(v)≥ Cλ^n (uniform expansion in E^u). * d_pf^n(w)≤ K·d_pf^n(v)(domination between E^cs and E^u). f is partially hyperbolic (PH) if both f and f^-1 are wPH: in this case there exists a df-invariant splitting TM=E^s⊕ E^c⊕ E^u such that vectors in E^u (resp. in E^s) are exponentially expanded (resp. contracted). The bundles E^s,E^u,E^c are denominated the stable, unstable and center bundle of f. See <cit.> for further information on Partially Hyperbolicity. We content ourselves reminding the reader that (weakly) partial hyperbolicity is a 𝒞^1 open condition.If f is PH, the Lyapunov exponents corresponding to E^s⊕ E^u are non-zero, so it is enough to study the Lyapunov exponents for vectors in E^c (these will be referred as center exponents). It is fair to say that so far the main focus of research in this topic has been the case when the center exponents have a definite sign (all positive or negative), or when there exist dominated splitting among the subspaces corresponding to Lyapunov exponents of opposite signs[If Λ⊂ M is f-invariant, we say that a df-invariant sum E⊕ F⊂ T_ΛM is a dominated splitting if there exists 0<K<1 such that for n≥ 0, p∈Λ and unit vectors v∈ E(p),w∈ F(p) it holds d_pf^n(v)≤ K·d_pf^n(w).]. See for example <cit.>.The map given in <cit.> is also PH, although of a different type. Its center bundle is two dimensional, with corresponding exponents of opposite signs, whereas it does not admit a dominated decomposition into one-dimensional sub-bundles, and even more, it is 𝒞^2 - robustly NUH. In the non-conservative case, results of the same type were obtained before by M. Viana <cit.>. A popular example of PH diffeomorphism are the so called PH skew products. Consider A:N→ N an Anosov diffeomorphism and let S:𝕋^d→𝕋^d be a differentiable map. Given a smooth function φ:N→𝕋^d, the skew product of A and S with respect to φ is the (bundle) map f:N×𝕋^d→ N×𝕋^d given asf(x,y)=(A(x),S(y)+φ(x)).We denote f=A×_φ S and call A the base map, S the fiber map and φ the correlation map. The manifold M:=N×𝕋^d is equipped with the product of any pair of metrics in N,^d. Since A is hyperbolic, natural domination conditions between dA and dS+dφ imply that f is PH, i.e. a PH skew product, with center bundleV= {0}× T^d.If TN=E^s_A⊕ E^u_A is the hyperbolic decomposition corresponding to A, we extend these bundles to N×𝕋^d using the same nomenclature (i.e., E^s_A=E^s_A×{0},E^u_A=E^u_A×{0}).Note that the inverse of a skew product as defined before is not necessarily a skew-product, but rather of the formf^-1(x,y)=(A^-1(x),ψ(x,y)).These types of systems are sometimes called fibered.For establishing positivity of center exponents of (families of) skew products we require some control in the dynamics along unstable directions of the base map. We will make the following hypotheses.Standing hypotheses for the rest of the article: The base maps of the skew products are linear automorphisms of the 2-torus, i.e. N=𝕋^2 and A∈ SL(2,ℤ). The correlation functions φ:^2→^2e are linear[Meaning that they are induced by linear maps ^2→^2e.].The linearity condition on the correlation map is not central, and is used to simplify further requirements. On the other hand, our methods use strongly conformality of the action of A on its unstable directions. The above hypotheses are strong,but even this case is not currently well understood and falls out the category of examples discussed in the literature (cf. third section); besides, we are more interested in the behavior of the map in the fiber directions. We will thus study families of PH skew products, where the fiber maps are given by a family {S_r=S_r,1⋯ S_r,e:^2e→^2e}_rsatisfying S-1,S-2, and where the correlation functions determine a weak but non vacuous interplay between the strong unstable directions coming from the base dynamics and the fiber directions. The simplest case is when e=1 i.e. there exists a single cone defined in the complement of its critical region where vectors are expanded under the action of the derivative. An example of this situation is given by the (uncorrelated) product f_r:^4→^4 with f_r=A× s_r, where A∈ SL(2,ℤ) is hyperbolic. Our techniques do no apply to these kind of map, although related cases are treated in Corollary A (cf. the next Section). It is known that 𝒞^2 conservative perturbations of f_r above are NUH, provided that r is small <cit.>. §.§ Coupled families In this part we state explicitly the conditions required on the correlation functions for the techniques in this article. If the reader so prefers, she/he can focus in the specific correlation function used in (<ref>), where all conditions are more transparent to check.Let {f_r=A_r×_φ_r S_r}_r be a family of skew products where{S_r=S_r,1×⋯× S_r,e:^2e→^2e}_r satisfies S-1,S-2, and A_r∈ SL(2,ℤ) are uniformly hyperbolic. Associated to A_r we have a decomposition into eigenspaces^2=E^u_A_r⊕ E^s_A_r, and we choose unit vectors e^u_A_r,e^s_A_r generating respectively E^u_A_r, E^s_A_r with A_r(e^u_A_r)=λ_r· e^u_A_r, A_r(e^u_A_r)=τ_r· e^s_A_r where 1<λ_r=1/τ_r. Note that since we are assuming that φ_r is linear, φ_r|E^u_A_r=φ_r(e^u_A_r) and φ_r|E^s_A_r=φ_r(e^s_A_r).For 1≤ i≤ e letP_i:^2e→^2_i be the projection onto the first coordinate of ^2_i, namelyP_i(x_1,…,x_2e)=(x_2i-1,0) We say that the coupling in {f_r}_r is adapted if A-1 * φ_r|E^s_A_r+dS_r^3/φ_r|E^u_A_r0, φ_r/λ_r0, φ_r|E^s_A_r·dS^-10. * There exists l∈ℕ such that λ_r/φ_r^l0,dS^-1_r^3l(dS_r^3l+d^2S_r^3l)/λ_r0 A-2 * min_1≤ i≤ eP_i∘φ_r|E^u_A_r>0. * max_1≤ı≤ eP_i∘φ_r|E^s_A_r+P_i∘ dS_r/P_i∘φ_r|E^u_A_r0. * K(r):=min_1≤ i≤ eP_i∘φ_r|E^u_A_r/max_1≤ i≤ eP_i∘φ_r|E^u_A_r 1. Above, it is tacitly assumed that dS_r≥ 1 (hence dS^-1_r≥ 1 since S_r preserves volume). Condition A-1 simply establishes domination among the relevant quantities. It implies that for large r the map f_r is PH (Corollary <ref>), and moreover the angles ∠(E^u_f_r,E^u_A_r),∠(E^u_f_r,E^s_A_r) converge to zero as r+ (cf. Lemma <ref>). In practice, the norms of dS_r,dS_r^-1,d^2S_r will behave polynomially in r while λ_r is exponential, so A-1 will be simple to check.As for condition A-2, it means that φ_r provides a interaction of the unstable directions of A_r with every S_i,r and all these interactions are comparable and close to conformal (for some suitable metric on the center bundle of f_r). The choice of picking the first coordinate in the definition of P_i is determined by the form of the critical region of S_i,r. In the case where 𝒞_i,r is a band in the other coordinatewe need to modify P_i accordingly. The reader will find no difficulty in adapting the results to thissituation. mainteo Consider the family {f_r=A_r×_φ_r S_r}_r with A_r∈ SL(2,ℤ) hyperbolic, {S_r}_r satisfying conditions S-1, S-2, and where the coupling is adapted. Then there exists r_0 such that for every r≥ r_0 the map f_r is PH and furthermore there exists Q(r)>0 and a full Lebesgue measure set NUH_r⊂ M satisfying: for every p∈ NUH_r there exists U(p)⊂ E^c_f(p) sub-space of dimension greater than equal to e, and so that v∈ U(p)∖{0}⇒χ(p,v)>Q(r). The same is true for any conservative map in a 𝒞^2 neighborhood 𝒰_r of f_r. One can estimate Q(r)≥min_1≤ i≤ e0.99σ/σ+1log(β_i(r)^6ζ_i(r)^1/σ). cf. the proof of Proposition <ref> in page prueba and <ref>. Let us point out to the reader that in the setting that we are studying, due to Oseledet's theorem there exist for μ almost every point p∈ M a natural number 1≤ k(p)≤ M, real numbers χ_1(p),…,χ_k(p) and subspaces E^1(p),⋯, E^k(p)⊂ T_pM such that 0≠ v∈ E^i(p)⇒χ(p,m)=χ_i(p). We call E^i(p) the multiplicity of the exponent χ_i at p. Our Main Theorem asserts that under its hypotheses, for large r it holds that for μ almost every point p the sum of themultiplicities corresponding to center exponents at p that are larger than Q(r) is at least e.The rest of the article is organized as follows. In the next section we show how to apply the stated results to the important case of random surface maps, and we discuss the existence of physical measures for such systems. We also consider several concrete examples, in particular (more general versions of) the random coupled standard maps system. This section also discusses how to apply the techniques to other classical realization of random maps when we interchange the base dynamics by a shift map. Then in the fourth section we present our main tool: admissible curves and adapted vector fields. These are used in the following section to establish the Main Theorem. We finish the article with a short appendix where some side technical considerations are examined.§ APPLICATIONS AND EXAMPLES Our Main Theorem can be applied to establish non uniform hyperbolicity of some coupled families {f_r=A_r×_φ_r S_r}_r. In this Section give examples and derive some consequences. §.§ Existence of physical measures and surface maps We recall that the basin of of attraction of a f - invariant measure μ is B(μ)={p∈ M:∀ h:M→ continuous ,1/n∑_k=0^n-1h∘ f^k(p)∫ h dμ}. An f - invariant measure μ is physical if it is ergodic and Leb(B(μ))>0. If (f,μ) is NUH, then a celebrated result of Pesin <cit.> implies that μ can be written (in the ω^∗ topology) as the converging series of a sequence {μ_n}_n≥ 1 of ergodic probability measures of f. The set of supports {supp(μ_n)}_n≥ 0 is a countable partition of M, hence there exists at least one μ_n for which its support has positive Lebesgue measure: this μ_n is a physical measure for f.Establishing the existence of physical measures is of utmost interest in smooth ergodic theory; most diffeomorphisms do not leave invariant any smooth volume, but physical measures at least detect the Lebesgue class (which comes from the differential/Riemannian structure of the manifold) in terms of its generic points, and thus they can be thought as a reasonable substitute for conservativity.Regrettably the list of known abundant examples (i.e., in an open class or at least, in parameterized families) having physical measures is not very large. Some illustrative results are <cit.> (physical measures for hyperbolic diffeomorphism), <cit.> (physical measures for the quadratic family),<cit.> (physical measures for a class of wPH maps with controlled dynamics along the dominated bundle) and <cit.> (physical measures for Hénon maps). In the significant particular case when S_r is a surface map we obtain the following. Assume that {f_r=A_r×_φ_r S_r}_r is an adapted family, with A_r∈ SL(2,) hyperbolic and {S_r:𝕋^2→𝕋^2}_r is a collection of𝒞^2 conservative maps satisfying conditions S-1,S-2. Let μ be the Lebesgue measure in ^4. a)There exists r_0 such that for every r≥ r_0 there exists Q(r)>0 satisfying: if (f_r,μ) is ergodic then for μ almost every p∈ M and every v∈ T_pM∖{0}: lim_n+|logd_pf^n_r(v)/n| >Q(r). Hence the map f_r is NUH, and in particular has a physical measure. The same is true for any μ ergodicin a 𝒞^2 neighborhood 𝒰_r of f_r. b) In general, if f_r is not ergodic but is 𝒞^3, then it is approximated in the 𝒞^2 category by stably (ergodic) NUH diffeomorphisms. Assume first that μ is ergodic for f_r. By Corollary 3.5 of <cit.>, for μ almost every p∈ M and every v∈ T_pM∖{0} it holdsχ_f_r(m,v)∈{χ^u_f_r,χ^s_f_r,χ^c,1_f_r,χ^c,2_f_r}, where χ^u_f_r,χ^s_f_r correspond to the exponents in E^u_f_r, E^s_f_r and χ^c,1_f_r≤χ^c,2_f_r are the center exponents. Since f_r preserves the volume μ, byProposition 4.3 in <cit.> it holds χ^u_f_r+χ^s_f_r+χ^c,1_f_r+χ^c,2_f_r=0. On the other hand, for every x∈^2 the Jacobian of the restriction f_r|_{x}×:{x}×→{A_r(x)}× is equal to one, hence again by the same Proposition we have that χ^c,1_f_r+χ^c,2_f_r=0 and thus χ^u_f_r+χ^s_f_r=0. Note that the family {f_r}_r satisfies the hypotheses of our Main Theorem, thus for r sufficiently large there exists Q(r)>0 so that χ^c,2_f_r≥ Q(r), which implies by (<ref>) that χ^c,1_f_r≤ -Q(r), and (f_r,μ) is NUH. To consider perturbations of f_r, we observe that since E^u_f_r, E^s_f_r are one-dimensional, and due to ergodicity of the system (f_r,μ), we can write χ^u_f_r=∫logd_pf_r|E^u_f_r dμ(p) χ^s_f_r=∫logd_pf_r|E^s_f_r dμ(p). The stable and unstable bundles in the partially hyperbolic class depends 𝒞^1 continuously on the map (cf. Theorem 2.15 in <cit.>), hence there exists 𝒩_r a 𝒞^1 neighborhood of f_r so that if ∈𝒩_r preserves μ and is ergodic, then * the sum of all its exponents (with respect to μ) is zero (by the same argument used for f_r cf. (<ref>)). * |χ^u_+χ^s_|<Q(r)/2 (by equality (<ref>) and continuity of the quantities). By the above, χ^c,2_+χ^c,1_<Q(r)/2 as well. If 𝒰_r is the 𝒞^2 neighborhood of f_r given in the Main Theorem, we hence deduce ∈𝒰_r∩ V_r⇒χ^c,2_>Q(r)⇒χ^c,1<-Q(r)/2<0 and (,μ) is NUH. This concludes the first part. For the second we use a Theorem of K. Burns and A. Wilkinson <cit.> that allow us to approximate any 𝒞^3 conservative skew product like f_r (in the 𝒞^2 topology) by conservative mapshaving a 𝒞^2 neighborhood where every μ preserving diffeomorphism on it is ergodic, which together with part a) implies the result.Part b) is an interesting consequence, since it does not require a priori knowledge of ergodicity of the maps. We remark that even though there exists a characterization of ergodicity for skew-products <cit.>, the conditions in practice are very difficult to check.More interestingly, the Main Theorem can be used to establish directly non uniform hyperbolicity of several examples, and then try to use this information (together with the fact that the exponents are uniformly separated from zero) to prove ergodicity. This has recently been achieved by D. Obata <cit.> for the original <cit.> map, who proves in fact that the map is 𝒞^2 - stably Bernoulli (a much stronger condition than ergodicity). His arguments are quite sophisticated. §.§ Examples: Random products of standard maps In this part we present several examples of applications of the Main Theorem. We also compare the results with some of the available literature.§.§.§ The Standard Map.Let us consider again the map f_r introduced in <cit.>,f_r(x,y,z,w)=(A^[2r]·(x,y),s_r(z,w)+P∘ A^[r]·(x,y)) P(x,y)=(x,0).To prove that (for large r) the map f_r is NUH, the arguments of the aforementioned article make extensive use of the following properties. * One can obtain sharp estimates on the form of the (one dimensional) strong unstable bundle. * The center bundle is two dimensional. Here we generalize the results of <cit.>, allowing higher-dimensional center behavior.We will start by showing that s_r,s_r^-1 satisfy conditions S-1,S-2, and that the coupling in {f_r}_r is adapted. We do so because we will be using similar types of couplings in our other examples, and because we are interested in dynamics related to the standard family.The proof of conditions S-1,S-2 for s_r essentially follow from the computations carried in Section 4 of <cit.>, and are recalled below. A direct computation shows thatd_(x,y)s_r=[ Ω_r(x,y) -1;10;]where Ω_r(x,y)=2+rcos(x). For z=(x,y) the vectors v⃗_r(z)=(1,Ω_r(z)), h⃗_r(z)=(Ω_r(z),-1) form an orthogonal basis of T_z, and one checks that d_zs_r(v⃗_r(z))=(0,1), d_zs_r(h⃗_r(z))=(1+Ω^2_r(z),Ω_r(z)).With this it is easy to verify that if X is a unit vector field in 𝕋^2,d_zs_r(X_z)≥ [r]·|sinθ^X(z)|·|cos x|-2where θ^X(z) is the angle ∠(X_z,v⃗_r(z)). We define the critical strip 𝐶𝑟𝑖𝑡(s_r) as the set of points𝐶𝑟𝑖𝑡(s_r)={z=(x,y)∈𝕋^2:b_1≤ x≤ b_2 or b_3≤ x≤ b_4}where 0<b_1<π/2<b_2<b_3<3π/2<b_4<2π are such thatcos b_1=cos b_4=1/√(r)cos b_2=cos b_3=-1/√(r).It holds that l(𝐶𝑟𝑖𝑡(s_r))≤8/√(r) (cf. (14) in <cit.>; note that 𝐶𝑟𝑖𝑡(s_r) is the union of two connected bands). We consider also the cone Δ_r:=ℝ·{(1,n)∈^2:|n|≤√(r)}. As for the sets ℬ_r^+, ℬ_r^-, they are treated in Lemma 6 of <cit.>: if CΔ^+_r=·{(1,n):n≥√(r)}, CΔ^-_r=·{(1,n):n≤ -√(r)} then ℬ_r^+={(x,y)∈: cos x<0}, ℬ_r^-={(x,y)∈: cos x>0}. It follows that l(ℬ_r^+), l(ℬ_r^-)=π.Using 𝐶𝑟𝑖𝑡(s_r) as critical region, the bands ℬ_r^+,ℬ_r^-and the cone Δ_r one can verify the following. The maps s_r,s_r^-1 satisfy S-1, S-2. Moreover, β(r) ≥ r^1/6-2 ζ(r) ≥ 1/2r In particular we can take σ=2. The bound for β(r) is spelled in Lemma 4 of <cit.>, while the bound for ζ(r) is simple to obtain from (<ref>). As for S-2, invariance of Δ_r is proved in Lemma 5 of <cit.>, while restitution of the expansion direction is implicit in Lemma 6 of the same work: indeed if v⃗∈CΔ_r⊂ T_z then v⃗=c·(0,1) and hence d_zs_r(v⃗)=c·(-1,0)∈Δ_r, or v⃗=c· (1,n) where either n≥√(r) (v⃗∈CΔ_r^+), or n≥√(r) (v⃗∈CΔ_r^+). Since d_zs_r(v⃗)=c·(2+rcos x-n,1) we deduce that if cos x and -n have the same sign (and r large), this vector makes a small angle with the horizontal axis in ^2 and thus is contained in Δ_r away from its boundary. It follows that if v⃗∈CΔ_r^+, z∈ℬ_r^+ or v⃗∈CΔ_r^-, z∈ℬ_r^- then d_zs_r(v⃗)∈Δ_r. Finally, note that Δ_r increases with r, which together with the previous remark implies S-2. For the inverse map one can use that s_r^-1=R∘ s_r∘ R where R(x,y)=(y,x). This well known fact is recalled in Lemma 1 of <cit.>Now we discuss the correlation function. The coupling in {f_r}_r is adapted. Let λ=A|E^u_A,τ=A|E^s_A. Since A_r=A^[2r] we obtain E^u_A_r=E^u_A, E^s_A_r=E^s_A and λ_r=λ^[2r],τ_r=τ^[2r]=1/λ_r. Observe that φ_r(x,y)=P∘ A^[r]·(x,y). The bundle E^u_A makes a positive angle with the line {(0,y):y∈}⊂^2, hence P|E^u_A>0 which in turn implies P∘φ_r|A_r>0. Note also that 0<P∘φ_r|E^u_A=φ_r|E^u_A≤A^[r]≤λ^[r]. Using this and the fact that ds_r,ds_r^-1,d^2s_r≤ 2r, both conditions A-1, A-2 follow. Recall that two diffeomorphisms f_1,f_2 of a manifold X are said to be differentiably conjugate if there exists a diffeomorphism h:X→ X (the conjugacy) such that f_1=h^-1∘ f_2∘ h. If f_1,f_2 are conservative and differentiably conjugate, then by the chain rule their Lyapunov exponents coincide Lebesgue almost everywhere. We will employ this remark to deal with the inverse, and in particular we use the fact that s_r is differentiably conjugate to its own inverse, i.e. it is reversible (cf. Lemma <ref> above).Indeed, this property of s_r is used in Lemma 1 of <cit.> to show that f^-1_r is differentiably conjugate to the mapf_r(x,y,z,w)= (A^-[2r]·(x,y),s_r^-1(z,w)+P∘ A^-[r]·(x,y)):to be precise f^-1_r=R∘f∘R where R:^4→^4 is the involutionR(x_1,x_2,x_3,x_4)=(x_1,x_2,x_4,x_3).Observe that R leaves invariant the bundle V, and thus the Main Theorem can be applied to f_r^-1. Reversibility of the center dynamics will also be used in the other examples todeal with the inverse map. We remark that this property for area preserving maps seems to be rather common. See <cit.>. One can check that {f_r^-1}_r satisfies condition A-3 of the Appendix, but regrettably it does not satisfy condition A-4, and hence we cannot use Corollary B of the Appendix to conclude directly that for r large the map f_r is NUH. We have proved the following. For large r the map f_r is 𝒞^2 - robustly NUH: for such r there exists 𝒰_r 𝒞^2 neighborhood of f_r with the property that if g∈𝒰_r is conservative then g is NUH. The map g has a physical measure and moreover for Lebesgue almost every p, it holds v∈ T_pM∖{0}⇒lim_n|logd_pf^n(v)/n|>3/5log r. The lower bound in the exponents is better than the one in <cit.> but worse to the one obtained by Obata <cit.>. See Remark <ref>. Note that for mapsin 𝒰_r above, the splitting between the center Oseledet's subspaces is not dominated (d|E^c_≃ ds_r|V, and the later cannot be dominated due to the existence of 𝐶𝑟𝑖𝑡). Compare with the result of J. Bochi and M.Viana <cit.> that states: for compact manifolds of dimension greater than one, 𝒞^1-generically in the space of conservative diffeomorphisms either there exists a zero exponent with multiplicity two, or the Oseledet's splitting is dominated, almost everywhere. Our methods seem to be particularly adequate to deal with the case where S_r is given by the generalized family of standard maps studied by O. Knill in <cit.> (x,y)→ (2x-y+rV(x),x)where V(x) is a periodic Morse potential. It is interesting to compare the lower bound on the topological entropy h_top(s_r)≥1/3log (C· r) given by Knill and our lower bound on the positive center exponent of f_r. By the Pesin formula <cit.>, for conservative surface maps the metric entropy with respect to the area is equal to the integral of the largest Lyapunov exponent, and since the two obtained bounds are comparable this can be taken as evidence for the suitability of Lebesgue measure to detect the majority of chaotic behavior in the dynamics of s_r. §.§.§ Higher dimensional examples: random products of (coupled) standard maps. We will now apply our techniques to coupled products of standard maps, thus giving examples of more general center (higher dimensional) dynamics. Let us go back to the map g_r:^2(e+1)→^2(e+1) given in (<ref>), g_r(z_0,z_1,⋯, z_e)=(A^[2r](z_0),s_r(z_1)+P∘ A^[r](z_0),⋯,s_r(z_e)+P∘ A^[r](z_0)). We start recalling that dg_r|V=ds_r×⋯× ds_r (the product taken e times); from this fact and by the computations of Lemma <ref> we get that conditions S-1, S-2 are satisfied for g_r. The corresponding correlation map φ_r:→^2e is of the form φ_r(z)=φ_r(z)×⋯×φ_r(z), where φ_r is the correlation function associated to f_r (considered in Lemma <ref>). From this it follows directly that the coupling in {g_r}_r is adapted. We can thus apply the Main Theorem to {g_r}_r and ensure the existence of Q(r)>0 (if r is sufficiently large), so that for Lebesgue almost every p, the sum of the multiplicity of center exponents at p bigger than Q(r) is at least e, and moreover this property is 𝒞^2 robust among conservative diffeomorphisms close to g_r. For the negative exponents, we consider Γ:^2(e+1)→^2(e+1) the map Γ(z_0,z_1,⋯, z_e)=(z_0,R(z_1),⋯, R(z_e)) where R:→ is the involution R(x,y)=(y,x). Then Γ is a differentiable involution, it preserves the bundle V, and one can verify that g^-1_r=Γ∘g_r∘Γ where g_r(z_0,z_1,⋯, z_e)=(A^[-2r](z_0),s_r(z_1)+P∘ A^-[r](z_0),⋯,s_r(z_e)+P∘ A^-[r](z_0)) (compare (<ref>),(<ref>)). Note that g_r has the same form of g_r thus by the Main Theorem applied to g_r, and by using Γ to conjugate every map sufficiently 𝒞^2 close to g_r to a map 𝒞^2 close to g_r^-1, we conclude the existence of a 𝒞^2 neighborhood of g_r^-1 so that for every conservative g in this neighborhood,for Lebesgue almost every p the sum of the multiplicity of center exponents of g at p that are bigger than Q(r) is at least e.Since the set of 𝒞^2 diffeomorphisms of M is a topological group when equipped with the 𝒞^2 topology, we finally deduce the existence of a 𝒞^2 neighborhood 𝒰_r of g_r so that every conservative g∈𝒰_r is NUH with respect to μ. This establishes Theorem <ref> of the Introduction. Now we look at the map g_r,τ:=A^[2r]×_φ_ru_r,τ considered in Corollary <ref>, where u_r,τ:^4→^4 is given by u_r,τ(x,y,z,w)=(2x-y+rsin(x)+τsin(x+z),x,2z-w+rsin(z)+τsin(x+z),z) and φ_r:→^4 is defined as above. Note that g_r,τ is a perturbation of g_r (with e=2) if τ is small. In particular, for large r and small τ, g_r,τ is PH and𝒞^2 robustly NUH, as claimed in the Corollary. The maps g_r, g_r,τ treated above provide new interesting higher dimensional examples to add to the list of known NUH systems, enabling us to construct partially hyperbolic diffeomorphisms with rich center dynamical behaviors. Other interesting examples can be obtained by considering some symplectic twist maps in ^2d, and in particular those of the form S_V(q,p)=(2q-p+∇ V(q),q) q,p∈^dwhere V∈𝒞^2(^d,). We illustrate this with the potential V_τ_1,τ_2,τ_3:^2→^2 whereV_τ_1,τ_2,τ_3(x,y)=τ_1cos(x)+τ_2cos(y)+τ_3cos(x+y).The resulting map S_V_τ_1,τ_2,τ_3 is the so called three parameter Froeschlé family <cit.>, and it is very similar to u_τ,r. Let us writeφ_r(x_1,x_2):=(P∘ A^[r](x_1,x_2),P∘ A^[r](x_1,x_2)): a slight variation in the argument used in the proof above (choosing τ_1,τ_2≈ r large, τ_3 small) yields the following. For every r sufficiently large there exists an open set in the parameter space E_r⊂{(τ_1,τ_2,τ_3):τ_i∈} and C>0 such that if (τ_1,τ_2,τ_3)∈ E_r then F_r,τ_1,τ_2,τ_3:=A^[2r]×_φ_rS_V_τ_1,τ_2,τ_3 is PH and satisfies for Lebesgue almost every p and v∈ T_pM∖{0}, lim_n|logd_pF_r,τ_1,τ_2,τ_3^n(v)/n|>Clog(r). The same holds for every conservative map in a 𝒞^2 neighborhood of F_r,τ_1,τ_2,τ_3.For further information on this family and other symplectic twist maps we refer the reader to <cit.>. To the extent that the author could check none of these examples can be treated by any available technique in the literature, and is his hope that he provided enough evidence to convince the reader of the versatility in the method presented. It is also worth noticing that for establishing our results it is not necessary to assume a priori ergodicity of the map. In particular, we do not need to perturb to guarantee accessibility as the methods based on the invariance principle <cit.> usually require. This is important by two reasons: first because our goal is understand concrete examples more than their perturbations (i.e. random products of the standard map, versusperturbations of random products of the standard map), and second because as we mentioned in the introduction, establishing NUH of the system could be used in some cases to establish ergodicity <cit.>.§.§ Cocycles over the shift Systems exhibiting similar dynamics with PH skew products are cocycles over expanding endomorphisms, and in particular over shifts spaces. These kind of maps are also very popular in the literature. As an example, we consider for a positive integer k the complete shift space in k symbolsΣ_k^+={0,…,k-1}^ℕEquipped with the natural product topology, it is a compact metrizable space. The dynamics is given by the shift map σ(ω)_n=ω_n+1, and we consider the Bernoulli measure μ_k=(1/k,…, 1/k)^ℕ on Σ_k^+, i.e. the product measure obtained by assigning weight 1/k to each symbol.Given a diffeomorphism S:𝕋^d→𝕋^d and a continuous function φ: Σ_k→𝕋^d we can define a (continuous) cocycle of matrices as follows: let G_φ:Σ_k×𝕋^d→Σ_k×𝕋^d be the mapG_φ(ω,x)=(σω,φ(ω)+S(x))and for n≥ 0 denote g^n(ω,x) the projection on the second coordinate of G_φ^n(ω,x). Then define∂_cG_φ^n(ω,x):=d_g^n-1(ω,x)S∘⋯∘ d_xS.This way we have determined a cocycle of matrices which could be interpreted as the derivative cocycle of G_φ. In a recent work by A. Blumenthal, J. Xue and L.S Young <cit.>, the authors considered a similar type of random cocycle over the infinite shift T:(-,)^ℕ→ (-,)^ℕ (>0 small), with fiber maps ψ_r(ω,(x,y))=s_r(x+ω_0,y), and other 2-dimensional conservative maps satisfying certain expanding conditions in the spirit of ours S-1, S-2. For these systems the authors prove the positivity (with precise bounds) of the largest exponent of the cocycle for ν^ℕ almost everywhere, where ν is the uniform measure on (-,). Their techniques are more probabilistic in nature than ours, and seem to depend on the two-dimensionality of the fiber maps (hence, one can deal with one Lyapunov exponent only).It is a simple exercise in dynamical systems to show that σ:Σ_k^+→Σ_k^+ equipped with μ_k is (measure theoretically) conjugate to the expanding map E_k:→, E_k(θ)=k·θ 2π with the Lebesgue measure on . Thus, instead of cocycles over Σ_k^+ we can equivalently consider cocycles over E_k.For k∈ℤ∖{1,0,-1} consider the multiplication map E_k. Smale's solenoid construction (described for example in Chapter 7 of <cit.>) permits us to find a diffeomorphismL_k:N:=× D→ N, D={(x,y)∈^2:x^2+y^2≤ 1} such that * L_k has an hyperbolic (expanding) attractor Λ⊂ N. * L_k(θ,(x,y))=(E_k(θ),ψ(x,y)) for some ψ:N→ D.Let us remind the reader that adiffeomorphism L:N→ N is said to have an hyperbolic attractor Λ if * Λ is an hyperbolic set for L. * There exists U⊂ N open such that Λ=∩_n∈ℕL^n(U).If Λis an hyperbolic attractor anddenotes its unstable lamination, one verifies easily thatx⊂Λ for every x∈Λ. See <cit.> Chapters 5 and 6. Furthermore, in case that L is of class 𝒞^2, there exists a particularly important invariant measure μ_SRB supported on Λ which can be characterized as follows: μ_SRB is the unique L - invariant measure such that for any sufficiently small lamination chart B corresponding to 𝒲^u_Λ, its conditional measures on x∩ B are absolutely continuous with respect to the induced Lebesgue measure, for almost every x∈ B. μ_SRB is the Sinai-Ruelle-Bowen measure of Λ. See <cit.> for further information on this topic.All the arguments used for the proof of the Main Theorem adapt directly to families of skew products {f_r=A_r×_φ_r S_r}_r where * A_r=L_r|Λ_r where L_r:^b→^b is a diffeomorphism of class 𝒞^2, and Λ_r is an hyperbolic attractor for L_r with one dimensional unstable bundle E^u_A_r. * There exists a continuous Riemannian metric on Λ_r such that dA_r|E^u_A_r is conformal with respect to the associated norm, i.e. there exists λ>1 such that dA_r(v)=λv for every v∈ E^u_A_r. * {S_r=S_r,1×⋯× S_r,e:^2e→^2e}_rsatisfies S-1, S-2. * The correlation functions φ_r:Λ_r→^2e are restriction of linear maps and properties A-1, A-2 are satisfied. In such case, our techniques give thatfor r sufficiently large there exists a μ_SRB× Leb full measure set NUH_r⊂Λ_r×^2e and Q(r)>0 such that for p∈ NUH_r the sum of the multiplicity of center exponents at of f_r at p that are bigger than Q(r) is at least e. Here we make a small abuse of language and call center exponents to those associated to vectors tangent to V={0}× T^2e (although the map f_r is not PH in general). The same property remains valid replacing S_r by a conservative diffeomorphism S_r':^2e→^2e, provided that S_r' is sufficiently 𝒞^2 close to S_r. Back to the solenoid, if E^u denotes the unstable bundle of A:=L_k|Λ, then for p=(θ,(x,y))∈Λ, the line E^u(p) is a graph over T_θ×{0}, hence there exists a Riemannian metric · on Λ so that dA(v)=kv for every v∈ E^u (in other words, dA|E^u is conformal). See Proposition 7.5 in <cit.>. Define the map t_r:×→× with t_r(θ,x,y)=(E_k^[2r](θ),s_r(x,y)+(E_k^[r](θ),0)).and write ∂_ct_r(θ,x,y) for the derivative of t_r|:{θ}×→{E_k^[2r](θ)}× at the point (x,y). There exists r_0 such that for every r≥ r_0 it holds for Lebesgue almost every θ∈,(x,y)∈^2 lim_n+log∂_ct_r^n(θ,x,y)/n>3/5log r Consider t_r:N×^2→ N×^2, t_r(θ,z,w,x,y)=(L_k^[2r](θ,z,w),s_r(x,y)+(E_k^[r](θ),0)). Arguing analogously as in Lemma <ref> one checks that the coupling functions {ϕ_r: Λ_r→^2}_r with ϕ_r(θ,z,w,x,y)=(E_k^[r](θ),0) satisfy conditions A-1, A-2, and thus by the Remark <ref> we conclude that for μ_SRB× Leb-a.e.((θ,z,w,x,y)∈ N×^2) it holds lim_n+log∂_ct_r^n(θ,x,y)/n>3/5log r where ∂_ct_r^n(θ,x,y) is the derivative of t_r^n|:{(θ,z,w)}×→{A_r^n(θ,z,w)}×at the point (x,y). As ∂_ct_r^n(θ,x,y)= ∂_ct_r^n(θ,x,y)(=ds_r(x,y)), we conclude lim_n+log∂_ct_r^n(θ,x,y)/n>3/5log rμ_SRB× Leb-a.e.(θ,z,w,x,y)∈ N×^2 If π:N×^2→𝕋×^2 is the projection map π(θ,z,w,x,y)=(θ,x,y), then π semi-conjugates t_r with t_r while π_∗(μ_SRB× Leb) is Lebesgue on ^3. From here it follows.Likewise, one can obtain similar results considering the product of e coupled standard maps instead of just one, establishing that the sum of the multiplicity of positive center exponents is at least e.Since E_k is conjugate to the one-sided shift in k symbols, Theorem <ref> provides a geometrical proof of the existence of non-zero Lyapunov exponents for cocycles over the shift. The type of system described above is similar to the ones considered by M. Viana in <cit.>, although those were non-conservative. It is possible that some of the techniques presented here can be adapted to non-conservative case as well, but the author has not pursued that enterprise in this paper. Besides the interest per-se in the dynamics of cocycles of standard maps, these systems appear naturally in physical and mathematical applications. Historically, they were introduced to study Arnold diffusion for systems of coupled oscillators <cit.>, albeit their research was mainly numerical. For a more up to date and rigorous study on this topic the reader is referred to the work of O. Castejón and V. Kaloshin <cit.>, where the authors analyze statistical properties of random products of standard maps appearing (as certain induced dynamics) in Arnold's original diffusion example. § ADMISSIBLE CURVES AND ADAPTED FIELDS The most important technical tools introduced in <cit.> are admissible curves and adapted fields. Roughly speaking, an adapted curve is a segment of the strong unstable manifold that makes approximately a complete turn in around every vertical torus {0}×_i while doing many more turns around the base torus 𝕋^2×{0}; an adapted field is a vector field along an admissible curve with very small variation. In that work however, the notion of admissible curve is tailored to the specific example considered. Here we present a more abstract definition. §.§ Partial Hyperbolicity Let {f_r=A_r×_φ_r S_r}_r be a coupled family where {S_r}_r satisfies S-1, S-2. We will be able to work from the beginning with perturbations of f_r. The map f=f_r:M=^2×^2e→^2×^2e is of the formp=(x,y)∈ M⇒ f(x,y)=(A(x),S(y)+φ(x))and we consider =f+h a small 𝒞^2 perturbation of it. Note that the derivative of f can be expressed in block form asd_pf=[A0;φ d_yS ]. Let us recall we are assuming that A=A_r∈ SL(2,) is hyperbolic with associated decomposition ^2=E^u_A⊕ E^s_A, where E^u_A=· e^u_A, E^s_A=· e^s_A, e^u_A=e^s_A=1 andλ=A|E^u_A=A(e^u_A)τ=A|E^s_A=A(e^s_A)=1/λ.The bundles E^u_A,E^s_A determine continuous bundles on M, which by an innocuous abuse of language will be denoted by the same letters. Observe that TM=E^u_A⊕(E^s_A⊕ V), with V={0}× T^2e. For convenience we will use in V=⊕_i=1^e^2_i the ℓ^ norm associated to this decomposition. DefineE=E_r=E^s_A⊕ Vso that TM=E^u_A⊕ E, and note df|E≤dS+φ|E^s_A+A|E^s_A. Define also ξ(r):=2φ|E^u_A/λ-df|E.By condition A-1 in the coupling, the above quantity is positive and converges to zero as r. Finally, consider the cone field of size ξ(r) centered around E^u_A in E^u_A⊕ EΔ^u(p)={(v,w)∈ E^u_A(p)⊕ E(p):w<ξv}∪{(0,0)}. There exists r_1>0 such that for every r≥ r_1, for every p∈ M it holds * d_pf(Δ^u(p))⊂Δ^u(f(p)), and * ∀ X∈Δ^u(p), λ(1-ξ(r)/1+ξ(r))X≤d_pf(X)≤λ(1+ξ(r))X. Let (v^u,w)∈Δ^u(p) with 0≠ v^u∈ E^u_A,w∈ E. We obtain d_pf(w)+φ(v^u) ≤d_pf|E·w+φ|E^u_Av^u≤ (ξ(r)d_pf|E+φ|E^u_A)·v^u<λξ(r)v^u= ξ(r)A(v^u) which shows the first part. As for the second, take X=(v^u,w)∈Δ^u(p) and use the previous inequality to compute d_pf(X)≤Av^u+d_pf(w)+d_pφ(v^u)≤λ(1+ξ(r))v^u≤λ(1+ξ(r))Xd_pf(X)≥Av^u-d_pf(w)+d_pφ(v^u)≥λ(1-ξ(r))v^u≥λ1-ξ(r)/1+ξ(r)X where in the last part we have used X≤v^u+w≤(1+ξ(r))v^u. It is a well known consequence of the above that there exists a df - invariant bundle E^u_f⊂Δ^u (see for example <cit.>), and in particular this implies that f is wPH. Similarly, computing the derivative of f^-1 we obtain d_pf^-1=[A^-1 0; -(d_y'S)^-1∘φ∘ A^-1(d_y'S)^-1 ], f(x',y')=(x,y)=pand thus if we define E'_r=E'=E^u_A⊕ V, ξ(r)=dS^-1∘φ|E^s_A/1-τdf^-1|E' and Δ^s(m)={(v,w)∈ E^s_A(m)⊕ E'(m):w<ξv}∪{(0,0)}we can proceed as before and deduce (by A-1) that f^-1 has an invariant expanding bundle E^s_f with τ(1-ξ(r)/1+ξ(r))≤ m(df|E^s_f)≤df|E^s_f≤τ(1+ξ(r)),hence f is PH. For future reference, we spell out this fact in form of a Corollary. There exists r_1>0 such that for r≥ r_1 it holds that f=f_r is PH with invariant splitting TM=E^u_f⊕ V⊕ E^s_f. Once the existence of E^u_f,E^s_f has been established, its expanding/contracting character is direct consequence of the previous Lemma and the inequality above. Since E∩ E'=V, and by hypotheses on the coupling the Whitney sums E⊕ E^u_f, E'⊕ E^s_f are dominated for f,f^-1, the rest follows. From now on we will assume r≥ r_1. Note that E^u_f⊂Δ^u, E^s_f⊂Δ^s and ξ(r),ξ(r)0, thus: the angles ∠(E^u_f,E^u_A), ∠(E^s_f,E^s_A) converge uniformly to zero as r+∞. Partial Hyperbolicity is a 𝒞^1 open condition (cf. Theorem 3.6 in <cit.>), thus for every r there exists ϕ(r)>0 such that if h_𝒞^1<ϕ(r) then f=f+h is PH. Its df - splitting TM=E_^s⊕ E^c_⊕ E^u_ converges to E^u_f⊕ V⊕ E^s_f as h_𝒞^1→ 0 in the corresponding Grassmanian. We defineE_=E^u_A⊕ E^c_. As E^c_f=V is differentiable and integrates to a foliation by torii,Theorem 7.1 of <cit.> guarantees that the center bundle of any sufficiently small𝒞^1 perturbation of f also integrates to a (non-necessarily smooth) foliation by torii.We deduce. The function ϕ(r) can be chosen so that for r≥ r_1, it holds ∠(E^u_,E^u_A))≤π/100∠(E^s_,E^s_A)≤π/100. It is also known (cf. Theorem 3.8 in <cit.>) that the invariant bundles ofare Hölder continuous, provided thatis 𝒞^2. We will choose ϕ(r) sufficiently small so that E_^c is θ - Hölder, with θ≈ 1.§.§ Curves tangent to E^uOur goal is have a qualitative description of the bundle E_^u, and for this we use the graph transform method. By the (un)-stable manifold theorem E^u_ is integrable to an -invariant foliation ={p}_p∈ M whose plaques can be obtained as graphs of functions from E^u_ to E_. The previous Corollary allow us to deduce that there exist δ>0 and a continuous family {Γ_p:E_A^u(p;δ)→ E(p)}_p∈ M such that for every p its local strong unstable manifold p;loc is of the formp;loc=p+graph(Γ_p)where Γ_p_𝒞^1<1. Here E_A^u(p;δ) denotes the δ - disc centered at zero inside E_A^u(p). Similar notation will be used for other bundles. Recall that K(r) is defined in condition A-2. There exists ϖ(r)>0, ϖ(r)0 with the following property. Consider p∈ M and X=e^u_A+X^E∈ E^u_(p) with X^E∈ E. Then for 1≤ i,j≤ e, 1-ϖ(r)≤ P_iX^E/P_jX^E≤1+ϖ(r). 0<(1-ϖ(r))P_i∘φ_r|E^u_A/λ≤ P_iX^E/X≤(1+ϖ(r))P_i∘φ_r|E^u_A/λ. Fix X=e^u_A+X^E∈ E^u_(p) and note that X^E=d_0Γ_p(e^u_A)∈ E. By invariance of the unstable bundle, X=d_^-1p(Y) where Y∈ E^u_(^-1p), hence of the form Y=a· e^u_A+Y^E where a∈, Y^E=a· d_0Γ_^-1p(e^u_A)∈ E. Since d_0Γ_^-1m≤ 1 it follows Y^E≤|a|, Y≤ 2|a|. Recall that E=E^s_A⊕ V, thus Y^E can be written as Y^E=Y^s+Y^v where Y^s∈ E^u_A, Y^v∈ V. Let π^u:TM→ E^u_A, π^E:TM→ E be the projections onto E^u_A, E respectively. As e^u_A=λ· a e^u_A+π^ud_^-1ph(Y), 1=λ· a e^u_A+π^ud_^-1ph(Y) and by using that π^ud_^-1ph≤ϕ(r) we get the following bound for |a|: 1/λ+2ϕ(r)≤|a|≤1/λ-2ϕ(r). On the other hand, X^E=a·φ_r(e^u_A)+φ_r(Y^s)+d_^-1pS_r(Y^v)+π^Ed_^-1ph(Y) hence fixing 1≤ i≤ e we obtain P_iX^E ≤ |a|·(P_i∘φ_r|E^u_A+2(P_i∘φ_r|E^s_A+P_i∘ d_^-1pS_r+ϕ(r)))≤P_i∘φ_r|E^u_A/λ-2ϕ(r)(1+2{P_i∘φ_r|E^s_A+P_i∘ d_^-1pS_r+ϕ(r)/P_i∘φ_r|E^u_A}) P_iX^E ≥ |a|·(P_i∘φ_r|E^u_A-2(φ_r|E^s_A+P_i∘ d_^-1pS+ϕ(r)))≤P_i∘φ_r|E^u_A/λ+2ϕ(r)(1-2{P_i∘φ_r|E^s_A+P_i∘ d_^-1pS+ϕ(r)/P_i∘φ_r|E^u_A}). Condition A-2 implies that by taking ϕ(r) small, the terms in braces converge to zero as r→+ and thus we deduce the existence of ϖ_1(r)>0 with ϖ_1(r)0 such that (1-ϖ_1(r))P_i∘φ_r|E^u_A/λ≤P_iX^E≤ (1+ϖ_1(r))P_i∘φ_r|E^u_A/λ. For 1≤ i, j≤ e we then have P_iX^E/P_jX^E≥ K(r)·1-ϖ_1(r)/1+ϖ_1(r) and thus K(r)·1-ϖ_1(r)/1+ϖ_1(r)≤P_iX^E/P_jX^E≤ K(r)^-1·1+ϖ_1(r)/1-ϖ_1(r). Since K(r)1, the above gives the first inequality in the lemma. Similarly, by (<ref>), X^E ≤ |a|·(φ_r|E^u_A+2(φ_r|E^s_A+d_^-1pS_r+ϕ(r)))≤φ|E^u_A/λ·1+2{φ|E^s_A+dS+2ϕ(r)/φ|E^u_A}/1-2ϕ(r)/λ X^E ≥ |a|·(φ_r|E^u_A-2(φ_r|E^s_A+d_^-1pS+ϕ(r)))≥φ|E^u_A/λ·1-2{φ|E^s_A+dS+2ϕ(r)/φ|E^u_A}/1+2ϕ(r)/λ Condition A-1 implies that the quantity φ|E^s_A+dS+2ϕ(r)/φ|E^u_A converges to 1 (assuming ϕ(r)≤ 1), thus there exists ϖ_2(r)>0 converging to zero as r→+ such that (1-ϖ_2(r))φ_r|E^u_A/λ≤X^E≤ (1+ϖ_2(r))φ_r|E^u_A/λ, which in turn implies 1-ϖ_3(r)≤X≤ 1+ϖ_3(r) where ϖ_3(r)=(1+ϖ_2(r))φ_r|E^u_A/λ. Note that ϖ_3(r)0 by condition A-1. Combining this with (<ref>) we finally get 1-ϖ_1(r)/1+ϖ_3(r)·P_i∘φ_r|E^u_A/λ≤P_iX^E/X≤1+ϖ_1(r)/1-ϖ_3(r)·P_i∘φ_r|E^u_A/λ. Inequalities (<ref>) and (<ref>) imply the conclusion of the Lemma, by defining ϖ(r) appropriately. Convention: From now on r is taken sufficiently large and ϕ(r) small to verify the hypotheses of the previous lemma. To avoid cluttering the notation, the quantity ϖ(r) will be subsequently re-defined to guarantee additional conditions, but always maintaining ϖ(r)0. An admissible curve for the mapis a curve γ: [0,2π]→ M tangent to E^u_ such that for some i∈1,…,e it holds |P_i∘dγ/dt(t)|=1. Ifis an admissible curve with |P_i∘dγ/dt(t)|=1 then its projection on _i makes exactly one turn. By the lemma above |P_j∘dγ/dt(t)|≈ 1 for any other 1≤ j≤ e, and the error in this approximation converges to zero as r +. We deduce that the projection ofinto _j completes almost a turn, with an error that can be taken arbitrarily small as r increases. A consequence of the lemma above is the following. There exists ϖ(r) with lim_r→+ϖ(r)=0 such that for any admissible curve γ its length Leb(γ) is bounded as 2π(1-ϖ(r))·λ/P_i∘φ|E^u_A≤ Leb(γ)≤ 2π(1+ϖ(r))·λ/P_i∘φ|E^u_A In particular, if γ,γ' are admissible curves, for sufficiently large r it holds (1-ϖ(r))·Leb(γ)≤Leb(γ')≤ (1+ϖ(r))· Leb(γ). Consider an admissible curve γ with |P_i∘dγ/dt(t)|=1. For every t the Lemma above gives 1/1+ϖ(r)·λ/P_i∘φ_r|E^u_A≤dγ/dt(t)≤1/1-ϖ(r)·λ/P_i∘φ_r|E^u_A and thus 2π/1+ϖ(r)·λ/P_i∘φ_r|E^u_A≤ Leb(γ)≤2π/1-ϖ(r)·λ/P_i∘φ_r|E^u_A. Re-defining ϖ(r) we get both results, by using the last part of A-2. §.§ Adapted fieldsWe turn our attention to some special class of vector fields parallel to E^c_. Recall that θ is the Hölder exponent of the center bundles (which can be taken uniform in a small neighborhood of f), and l∈ℕ is given in condition A-1. LetC_0=1/λ^θ(1-1/2l) An adapted field (γ,X) forconsists of an admissible curve γ and a unit vector field X along γ satisfying the following. * X is tangent to E_^c. * X is (C_0,θ)-Hölder along γ. This means that if d_γ denotes the intrinsic distance in γ, it holds p,p'∈γ⇒X_p-X_p'≤ C_0· d_γ(p,p')^θ. Even though the length of the admissible curves is rather large, the bound on the constant C_0 makes the variation of any adapted field to be very small. Using Corollary <ref> one establishes the following without any trouble (recall that l is defined in condition A-1). If (γ,X) is an adapted field forand r is sufficiently large, then for every p,p'∈γ it holds X_p-X_p'<(λ^1/2l/φ)^θ.In particular the variation of X converges to zero as r+, due to A-1.We will now show that the set of adapted fields has certain -invariance. Let (,X) be an adapted field. If k≥0 we can write the curve ^k∘γ as^k∘γ=γ_1^k∗⋯γ_N_k^k∗γ_N_k+1^kwhere N_k=N_k(γ) is an integer, γ^k_j are admissible curves for j=1,…, N_kand γ^k_N_k+1 is a segment of an admissible curve[ Here ∗ denotes concatenation of paths.]. Even more, by the Remark <ref> the curves ^k_j can be taken so that they make a complete turn around each of the coordinate torii _i. If X is a vector field we denote _∗X=d∘ X∘^-1, the push-forward by .Now let Y^k:=(^k)_∗X/(^k)_∗X;we have the following. For sufficiently large r it holds the following. If (,X) is an adapted field and k≥ 0 then for every1≤ j ≤ N_k the pair (^k_j,Y^k|_j^k) is an adapted field. This is similar to Lemma 2 in <cit.>. The proof is given below for completeness. Arguing by induction it is enough to show the claim for k=1. Denote Y:=Y^1 and observe ∀ p,p'∈ Md_p(X_p) -d_p(X_p')≤ 2dS·X_p-X_p'≤ 2dS· C_0· d_γ(p,p')^θ and (choosing ϕ(r) small enough), d_p(X_p')-d_p'(X_p')≤ 2d^2S· d_γ(p,p')≤ 3d^2S· d_γ(p,p')^θif d_γ(p,p')< 1 3dS≤ 3dS· d_γ(p,p')^θif d_γ(p,p')≥ 1 hence ∀ p,p'∈ M, d_p(X_p)-d_p'(X_p')≤(3max{dS,d^2S}+2dSC_X)· d_γ(p,p')^θ. On the other hand, by triangular inequality, for p,p'∈^1_j Y_p-Y_p'=1/_∗X_p_∗X_p'_∗X_p'_∗X_p -_∗X_p_∗X_p'≤1/_∗X_p_∗X_p'(_∗X_p'_∗X_p-_∗X_p'_∗X_p'+_∗X_p'_∗X_p'-_∗X_p_∗X_p')≤2/_∗X_p_∗X_p-_∗X_p'=2/_∗X_pd_^-1p(X_^-1p)-d_^-1p'(X_^-1p') Putting both inequalities together we deduce for p,p'∈^1_j Y_p-Y_p' ≤2/_∗X_p(3max{dS,d^2S}+2dSC_0)· d_γ(^-1p,^-1p')^θ≤ 2dS^-1(3max{dS,d^2S}+2dSC_0)1/λ^θd_γ(p,p')^θ=(6max{dS^-1dS,dS^-1d^2S}/λ^θ+4dS^-1dS/λ^θC_0)d_γ(p,p')^θ<(C_0/2+C_0/2)d_γ(p,p')^θ=C_0· d_γ(p,p')^θ if r is sufficiently large and θ close to 1, by A-1.§ POSITIVITY OF THE CENTER EXPONENTSTo establish the existence of positive Lyapunov exponents in the E^c_ directions we will study the quantities I_n(γ,X):=1/|γ|∫_γlogd_p^n(X_p)dγ n∈ℕwhere (γ,X) is an admissible curve and dγ denotes the intrinsic Lebesgue measure in γ. By our choice of ϕ(r), the center direction E_^c ofis close to E^c_f=V, and in particular there exists a bundle isomorphism T_:E_^c→ V that can be chosen to depend continuously with respect to the map(in particular T_Id:V→ V). We will write E_i,^c=T_^-1(^2_i)and observe that E_^c=⊕_i=1^e E_i,^c although this decomposition is not d invariant in general.The cone fields Δ_i,r, 1≤ i≤ e associated S_r are extended (with the same nomenclature) to cone fields in M,and by usingT_ we obtain cone fields Δ_i,⊂ E_i,^c.We write J^u_^-k:=|d^k|E^u_| for the unstable Jacobian of , andrecall the following classical Lemma[cf. Lemma 8 and Corollary 6 in <cit.> : observe that by Corollary <ref> the length of the admissible curves is bounded.]. There exists a constant ℰ=ℰ_r such that for every admissible curve γ it holds ∀ p,p'∈,1/ℰ≤J^u_^-k(p)/J^u_^-k(p')≤ℰ. Therefore for every measurable set A⊂γ and k≥ 0 we have 1/ℰ·Leb(A)/Leb()≤Leb(^-kA)/Leb(^-k)≤ℰ·Leb(A)/Leb(). Moreover, if r sufficiently large and ϕ(r) small it holds ℰ(r)≈ 1. The last part follows by Lemma <ref> and continuous dependence of the partially hyperbolic splitting on the map.The fundamental Proposition is the following. Suppose that there exist C>0, a full Lebesgue measure set M_0 and continuous cones Δ_i,⊂ E_i,^c, 1≤ i≤ e with the following properties. * M_0 is saturated by the foliation ^u_. That is, M_0 consists of complete unstable leaves of ^u_. * Given an adapted field (,X) with γ⊂ M_0and satisfying X_p∈Δ_i,(p) for all p∈γ it holds lim inf_nI_n(γ,X)/n≥ C Then for Lebesgue almost every point, the sum of the multiplicity of center exponents bigger than equal C/ℰ is at least e.This generalizes Proposition 5 in <cit.> to our setting, and even improves the lower bound on the exponents.Fix 1≤ i≤ e and consider the set B_i={p:∀ v∈ E_i,^c∖{0}, χ(p,v)<C/ℰ}. Write B_i=∪_j=1^ B_i,j where B_i,j={p:∀ v∈ E_i,^c∖{0}, χ(p,v)<C/ℰ(1-1/j)} and assume for the sake of contradiction that μ(B_i)>0; then for some j it holds μ(B_i,j)>0.By absolute continuity of the foliation ^u_ (see for example <cit.>) and the hypothesis on M_0 there exists an interval L inside an unstable leaf with L⊂ M_0 and so that L∩ B_i,j has positive intrinsic Lebesgue measure in it. Take a density point b∈ B_i,j∩ L for the measure in L. Given an admissible curve γ we call p_∈γ its center if Leb([γ(0),p_])=Leb(γ)/2, where [γ(0),p_] denotes the (oriented) interval inside γ. For ϵ>0 small and some k large to be specified later consider ^:[-,]→ M, ^(t)=^-k∘_k(t) where _k is the admissible curve with center ^k(b). Note that Leb(^) decreases with k, and even though this curve is not necessarily symmetric with respect to b, the ratio of the length of the intervals [^(-),b],[b,^(ϵ)] is close to one, by Lemma <ref> and almost conformality ofon its unstable foliation. Thus by Lebesgue's differentiation theorem,Leb(γ^ϵ∩ B_i,j)/Leb(γ^ϵ)1, which implies if γ^ϵ small enough (or equivalently k sufficiently large) Leb(^∩ B^c_i,j)/Leb(^)<C/j·ℰ·d|E^c_. Note also that for every point p∈_k one has J^u_^-k(p)≥Leb(γ^ϵ)/ℰ· Leb(_k) Take v∈ T_bM such that d_b^k(v)∈Δ_i,(b). We claim that χ(b,v)≥C/ℰ(1-1/j), which gives a contradiction since b∈ B_i,j. By contradiction, suppose that v is as before and extend it to a continuous vector field X over ^-k∘_k(t) with the property that (_k,_∗^kX)is an adapted field satisfying _∗^kX∈Δ_i, (this is possible since Δ_i, is continuous). Consider for p∈γ^ϵ the quantity χ(p)=lim sup_n 1/nlogd^n ∘ X∘^k(p). We compute, using (reverse) Fatou's Lemma ∫_^χ d^= ∫_^k^χ∘^-k J^u_^-kd(^k^)≥Leb(γ^ϵ)/ℰ· Leb(_k)∫__kχ∘^-k d(_k)≥Leb(γ^ϵ)/ℰ· Leb(_k)· C· Leb(_k)=C/ℰ· Leb(γ^ϵ) On the other hand, since χ(p)<C/ℰ(1-1/j) for p∈ B_i,j, ∫_^χ d^=∫_^∩ B_i,jχ d^+∫_^∩ B_i,j^cχ d^< C/E(1-1/j)Leb(^∩ B_i,j)+d|E^c_Leb(^∩ B_i,j^c)<C/ℰLeb(^) which is absurd. We have thus proved that μ(B_i)=0. For a center Lyapunov exponent χ_i^c denote mult_i its multiplicity; then μ({p∈ M:∑_χ^c_i(p)≥ C/ℰmult_i(p)≥ e})=μ((∪_i=1^e B_i)^c)=1. If the reader wants to consider the more general case (cf. Remark <ref>) S_i,r:^d_i→^d_i, d_i≥ 2 where Δ_i,r is a cone centered around a subspace in W_i⊂ E_i,^c, the set B_i in the previous proof has to be replaced by B_i={p∈ M:∑_χ^c_i(p)< C/ℰmult_i(p)> E_i,^c- W_i} and the argument follows along the same lines to give that for almost every p there exists at least ∑_i^e W_i center positive exponents larger that C/ℰ. §.§ Study of the integralRecall (<ref>) and the definition of Y^k given in (<ref>). One can write I_n(,X)=∑_k=0^n-11/||∫_logd_^kp(Y^k∘^k(p))d= ∑_k=0^n-11/||∫_f^klogd_p Y^kJ^u_^-kd, therefore by (<ref>)I_n(,X)= ∑_k=0^n-1(R_k+∑_j=0^N_k1/||∫__j^klogd_p(Y^k)J^u_^-kd_j^k ),with R_k= 1/||∫_^k_N_k+1logd_p(Y^k)J^u_^-k d_N_k+1^k (observe that byLemma <ref>, for every 0≤ k<n,0≤ j≤ N_k the pair (γ_j^k,Y^k) is an adapted field). It will be convenient to introduce the following notation. If (γ,X) is an adapted field then I(γ,X):=1/|γ|∫_γlogd_p(X_p)dγ.Using Corollary <ref> we deduce that for every j, k|_j^k|/||≥ 1-ϖ(r) and|_N_k+1^k|/||≤ 1+ϖ(r),thus (again, re-defining ϖ)I_n(,X)≥∑_k=0^n-1(R_k+(1-ϖ(r)) ∑_j=0^N_kmin__j^k (J^u_^-k) · I(_j^k, Y^k) ) ). It holds lim_n→∞1/n∑_k=0^n-1 |R_k|=0. We compute |R_k| ≤ (1+ϖ(r))·max__N_k+1^k | J^u_^-k|·logd |E^c_, This implies that |R_k| converges to zero as k goes to infinity, hence so does its average. It follows, lim inf_nI_n(,X)/n≥(1-ϖ(r))lim inf_n1/n∑_k=0^n-1∑_j=0^N_kmin__j^k (J^u_^-k) · I(_j^k, Y^k)).§.§ Good and bad vector fieldsWe will now check that for each i the cone Δ_i,⊂ E_i,^c induced by Δ_i has the property that if (,X) is an adapted field with X∈Δ_i, then the right hand side term in the previous inequality is positive, thus showing (thanks to Proposition <ref>) the existence of e positive Lyapunov exponents. Only now the more specific aspects of the dynamics S_r (the existence of the cones) enters in consideration.There exists a positive constant Q=Q(r)>0 such that for every k≥ 0, for every admissible field (γ,X) satisfying X∈Δ_i,, it holds ∑_k=0^N_kmin__j^k(J^u_^-k· I(_j^k, Y^k) )≥ Q.The above proposition will be proven through a series of Lemmas. A vector X_p∈ E_^c(p) can be written uniquely asX_p=X_p,1+⋯ X_p,e X_p,i∈ E_i,^c(p). Recall that in V=E^c_f we are using the ℓ^ norm in associated to the decomposition V=⊕_i=1^e^2_i. Using T_, we induce the correspondingℓ^ norm in E_^c associated with the decomposition E_^c=⊕_i=1^e E_i,^c(p), i.e. X_p=max_i=1,⋯, eX_p,i.We say that X_p,i is a leading component of X_p if X_p,i=X_p. By Corollary <ref> we deduce the following. For r sufficiently large it holds that if (,X) is an adapted field forand for some p∈ the vector X_p,i is a leading component of X_p, then p'∈⇒X_p',i≥ 1-ϖ(r).Let (,X) be an adapted vector fieldfor . * We say that X_·,i is a leading component of X if there exists p∈ such that X_p,i is a leading component of X_p. * X is called good if it has some leading component X_·,i satisfying ∀ p∈ X_p,i∈Δ_i,(p). Otherwise it is called bad. The functions β_i(r)>ζ_i(r) are specified in condition S-2. For r sufficiently large and ϕ(r) small it holds for every adapted field (,X) forwith leading component X_·,i, * I(,X)≥ (1-10^-23)logζ_i(r). * If moreover (,X) is good, then I(,X)≥ (2-10^-23)πlogβ_i(r). Denote by π_i: E^c_→ E_i,^c the projection. Since the bundles _i⊂ V are df-invariant, for ϕ(r) sufficiently small we have that for i≠ j π_j∘ d∘π_i≈ 0, and inf_p m(π_i ∘ d_p∘π_i )≈inf_p m(d_pf|^2_i)= ζ_i(r). One then has d_p(X_p)≥π_i∘ d_p(X_p)≈π_i∘ d_p(X_p, i)≥ a(r)ζ_i(r) where a(r)1; to conclude the first part it is enough then to choose ϕ(r) is sufficiently small so that a(r)≥1/√(ζ_i(r)). Arguing similarly we also obtain that if (,X) is good then for p∉𝒞_i, d_p(X_p)≥a'(r)β_i(r) with a'(r)1; note that the projection ofon _i may be doing slightly more than one turn (cf. Remark <ref>), but since the length of this additional part converges to zero as r +, we can assume as a worse case scenario situationthat the additional part is contained in 𝒞_i,r, and hence I(,X) ≥ (2π-2l(𝒞_i))(a'(r))·logβ_i(r)-2l(𝒞_i)(1-10^-23)·logζ_i(r)≥ (2-10^-23)πlogβ_i(r) if r is sufficiently large (hence l(𝒞_i) small), and ϕ(r) small. Given an adapted vector field (,X) and k≥ 0 we defineG_k=G_k(,X)={(^k_j,Y^k|_j^k) good adapted field: 1≤ j ≤ N_k}B_k=B_k(,X)={(^k_j,Y^k|_j^k) bad adapted field: 1≤ j ≤ N_k}cf. Lemma <ref>. Note # G_k+# B_k= N_k. For every adapted field (,X) and every positive integer k≥ 0, it holds 1-10^-23≤∑_j∈ G_kmin__j^kJ^u_^-k+ ∑_j∈ B_kmax__j^kJ^u_^-k≤ 1+10^-23. provided that ϕ(r) is small and r large. By Lemma <ref> and <ref> 1 =1/||∫_d=1/||∑_k=1^N_k+1∫__j^kJ^u_^-kd_j^k ≥(∑_j∈ G_k|_j^k|/|γ|min__j^kJ^u_^-k+ ∑_j∈ B_k∪{N_k+1}|_j^k|/|γ|min__j^kJ^u_^-k)≥ (1-ϖ(r)) (∑_j∈ G_kmin__j^kJ^u_^-k+ 1/ℰ∑_j∈ B_kmax__j^kJ^u_^-k). Using that ℰ 1 the first inequality follows. The second one is similar. The next lemma takes care of the transitions between good and bad vector fields. Recall the definition of R,l(𝒞_i,r) given in S-1 and denoteρ(r):=e·max_1≤ i≤ el(𝒞_i,r)/2π.For sufficiently large r and correspondingly small ϕ(r) the following holds. (a) If (,X) is a good adapted field then there exists a relatively open set _g⊂ of length |_g|≥ (1-ρ(r))|| such that if ^-1^1_j⊂_g then(^1_j,_∗X/_∗X) is good. (b) If (,X) is a bad adapted field then there exists a relatively open set _bg⊂ of length |_g|≥ 0.99R|| such that if ^-1^1_j⊂_bg then(^1_j,_∗X/_∗X) is good. We first deal with the case =f. Write proj_i:M→_i the projection and consider (,X) a good adapted field such that X_·, k is a leading component.By hypothesis S-2 on the map S_k, the vector field (^1_j,Y=f_∗X) has its k-th component inside Δ_k provided that proj_k(f^-1^1_j)∉𝒞_k.It is no loss of generality to assume that for 1≤ i≤ e proj_i'(p)∉𝒞_i', ∀ 1≤ i'≤ e⇒d_pS_i|CΔ_i<min_i'≠ i m(d_pS_h|Δ_i') otherwise Δ_i can be enlarged while preserving its expanding character. Define _g={p∈: proj_i(p)∉𝒞_i, 1≤ i≤ e}. Then |_g|≥ (1-ρ(r))||, and by the discussion above if f^-1^1_j⊂_g then either the Y_k is its leading component and thus (^1_j,Y) is good, or the leading component of Y is Y_k', where X_k' was expanded and hence was inside Δ_k'. In any case Y has a leading component Y_i inside a cone Δ_i which implies that (^1_j,Y) is good. This completes the proof of the first part for the unperturbed case. For a perturbation we denote π_i: E_^c→ E_i,^c the projection, and notice that π_i(_∗π_iX) is uniformly close to T_f_∗T_^-1(π_iX), thus we can reduce to the previous case. To deal part (b) we first consider =f: we know that for all leading components X_·,i of X there exists p_0 such that X_p_0∈CΔ_i^ς_i(p_0), ς_i=+,-. Now for any other p∈ Im(), d_pf(X_p)=d_pf(X_p-X_m_0)+d_pf(X_p_0) and thus by Corollary <ref> the first term of right hand side is very small, hence d_pf(X_p)≈ d_pf(X_p_0). Consider ℬ_i,r^+, ℬ_i,r^-, the bands defined in condition S-1, and note that if proj_i(^-1^1_j)⊂ℬ_i,r^ς_i then the i-th component of (^1_j,Y) is properly contained into Δ_i, away from its boundary. Define _bg=_g∩⋃{proj_i^-1ℬ_i,r^ς_i:i/ X_ileading}and note that for r large |_bg|≥ 0.99R||. If ^-1^1_j⊂_bg, then we have two possibilities: * exists i such that X_i,Y_i are leading components of X,Y. Then by the previous argument together with S-2 we have that Y_i is completely contained in a expanding cone, hence (^1_j,Y) is good, or * the leading components of Y come from non-leading components of X. In this case necessarily these non-leading components are in expanding cones, hence by the invariance of these cones we also deduce that (^1_j,Y) is good. The result follows. Arguing as for the first part we can deal with the perturbative case. Compare Lemma 12 in <cit.>.Consider now a good adapted field (,X) with X∈ E_i,^c. We can estimate ℰ∑_j∈ G_1min__j^1J^u_^-1≳ (2π-l(𝒞_i,r))1/ℰ∑_j∈ B_1max__j^1J^u_^-1≲ l(𝒞_i,r).From this we deduce that for sufficiently large r (and ϕ(r) small), it holds∑_j∈ G_1min__j^1J^u_^-1>σ∑_j∈ B_1max__j^1J^u_^-1where σ is the natural number given in condition S-1. We continue to work with r so the above holds.Armed with the two previous lemmas now we will establish the following, which almost immediately implies Proposition <ref>. If r is sufficiently large and its corresponding ϕ(r) is sufficiently small, then the followingholds for every (,X) good adapted field for : * ∑_j∈ G_kmin__j^kJ^u_^-k≥ (1-10^-23)σ/σ+1. * ∑_j∈ G_kmin__j^kJ^u_^-k≥σ∑_j∈ B_kmax__j^kJ^u_^-k. We start noticing that the first part is consequence of the second together with Lemma <ref>. We argue by induction. The base case is just the hypothesis, so assume that we have established the claim for k≥ 0. By Lemma <ref> ∑_j∈ G_k+1 |γ_j^k+1| min_γ_j^k+1 J^u_^-k-1≥1/ℰ∑_j∈ G_k+1∫_γ_j^k+1 J^u_^-k-1 d(^k+1_j)≥1/ℰ∑_j∈ G_k+1∑_t∈ G_k∫_γ_j^k+1∩(γ_t^k) J^u_^-k-1 d(^k+1_j)=1/ℰ∑_t∈ G_k∑_j∈ G_k+1∫_^-1(γ_j^k+1)∩γ_t^k J^u_^-k d(^k_t)≥1/ℰ∑_t∈ G_kmin__t^k (J^u_^-k)∑_j∈ G_k+1|^-1(γ_j^k+1)∩γ_t^k|≥1/ℰ∑_t∈ G_kmin__t^k(J^u_^-k)·(1-ρ)·|γ_t^k| where in the last line we have used Lemma <ref>. Since the length of admissible curves is comparable (cf. Corollary <ref>), we finally obtain ∑_j∈ G_k+1min_γ_j^k+1 J^u_^-k-1≥1-ϖ/ℰ(1+ϖ)(1-ρ)·∑_t∈ G_kmin__t^k(J^u_^-k) On the other hand, and arguing in the same way ∑_j∈ B_k+1 |γ_j^k+1| max_γ_j^k+1 J^u_^-k-1≤ℰ∑_j∈ B_k+1∫_γ_j^k+1 J^u_^-k-1 d(^k+1_j)≤ℰ∑_j∈ B_k+1∑_t∈ G_k∫_γ_j^k+1∩(γ_t^k) J^u_^-k-1 d(^k+1_j)+ℰ∑_j∈ B_k+1∑_t∈ B_k∫_γ_j^k+1∩(γ_t^k) J^u_^-k-1 d(^k+1_j)+ℰ∫_(γ_N_k+1^k)) J^u_^-k-1 d((γ_N_k+1^k))=ℰ∑_t∈ G_k∑_j∈ B_k+1∫_^-1(γ_j^k+1)∩γ_t^k J^u_^-k d(^k_t)+ℰ∑_t∈ B_k∑_j∈ B_k+1∫_^-1(γ_j^k+1)∩γ_t^k J^u_^-k d(^k_t)+ℰ∫_γ_N_k+1^k J^u_^-k d(γ_N_k+1^k)≤ℰ^22ρ∑_t∈ G_kmin__t^k(J^u_^-k)·|γ_t^k|+ℰ·2π-0.99R/2π∑_t∈ B_kmax__t^k(J^u_^-k)|γ_t^k|+ℰmax_γ_N_k+1^k(J^u_^-k)· |γ_N_k+1^k|. Choose ϕ(r) sufficiently small so that for every k≥ 0, max_M(J^u_^-k)≤(1-10^-23)σ/σ+1λ^-k/2 and use Lemma <ref> with the induction hypotheses to conclude ∑_j∈ B_k+1max_γ_j^k+1 J^u_^-k-1≤1+ϖ/1-ϖ·ℰ·( 2 ℰρ+2π-0.99R/2πσ+λ^-k/2)∑_t∈ G_kmin__t^k(J^u_^-k) and hence, putting together (<ref>), (<ref>) and using that ℰ 1, ∑_j∈ G_k+1min_γ_j^k+1 J^u_^-k-1/∑_j∈ B_k+1max_γ_j^k+1 J^u_^-k-1 ≥(1-ϖ/1+ϖ)^21-ρ/ℰ^22π/4πℰρ+2π-0.99R/σ+2πλ^-k/2>(1-10^-23)2π(1-ρ)/(2π-0.99R)·σ>σ if ϖ(r),ρ(r) are sufficiently small. Finally we are ready to finish the proof of Proposition <ref>. Define β:=min_1≤ i≤ eβ_i, ζ:=min_1≤ i≤ eζ_i and let (,X) be a good adapted field forwith leading component X_i∈Δ_i,. We compute using Lemma <ref> and Proposition <ref> ∑_j=0^N_kmin__j^kJ^u_^-k· I(_j^k, Y^k)=∑_j∈ G_kmin__j^kJ^u_^-k· I(_j^k, Y^k)+∑_j∈ B_kmin__j^kJ^u_^-k· I(_j^k, Y^k)≥(2-10^-23)πlogβ(r)+1/σ(1-10^-23)logζ_i(r))∑_j∈ G_kmin__j^kJ^u_^-k· I(_j^k, Y^k)≥(6logβ(r)+1/σlogζ(r))(1-10^-23)σ/σ+1=(1-10^-23)σ/σ+1log(β(r)^6ζ(r)^1/σ) and the later quantity is positive, if r large enough by the last part of S-1. § ACKNOWLEDGMENTS The results here presented are based on previous joint work with Pierre Berger, and are deeply influenced by several discussion that the author had with him during these times. I would like to thank Pierre for sharing his ideas with me.Initial stages of this paper were prepared while I was visiting PUCV-Valparaiso; I would like to thank Carlos Vazquez for his encouragement and generosity in these moments. Also, I would like to thank Enrique Pujals and Jiagang Yang for encouragement and the interest deposited in this project.Finally, I would like to express my sincere thanks to the referees who not only gave me many suggestions to improve the presentation and caught inaccuracies-plain errors, but also gave me ideas on how to improve the results appearing on previous versions. § APPENDIX Given a coupled family {f_r=A_r×_φ_r S_r}_r over an hyperbolic base,it is desirable to have some conditions that will imply non-uniform hyperbolicity of the family instead of only positive exponents along the fiber direction. We discuss a possible approach here.We assume that both {S_r}_r,{S_r^-1}_r satisfy S-1,S-2, A_r∈ SL(2,)hyperbolic, and define the following conditions. A-3 * dS^-1_r^3·dS_r/λ_rφ_r|E^s_A_r0,dS^-1_r·dS_r·φ_r|E_A_r^u/λ_r0. * There exists q∈ℕ such that dS_r^3qd^2S_r^-1^3q/λ_r0. A-4 * min_1≤ i≤ eP_i∘ dS_r^-1∘φ_r|E^s_A_r>0. * max_1≤ı≤ eP_i∘ dS_r^-1∘φ_r|E^u_A_r+P_i∘ dS_r^-1/λ^2P_i∘ dS^-1∘φ_r|E^s_A_r0. * min_1≤ i≤ eP_i∘ dS^-1∘φ_r|E^s_A_r/max_1≤ i≤ eP_i∘ dS^-1∘φ_r|E^s_A_r 1. We say the the coupling in a family of skew-products {f_r=A_r×_φ_r S_r}_r is bi-adapted if it is adapted and moreover the family satisfies A-3,A-4 above. In this case we also say that {f_r} is a bi-adapted family.The lack is symmetry between these and A-1,A-2 comes from the different form of df and df^-1; compare (<ref>) with (<ref>).It is direct (although somewhat tedious) to check that if f_r verifies A-1 to A-4 then df_r^-1 verifies A-1,A-2.We can thus apply our Main Theorem to both {f_r}_r, {f_r^-1}_r and deduce the following. Assume that {f_r=A_r×_φ_r S_r:M=^l×^2e→ M}_r is a bi-adapted family with A_r∈ SL(2,) hyperbolic and the families {S_r}_r,{S_r^-1}_r satisfy conditions S-1,S-2. Then there exists r_0 such that for every r≥ r_0 there exists Q(r)>0 satisfying for Lebesgue almost every m∈ M v∈ T_mM∖{0}⇒lim_n|logd_mf^n(v)/n| >Q(r). In particular f_r is NUH and has a physical measure.The same holds for anyin a 𝒞^2 neighborhood 𝒰_r of f_r. The previous Corollary is given for completeness. In practice however, checking A-4 could be difficult since it depends on the relation between dS_r^-1 and dφ_r|E^u_A, and this control may not be achievable. This iswhy in the examples given in Section 3 we appeal to other arguments to deal with the inverse map.alpha | http://arxiv.org/abs/1705.09705v5 | {
"authors": [
"Pablo D. Carrasco"
],
"categories": [
"math.DS",
"37D30, 57R30"
],
"primary_category": "math.DS",
"published": "20170526201221",
"title": "Random Products of Standard Maps"
} |
Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, JapanThe structural and electronic properties of amorphous silicon () are investigated by first-principles calculations based on the density-functional theory (DFT), focusing on the intrinsic structural defects. By simulated melting and quenching of a crystalline silicon model through the Car-Parrinello molecular dynamics (CPMD), we generate several differentsamples, in which three-fold (), five-fold (), and anomalous four-fold () defects are contained. Using the samples, we clarify how the disordered structure ofaffects the characters of its density of states (DOS). We subsequently study the properties of defect complexes found in the obtained samples, including one that comprises threedefects, and we show the conditions for the defect complexes to be energetically stable. Finally, we investigate the hydrogen passivation process of thedefects inand show that the hydrogenation ofis an exothermic reaction and that the activation energy for a H2 molecule to passivate twosites is calculated to be 1.05 eV. N/A Analysis of single and composite structural defects in pure amorphous silicon: a first-principles study Yu-ichiro Matsushita December 30, 2023 ======================================================================================================= § INTRODUCTIONAmorphous materials, which lack long-range structural orders but still keep short-range orders, have been investigated for almost a half century <cit.>. An important problem in the physics of such disordered materials is the understanding of their structural characteristics that incorporates the short-range order in the disordered atomic network and of their influence on the electronic properties. Despite the substantial progress made in the past, our understanding of how the structural characteristics affect the electronic properties of disordered materials is still incomplete.Amorphous silicon () is an important example of such amorphous materials. It shows attractive properties that are distinct from crystalline silicon (), such as the high light absorption coefficient and the large bandgap. Its low deposition temperature and low fabrication cost, as well as those physical properties, allow the industry to use this material for thin-film devices including solar cells and transistors. Upon miniaturization of electronic devices, clarification of how the atom-scale structural characteristics influence its electronic properties ofis crucial also from a technological viewpoint.Accurate determination of the local atomic structures ofis a prerequisite of any quantitative theoretical approach to its physical properties. In the early days, continuous-random-network models <cit.> were used to consider the local structures of amorphous materials. Then molecular dynamics (MD) <cit.> or Monte Carlo <cit.> techniques combined with empirical interatomic potentials <cit.> are used to obtain the radial distribution functions of . However, the validity of the empirical potentials is always an issue and a “try and error” approach has been continued.The MD approach based on the first principles of quantum theory gets rid of the problem of the interatomic potentials. Car and Parrinello invented a scheme in which the electron-electron interaction is treated in the density-functional theory (DFT) <cit.> and Hellmann-Feynman forces are used to track the dynamics of ions <cit.>. In this scheme, a fictitious mass of the wavefunction (Kohn-Sham orbital) is introduced to perform efficient first-principles MD (Car-Parrinello molecular dynamics, CPMD) simulations. The CPMD scheme has been applied to<cit.> and the obtained radial distributions up to the nearest neighbor distance agree with the experiments <cit.> satisfactorily. Other Born-Oppenheimer MD simulations based on the DFT have been performed forand the nature of the short-range order has been partly clarified <cit.>. These MD simulations show that while the Si atoms mostly form four-foldconfigurations, whose bond angles are around 110 deg, there are three kinds of structural defects inas well: The three-fold (), the five-fold (), and the anomalous four-fold () sites <cit.>. The difference between thesites and thesites is that the bond angles of the latter are strongly distorted than those of the former. These structural defects may induce deep levels in the energy gap and be responsible for the conduction and valence band tails <cit.>. The deep levels are indeed observed by the electron paramagnetic resonance (EPR) measurements <cit.> and identified as eitherorconfiguration <cit.>.A problem in the first-principles MD simulations described above is the high defect density appearing in the simulation cell: The density is an order of which is higher by two orders of magnitude than the experimentally determined value <cit.> 10^19 cm^-3. In the CPMD simulations,is prepared by heatingto melt and then quenching the obtained liquid. The discrepancy in the defect density from the experimental situation is mainly due to the unrealistic quenching rate in the CPMD simulations, which is usually more than 100 K/ps. In this work, we cool liquid silicon with the speed of ∼ 10 K/ps, which is slower than those used to preparesamples in the past, and show that defect-free structures can be certainly obtained by the CPMD. Furthermore, using the generated samples, we tackle several questions inthat have not been understood fully.Theoretical efforts were made to clarify the relationships between the structural and the electronic properties of<cit.>, and it was revealed that the disordered network of Si atoms is a key to understanding its electronic states. In this paper, we make a quantitative analysis of the electronic properties ofand make a clearer explanation of how the geometric properties ofaffects its density of states (DOS) in association with its atomic configurations.One of the important things that characterize the structural property ofis the formation of the structural defects. It is commonly assumed that thedefects are prevalent , and previous researches have paid less attention to the other two types of the defects. Moreover, little is known about the complexes that these structural defects possibly compose. Study of defect complexes is important in that they might help us to understand the how defects are spacially distributed or how likely they are to be stable in a particular configuration. To give further insights into these problems, we perform calculations focusing on the defect complexes found in oursamples.Another topic relevant to the defects is the effect of hydrogenation. Defects inare the origins of deep levels in the mid-gap, which contribute to lowering the mobility ofand thus to degrading its quality as a material for semiconductor devices. Therefore, industrially fabricatedcontains a large number of H atoms which passivate the defects in the structure with. The deep levels are known to be made up of two electronic states: the dangling bonds and the floating bonds, the latter of which is a state derived from thedefects <cit.>. While H atoms are believed to mostly passivate thedefects, not much attention has been paid to the passivation ofdefects. As some researchers have mentioned earlier <cit.>, however, thedefects might be prevalent inand play a major role in forming the deep levels. We provide discussions on how likely thedefects are to be passivated by H atoms from an energetic point of view.The outline of this paper is as follows. In Section <ref>, the calculation methods and the process of generatingsamples are described. In Section <ref>, the structural properties of the obtained samples are investigated. In Section <ref> we analyze the correspondence between the DOS and the structural properties of . In Section <ref>, we take a closer look at the electronic states near the Fermi energy, particularly focusing on the defect levels. In Section <ref>, the stabilities of the defects are studied. In Section <ref>, hydrogenation effect of thesites are discussed. A summary and conclusions are given in Section <ref>. § COMPUTATIONAL DETAILSWe have used our RSDFT (Real-Space Density-Functional Theory) package <cit.>, in which the Kohn-Sham equation based on the DFT <cit.> is calculated under the real-space scheme <cit.>. In the real-space scheme, discrete grid points are introduced in the real space, and the wavefunction is expanded on the mesh in the real space. To simulate the structures of , CPMD calculations have been done using RS-CPMD (Real-Space Car-Parrinello Molecular Dynamics) code, which is incorporated in RSDFT package. In this work, we have set the mesh size as 0.41 Å, which corresponds to the cutoff energy of 31.7 Ry. We have used PBE exchange-correlation functional <cit.> and norm-conserving pseudopotential in both static and dynamic calculations. Brillouin zone (BZ) sampling has been done for the Γ point. We have confirmed that the total energy of the systemconverges within 0.2 eV/cell with our calculational settings.Amorphous samples have been obtained by melting and quenching a crystalline structure through the CPMD simulations. As for the initial structure, we have prepared a 3 × 3 × 3 supercell containing 54 Si atoms/cell. The volume of the supercell has been fixed as 1.08 [nm^3], which is consistent with the experiments. The time step in the simulations has been set to be 0.1 fs, and the temperature has been controlled by velocity scaling.We have started heating the system from 500 K and increased the temperature with the heating rate of 125 K/ps until it reaches 1700 K. At 1700K, we have heated for another 5.8 ps to sufficiently liquify the system. We have subsequently cooled the system until the temperature has reached 500 K. Here, we have employed four different cooling rates separately: 20.0, 16.7, 14.3, and 12.5 K/ps. We note that these speeds are slower than what have ever been applied for the CPMD simulation ofreported by other groups. Finally, we have relaxed the structures of the final step of the cooling to obtain the stable atomic configurations. We refer to the samples obtained from each cooling rate 20.0, 16.7, 14.3, and 12.5 K/ps as 20, 16, 14, and 12, respectively. After the relaxation, each sample has been heated at 300K for 2.0 ps to calculate the radial distribution and the angle distribution, both of which are time-averaged functions of the atomic positions. § STRUCTURES OF THE OBTAINED SAMPLESThe radial distribution g_r and the bond angle distribution g_a of each sample are shown in Figure <ref>. All the four calculated plots of g_r are in good agreement with the experimental result <cit.>. The sharp peaks are located at 2.31, 2.33, 2.32, and 2.36 Å for 20, 16, 14, and 12, respectively. Considering the experimentally obtained first-neighbor distance in , 2.35 Å, these indicate that the short-range order is preserved in every amorphous sample. Furthermore, we can clearly observe the second and third peaks near r = 3.8 and 5.8 Å. These broad peaks reflect the deviation in the bond length and the bond angle.Every plot of g_a shown in Figure <ref> (b) has a broad peak around 100 deg. The peak position is close to the bond angle in , 109.5 deg, which indicates that the majority of the Si atoms in the samples retain the nearly-tetrahedral bonds.Figure <ref> (c) shows the distribution of the rings composed of n (n=3,…,8) Si atoms in each sample. We find that dominant rings are those composed of five or six atoms in every sample. We also notice that three-membered rings are found only in 20 and 12, which can be associated with the small peaks around 60 deg in Figure <ref> (b). Defects found in the three samples are illustrated in Figure <ref>. In this paper, we define asite as one which satisfies the following inequalityΩ = ∑_(i,j,k)Ω_ijk < 4π.Here, Ω is the “solid angle” [sr] at thesite which can be described using Figure <ref>. Figure <ref> schematically shows asite, named O, and the four neighboring sites A_1, A_2, A_3, and A_4. P_i (i=1,2,3) is the point where a vector OA_i passes through the unit sphere S, represented by the dashed circle. Ω_123 is the area of the spherical triangle P_1P_2P_3, colored by orange. Ω is the sum taken over for all the combinations of the four neighboring sites. By this formulation, for those that retain nearly-tetrahedral bonds, such as the one illustrated in Figure <ref>, Ω is equal to 4π. On the other hand, those that contain heavily distorted bonds, such as the one shown in orange in Figure <ref>, satisfy Eq. (<ref>).The numbers of the defects (N_, N_, N_), the defect concentration ρ_d, and the total energy E_tot of each obtained sample are listed in Table <ref>. All the three kinds of the defects are incorporated into the calculation of ρ_d. Table <ref> reveals that 14 contains no defect, in contrast to the other three structures. We also find thatdefects are prevalent in all the defect-containing samples.E_tot of 14, the defect-free structure, is found to be the lowest of all. The highest is that of 12, which contains the largest number of the defects of all the samples. This implies that the existence of the defects increases the energy and thus decreases the energetic stability of the structure. Furthermore, although it is expected that slower cooling rate makes the smaller amount of the defects, which is why we have introduced very slow cooling rates for this study, the defects are most abundant in 12, which has been produced with the slowest cooling speed. We have found that this is of statistical occurrence as described below.We have additionally generated ten differentsamples employing the same cooling rate for 20. The numbers of the defects and the total energy of each obtained sample are listed in Table <ref>. The variance of E_tot is 0.78 eV, and the difference between the maximum and the minimum energy is more than 3.0 eV. Moreover, two samples are found to be free of defect, while the others contain 2 to 5 defect sites. This way, we have demonstrated that the number of the defects and the total energy fluctuate even with the fixed cooling rates, and this suggests that the inconsistency between the total energy and the cooling speed in the former four samples can be explained as a statistical error. § THE DENSITY OF STATESFigure <ref> shows the density of states (DOS) of each obtained sample. Every plot has a strong peak at ∼ -2 eV, which we shall call a “high-energy” peak, and a weaker, broader hump below -6 eV, which we shall call a “low-energy” hump. These features make the DOS ofdistinct from that ofshown in Figure <ref>. In the valence bands, the DOS ofpossesses two low-energy peaks below -6 eV, with a valley in between, and a high-energy peak around -2 eV. Joannnopoulos et al. <cit.> made the following explanation for the difference of the shape of DOS betweenand . They compared several polytypes ofand found that the DOS of ST-12 structure, which contains a five-membered ring in a unit cell, is similar to that ofin that the two low-energy peaks that are used to exist inDOS are merged into one broad hump. Then they concluded that the existence of odd-membered rings in the structure contributes to filling the gap between the two peaks at the lower energy.Here, we give two perspectives on this argument. Firstly, odd-membered rings are not necessary to fill the valley in the DOS. In fact, a polytype ofcalled bct-Si <cit.>, which contains only even-membered rings in the geometry, does not hold a valley in the low-energy part of its DOS; it is filled by another peak, as shown in Figure <ref>.The second argument, which is more physically fundamental, is that the shape of DOS strongly depends on the symmetry of the system and thus its band structure. We takeas an example here. The correspondence between the DOS and the band diagram is illustrated in Figure <ref>. We find that the position of the two low-energy peaks in the DOS ofcorrespond to the points in the band diagram where the energies of two bands flatten at the L point, a symmetric point in the BZ. We can understand that this is caused by Bragg reflection in the BZ.The DOS satisfies the following equation:DOS(E) ∝∫dS/| ∇_k E(k) |,where S, k, and E represent the iso-energy surface in the reciprocal space, a reciprocal vector, and the energy. Obviously from Eq. (<ref>), the DOS becomes larger with smaller |∇_k E(k)|, and it should be peaky near |∇_k E(k)| ≃ 0, which is the effect of Bragg reflection. The two peaks guided with dashed lines in Figure <ref> certainly demonstrate this. One consequence of Eq. (<ref>) is that if the system loses its symmetry, | ∇_k E(k) | no longer depends on k and thus the peaks in the DOS of the original structure will be vague or be totally lost. This explanation is well fitted to what we have observed in the DOS of : In contrast to , symmetry is lost inand therefore Bragg reflection does not occur, ending up having no strict peak in its DOS. This is one of the physical reasons that the DOS ofhas a hump instead of peaks.To summarize the point, the change of the DOS fromtocan be explained as the result of the loss of the symmetry of the system, rather than the emergence of the odd-membered rings.This conclusion, however, does not explain how exactly the atomic configuration ofaffects its DOS. In the following paragraphs, we clarify this point by analyzing the DOS in detail in association with the geometry.In Figure <ref>, we show partial density of states PDOS_s and PDOS_p, which are DOS projected onto either s or p atomic orbitals, each defined asPDOS_s(E) =∑_i^all atoms∑_j^occupied states|⟨ϕ^i_s|ψ_j⟩|^2 δ(E-E_j) PDOS_p(E) =∑_i^all atoms∑_j^occupied states|⟨ϕ^i_p|ψ_j⟩|^2 δ(E-E_j),where ϕ^i_s, ϕ^i_p are the s and the p orbital of the i-th atom, and ψ_j is the one-electron wavefunction of the j-th state. For all the four samples, we observe that PDOS_s has a broad peak in the low-energy region, which implies that the low-energy hump of the DOS ofis mostly made up of the contribution of the s orbitals. One possible reason for PDOS_s to have such a broad peak is that it has resulted from the variation in the Si-Si bond length, since the splitting of the bonding and antibonding energy becomes larger if two neighboring atoms get closer to each other, and vice versa.To examine this idea, we further decompose the PDOS_s into PDOS^i_s, which represents the contribution of atom i to PDOS_s, and find a correspondence between the geometrical configuration of atom i and the shape of PDOS^i_s. PDOS^i_s is defined asPDOS^i_s(E) =∑_j^occupied states|⟨ϕ^i_s|ψ_j⟩|^2 δ(E-E_j).Compared with Eq. (<ref>), this satisfies PDOS_s(E) = ∑_i^all atomsPDOS_s^i(E).Figure <ref> shows PDOS^i_s for each sample. In each plot, the green (purple) curve is for the atom named “long” (“short”), whose bond length with its neighboring atoms is the longest (shortest), r_max (r_min), in the sample. The ratio r_min/r_max of each sample is found to be within the range of 0.90 and 0.94. To quantify the distribution of each PDOS^i_s, we define its “center of mass” E_p^i asE_p^i = ∫_E_bottom^E_cutPDOS^i_s(E)· EdE/∫_E_bottom^E_cutPDOS^i_s(E)· dE.Here, we have set E_bottom as -12.5 eV and E_cut as -4 eV to avoid the influence of the sharp peak near the Fermi energy that are originated from the defects. Obtained values are presented in Table <ref>. We find that in every sample, E_p^short is lower than E_p^long. This implies that the peak position of PDOS^i_s slides farther from the Fermi level with the shorter bond length of the atom. These observations are consistent with the notion above and indicate a certain dependence on the bond length. Therefore, we conclude that the variation of the Si-Si bond length inbroadens the peaks of PDOS_s, which is the sum of all PDOS^i_s, and this finally results in the emergence of the low-energy hump in the DOS. § ELECTRONIC STATESWe take a look at the states near the Fermi energy, which are shown in Figure <ref>. For the defect-containing structures (20, 16, and 12) we observe small peaks, while a clear gap is seen for 14. The mid-gap states of 20, 16, and 12 are understood by examining their electronic states shown in Figure <ref>. We immediately find that all the states are localized around the defects in each sample.For 20 and 16, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are found to be localized at thesites. In contrast to the dangling bond found at thesite and thesite in 12 (Figure <ref> (c)), the electronic states are rather delocalized around thedefects, as pointed out in the past <cit.>. In 12, four localized states are found at ,andsites. We find that not all the defects are associated with the mid-gap states. Thesite in 20, pictured in Figure <ref> (a), makes no contribution in the gap states. For 12, however, we find that the HOMO and the second highest occupied molecular orbital (HOMO-1) are due to the existence of thesite. The important difference of thesite between 12 and 20 is that whereas in 20 it retains the nearly-tetrahedral bonds, in 12 it is heavily distorted, forming pyramid-like bonds with the neighboring atoms.Using 14, we are capable of evaluating the bandgap of . From the DOS shown in Figure <ref>, calculated using PBE functional for the exchange-correlation energy, we obtain the bandgap of 0.96 eV. To evaluate it more precisely, another calculation has been performed using HSE functional <cit.>, which produces the bandgaps of covalent materials with higher precision <cit.>. The same calculation has also been done for themodel. Obtained bandgaps are listed in Table <ref>, along with experimental data <cit.>. The calculated bandgap of 14 is 1.41 eV, which is larger than that ofby 0.39. The result is in good agreement with the experiments. § STABILITY OF COMPLEX DEFECT STRUCTURESIn 12, which contains four defect sites per supercell as illustrated in Figure <ref> (c), we find that the twosites and onesite comprise a defect complex, forming a three-membered ring. This result clearly shows the possibility of a defect complex consisting of twosites and onesite.We can say that this defect complex is made up of threesites because of the following reason. While theatom in 12 is bonded with its four neighboring atoms, it holds a dangling bond at the same time, which means that it is capable of forming another chemical bond with another atom using the unbonded hand. Therefore, we can regard thesite as a kind ofsite that lacks one neighboring atom and holds a dangling bond instead. Using this notion, then, we can treat the defect complex in 12 as a triangle made up of threesites. For the detailed analysis, we investigate the electronic properties and the stabilities of the defect complex observed in 12 by modeling “defect-only” structures as described below.We consider three simple molecules composed of onlysites: SiH5, Si2H8, and Si3H9. Each of them, illustrated in Figure <ref> - <ref>, is a model of an isolatedsite, a neighboringsites, and a three-membered ring ofsites, respectively. Each Si atom is accompanied with five atoms of H and Si in which H atoms compensate for the missing neighboring bonds that exist in the actual structure of . By analyzing each model, we make the following discussions for the energetic stability of thedefect complexes. In the calculations below, to perform reliable calculations for systems containing H atoms, we have set the mesh spacing of the real space as 0.27 Å, which corresponds to the cutoff energy of 72.5 Ry. §.§ SiH5First, we clarify the electronic states of SiH5. The relaxed structure of SiH5 has a hexahedron-like shape, in which the five H atoms are covalently bonded to the Si atom in a stable manner, as shown in Figure <ref> (a). The wavefunction of the HOMO presented in Figure <ref> (b) indicates that it is mostly the s orbitals of the H atoms that comprises the HOMO and that it is a non-bonding orbital, which makes no contribution to the chemical bonds in SiH5. The result is not only consistent with the tight-binding analysis using parameters in Ref. chadi1975, but also with the character of the HOMO and LUMO wavefunctions of 20 and 16, which we have confirmed to have delocalized amplitudes around thesites, as shown in Figure <ref> (a) and (b). §.§ Si2H8Next, we show that the dimer Si2H8 is unstable. We have calculated the dependence of the total energy of Si2H8 on the distance between the Si atoms, represented as d in Figure <ref> (a), by performing structural relaxations while fixing d at several values. The results are shown in Figure <ref> (b), from which we immediately find that the total energy becomes smaller with longer d, meaning that neighboring twosites is energetically unstable. §.§ Si3H9We finally show that Si3H9 is a stable configuration in the positively charged +1 state. We have found that in the neutral charge, where the total number of the valence electrons is an odd number, the HOMO is an anti-bonding state occupied with an electron and that it weakens the chemical bonds between the Si atoms. Figure <ref> (a) shows its relaxed structure. Although it retains the triangular shape, the Si-Si bond indicated by an arrow is 3.25 Å, which is much longer than what we have observed insamples.Another calculation has shown that by removing an electron from the structure, making it the positively charged state, the Si-Si bonds strengthens. The relaxed structure of Si3H9 in the positively charged state is presented in Figure <ref> (b). The three Si atoms form a nearly-equilateral triangle, whose bond lengths and angles are almost 2.62 Åand 60.0 deg, respectively. These traits are close to that of the triangle in 12, where the atomic distances are within the range between 2.43 and 2.63 Å and the bond angles between 54.7 and 62.9 deg.Therefore, we conclude that Si3H9 is stable when it is positively charged and that the charge state of defect complex in 12 is thought to be close to that.To summarize the point, by introducing three simplified models of thedefects, we have found that thesites are likely to be formed in isolation from each other or as a positively charged trimer in a similar way as in 12, and that they are unlikely to form a dimer. § THE EFFECTS OF HYDROGENATINGIn the previous sections we have observed that in our samples,is the most abundant of all the three kinds of the defects. In this section, we make a quantitative analysis of the effect of passivatingsites using H atoms, and examine whether this reaction energetically reasonable.We have chosen 20 as the target system for hydrogenation. To generate Si-H bonds in it, we have manually inserted two H atoms near thesites and relaxed the structure. Figure <ref> shows the defects in 20 before (left) and after (right) hydrogenation. Si atoms are indexed by integers n, which we shall call each of them Sin. We have found that two H atoms have broken into Si5-Si9 and Si3-Si10 bonds and that they have changed eachsite into asite, leaving the other bonds unchanged. Considering that the lengths of both the Si5-Si9 and the Si3-Si10 bonds before hydrogenation, 2.68 and 2.45 Å respectively, are longer than the first peak of the radial distribution, 2.31 Å, these two Si-Si bonds are thought to be weak enough to allow the H atoms to break them and newly form Si-H bonds that are stronger than themselves.The DOS before and after the hydrogenation are presented in Figure <ref>. We notice that the mid-gap peak, derived from thedefects, has been completely removed after the hydrogenation, which is consistent with the computational result in the past study <cit.>.To clarity whether this type of transition is likely to occur from the energetic viewpoint, we have calculated the activation energy for a hydrogen molecule to passivate thesites by applying the Nudged Elastic Band (NEB) method <cit.> implemented in VASP code <cit.>. The NEB method provides a geometric pathway from the initial to the final state, assuring the continuity of each reaction step by imposing restrictions between different steps.We have prepared a structure which contains a H2 near thedefects in 20, and set its optimized geometry as the initial step of the NEB calculation. The final geometry has been chosen to be the one shown on the right in Figure <ref>. Eight discrete steps have been imposed between the initial and final states. The calculated total energy at each reaction step is shown in Figure <ref> with respect to that of the final step. From the comparison of the total energy of the initial and the final step, we confirm that this is an exothermic reaction of 0.41 eV, which implies that the H2 molecule certainly terminates thedefects, leading to a stable atomic configuration. The activation energy for a H2 to passivatesites is found to be 1.05 eV. We find that the energy achieves the maximum at Step 4, whose geometry is shown in the top-right of Figure <ref>. At Step 4, H atoms are still close to each other and retain the H-H bond with the distance of 0.91 Å. The Si3-Si5 bond, in contrast, has been lost. The results suggest that the energetic barrier 1.05 eV has been used to cut the bond between thedefects. In the next step, illustrated in the bottom-left of Figure <ref>, each H atom approaches the different Si atoms, namely Si3 and Si5, each of them forming a Si-H bond. These are comparable to the diffusion energy of H in :H ∼1.5 eV <cit.>. Therefore, the termination we have observed in our calculations is a reasonable one.§ CONCLUSIONIn summary, we have performed first-principles calculations for . By CPMD simulation of melting and quenching of the crystalline structure, we have obtained four samples of , including one that contains no structural defect. The radial distributions of all the samples have been in good agreement with the experiments, and the angle distributions and the DOS have been consistent with the calculation result of the past studies. We have found that three-fold (), five-fold (), and anomalous four-fold () sites are generated in the samples and that thedefects are more abundant than the other kinds of defects. We have studied the origin of the transition of the shape of the DOS fromandand concluded that it is determined by the symmetry of the structure and is also certainly affected by the atomic configurations. We have confirmed the emergence of the mid-gaps states in those that contain defects. For the defect-freesample, the bandgap has been found to be 1.41 eV from the HSE calculations, which is in good agreement with the experiments. We have found a defect complex in a sample that consists ofandsites, and clarified that thesites can be stable in isolation from each other or by forming a trimer with a positive charge, and that the dimer ofsites is energetically unstable. Finally, the effect of hydrogenation on thesite has been investigated by introducing H atoms. We have found that two differentdefects have been hydrogenated through the exothermic reaction, and that the mid-gap states derived from thesite disappear. The activation energy for a H2 to passivate twodefects has been determined to be 1.05 eV. We would like to thank Professor Atsushi Oshiyama and Hirofumi Nishi for helpful discussions. This work has been supported in part by Ministry of Education, Culture, Sports, Science and Technology. 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"authors": [
"Yoritaka Furukawa",
"Yu-ichiro Matsushita"
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"published": "20170525171544",
"title": "Analysis of single and composite structural defects in pure amorphous silicon: a first-principles study"
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New Optimal Binary Sequences with Period 4p via Interleaving Ding-Helleseth-Lam Sequences Wei Su, Yang Yang, and Cuiling Fan W. Su is with School of Economics and Information Engineering, Southwestern University of Finance and Economics, Chengdu, China. Y. Yang and C.L. Fan are with the School of Mathematics, Southwest Jiaotong University, Chengdu, China. Email: [email protected], [email protected], [email protected] received May 28, 2017. December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================10000 =10000 The standard model of memory consolidation foresees that memories are initially recorded in the hippocampus, while features that capture higher-level generalisations of data are created in the cortex, where they are stored for a possibly indefinite period of time. Computer scientists have sought inspiration from nature to build machines that exhibit some of the remarkable properties present in biological systems. One of the results of this effort is represented by artificial neural networks, a class of algorithms that represent the state of the art in many artificial intelligence applications. In this work, we reverse the inspiration flow and use the experience obtained from neural networks to gain insight into the design of brain architecture and the functioning of memory. Our starting observation is that neural networks learn from data and need to be exposed to each data record many times during learning: this requires the storage of the entire dataset in computer memory. Our thesis is that the same holds true for the brain and the main role of the hippocampus is to store the “brain dataset”, from which high-level features are learned encoded in cortical neurons. § MEMORY AND LEARNING SYSTEMSdifferent types of memory and learning In the brain different types of memory are implemented. Based on duration, memory can be classified in short-term (or working) memory and long-term memory <cit.>. Based on content, memory is classified in declarative (further subdivided in episodic and semantic) and implicit or procedural <cit.>. Regarding learning, the number of ways in which the brain can learn exceeds the power of any classification system.hippocampus and cortex The hippocampus (a seahorse-shaped brain structure located in the medial temporal lobe) and the cortex (a 3 mm-thick layer of tissue distributed on the surface on the brain)are both involved in the process of memory formation. Overall, the empirical evidence seems to hint that the hippocampus stores complete and unprocessed memory records for a short time, while the cortex develops features capturing high-level generalisations of data <cit.>, that are stored for longer (possibly indefinite) periods. These generalisations may correspond to the schemas of Piaget's developmental theory <cit.>. neural networks Artificial neural networks (simply neural networks from now on) are computational models that take inspiration from the structure of biological neural networks. A deep neural network (Fig. <ref>) is a particular type of neural network (called “multilayer perceptron”) composed of a number of computational units called neurons, grouped in layers stacked on top of each other (input and output layers are always present, the number of intermediate layers defines the network's depth). deep learning, convolutional Deep neural networks represent the state of the art in many artificial intelligence applications, such as computer vision, speech recognition, natural language processing. For visual classification tasks, the best results are achieved by convolutional neural networks trained with “back-propagation” algorithm <cit.>, which appears to be less prone to the vanishing gradient problem <cit.> when huge amounts of data are used. A recent method <cit.> has achieved human-like performance on the ImageNet dataset using a network with 152 layers.mismatch Neural networks work well in a growing number of applications, yet their structure seems to lack biological plausibility. If neural networks were a good model for brain architecture, the brain should be characterised by a striped colour pattern, with an alternation of grey and white regions. The topological structure of Fig. <ref> could also map to an irregular, spaghetti-like geometrical structure in the brain: in this case, we would expect to observe a uniform (noisy) colour pattern. What we see, instead, is a distribution of grey matter concentrated on the brain's surface and in the central core, with the intermediate regions filled by white matter (Fig. <ref>).objective and structure Despite these apparent contradictions, we will argue that the structure of neural networks and the functioning of the associated algorithms are fully compatible with brain architecture. The rest of the paper is organised as follows: based on the known facts about the working of biological memory summarised in section 2 and on the functioning of neural networks described in section 3, we will present a proposal for brain architecture in section 4, which includes an unsuspected role for the hippocampus; section 5 discusses some implications for memory and learning in normal and pathological conditions; section 6 draws the conclusions and outlines future research directions. This work is intended for a readership of both neuro- and computer scientists: therefore concepts are presented with sufficient background information.§ MEMORY AND LEARNING IN THE BRAIN: HIPPOCAMPUS AND CORTEX hippocampus Hippocampi are located (more or less) in the centre of each brain hemisphere, in a region where fibres carrying multiple sensory inputs converge <cit.>. Each hippocampus is directly connected to a portion of cortex called entorhinal cortex, which functions as a connection hub to and from the rest of cortex. The size of the entorhinal cortex is relatively small and (presumably) only allows a subset of cortical fibres to reach the hippocampus <cit.>.patient H.M. A decisive contribution to our understanding of memory processes came from studies conducted on patient H.M., who had large portions of both medial temporal lobes (including the hippocampi) removed at age 27, in an attempt to cure severe epilepsy. The surgical procedure was successful in treating epilepsy, but left he patient with an almost complete anterograde amnesia and a graded retrograde amnesia. In other words, H.M. was unable to form new declarative memories and his recollection of recent past events was impaired, while older memories were intact <cit.>.standard model Based on these and other studies, the “standard model” of memory formation and consolidation was proposed and consolidated <cit.>. The model foresees that new (declarative) memories are first stored in the hippocampus and then, in a process that can last decades, gradually transferred to the cortex, where they are stored indefinitely. Once the transfer is complete, memories are retained even if the hippocampus is removed or damaged. neurogenesis, physical activity The vast majority of brain neurons are generated during embryonic development. However, new neurons are created also in specific regions of the adult brain. The generation of neurons in the dentate gyrus of the hippocampus is a process that continues for the whole duration of life <cit.>, favoured by physical exercise <cit.>. It is natural to think that such neurons are involved in the process of memory formation in this brain region.recent model A recent model <cit.> suggests that the instantiation of the cortical representation of memory “engrams” <cit.> occurs from the very beginning. New memories, instead of being first recorded in the hippocampus and then gradually copied or moved to the cortex, would be written in both places in parallel. The cortical representation would be immature at first and develop with time to more mature forms.§ MEMORY AND LEARNING IN NEURAL NETWORKS Mnist dataset Neural networks learn from data, organised in a dataset, structured as a collection of records. The upper part of Fig. <ref> shows a sample of Mnist <cit.>, a dataset commonly used to train neural networks, composed of 60000 images representing hand-written characters from 0 to 9, where each record of the dataset is a set of numbers (the dataset variables) encoding the grey shades of the 28x28 individual pixels that compose an image. From the dataset, the network learns higher-level features, representing oriented edges or simple shapes, combination of simple shapes, and the “concepts” of digits (lower part of Fig. <ref>).word dataset Datasets can contain any kind of elements: real numbers, categorical values, words, etc. An example of word dataset is shown in Table <ref>, in which records correspond to fruits on sale at the local market in some days. In this case, high-level features may correspond to “rules” that occur more frequently than expected by chance, e.g.: “round fruits occur on odd days”, “non-round fruits occur on even days”, or “there is an alternation of occurrence of round fruits and non-round fruits”, etc. learning For learning, dataset variables are mapped to neurons of the neural network input layer, while high-level features are encoded in neurons of successive layers. Learning consists in modifying the parameters that define the strength of connections between neurons which, collectively, represent the memory of the network. Neural networks can learn to perform a variety of tasks (e.g., recognition, classification, etc.): once learning is finished, such tasks can be performed on new data very quickly. learning procedure The typical learning procedure is divided in two parts: training (in which connection parameters are optimised based on the dataset) and test (in which the network is tested on data not used for training). The training part is in turn structured as a cycle composed of two phases: in the “change” phase the algorithm brings small changes to connection parameters, while in the “assessment” phase the network performance is measured on the dataset, yielding a performance score. This two-phase cycle (called “epoch” in the machine learning jargon) is repeated a number of times, until the score reaches a satisfactory value. supervised /unsupervised With “back-propagation” (the most commonly used supervised learning algorithm), the score depends on the percentage of records classified correctly for output neurons and on the “propagated” gradient of the classification error for neurons of intermediate layers; with unsupervised learning algorithms, other scores are used. While in supervised learning all neurons are optimised at the same time, in unsupervised learning, e.g. “contrastive divergence” <cit.>, neurons are usually optimised one layer at a time: first layer 1 neurons, then layer 2 neurons, etc. learning cycle, data structures Let us suppose that unsupervised learning is used and layer 1 neurons are being optimised: Fig. <ref> shows all data structures needed in the training process. The assessment phase procedure requires that: i) input neurons are exposed to all data records, ii) outputs of layer 1 neurons are calculated for all data records and iii) the score is calculated for the neurons under optimisation. need to read /write This requires that the entire dataset is read at each assessment phase during training (first row block in Fig. <ref>). Furthermore, computing the outputs of layer 1 neurons (needed to obtain the score) requires that these outputs are also written (the second row block in the figure). These additional rows are strictly-speaking not part of the dataset, but correspond to the new features learned, whose values must be calculated for each record (each column). extended dataset It is convenient to define the extended dataset as the union, for each record, of all values of dataset variables and successive layers' neurons. With this definition, we can conclude that the training procedure requires that the extended dataset be accessed in both reading and writing at each assessment phase during training. This, in turn, requires the storage of the extended dataset for the entire duration of the process. need for dataset storage Some training algorithms may be able to process data “on the fly”, with each record seen once and then discarded. In principle this is possible, but the most commonly used algorithms, proved effective in practical applications, need to be exposed to data many times. We will argue that this need is shared also by the “algorithms” operating in the brain.§ IMPLEMENTATION OF DEEP NETWORKS IN THE BRAIN features and associations Understanding and interpreting reality is the main objective of brain activity, and reality can be conceptualised in terms of features, defined as descriptors or qualities that can be used to characterise a given situation. Features can be simple visual qualities, such as “red colour”, “round shape”, “vertical orientation”, or correspond to more complex visual-motor characteristics, such as “dance movements”, “hiding an object with the hand”, etc. Therefore, in our interpretation, the concept of feature encompasses all components of mental life, ranging from simple perceptual elements to the most complex and abstract ideas. The model of reality is constructed by establishing meaningful associations among features: such associations become features in their own right, and can take part in further associations.features and associations, physical implementation Our hypothesis is that features of any complexity are implemented in the brain by single neurons, an idea already entertained <cit.>. A neuron may represent a basic visual or auditory feature or the most complex philosophical thought: the complexity of a feature is given by its relation with all other features. We further conjecture that an association among features A, B and C is physically built by recruiting a new neuron N and linking through synapses the axons of neurons A, B and C to the dendrites of N, which encodes a new feature representing the association.cortical neurons, lateral connections Neurons encoding features are located in the cortex (Fig. <ref>). Visual features, for instance, are mapped to the occipital cortex, auditory features are mapped to the auditory cortex, tactile features are mapped to the parietal cortex, etc. Neurons are connected with each other through “lateral” connections (light blue belt around the cortex in the figure) and are arranged in layers. The neuronal organisation is not “strictly layered” as for the network in Fig. <ref>, in which layer n neurons' inputs are only connected with layer n-1 neurons' outputs. In this case layer n neurons' inputs are connected with the outputs of neurons of all previous layers, down to sensory input neurons. We assume that the “cabling” linking all cortical neurons is laid out during embryonic development, before learning starts.cortical neurons, radial connections The axons of all cortical neurons project also to the hippocampus, where their tips are very close to each other, much closer than their respective cellular bodies (also these connections are pre-set before the start of the learning process). When a set of neurons fire together, their axons make synaptic connections to the dendrites of a hippocampal neuron generated in the dentate gyrus (Fig. <ref>): in this way, a new record of the “brain extended dataset”, encoding the co-occurrence of cortical activations, is created. subset allowed to write /addition At each moment in the course of life, only a subset of neurons can write in the hippocampal dataset: in the beginning, this is done by sensory neurons only as all other neurons are not optimised. As upper layers' neurons become optimised, they also start recording their activations in the hippocampus. In the case of visual learning, for example, the hippocampal dataset is initially written by sensory neurons representing pixel intensity values. Once the concepts of digits become available as a result of learning, they are employed to create future records.subset allowed to write /deletion We may also hypothesise that “obsolete” neurons are gradually replaced by newer neurons and their axons gradually lose access to entorhinal cortex and hippocampus. This is consistent with the limited “bandwidth” allowed by the entorhinal cortex, whose size (presumably) only allows a subset of cortical axons to reach the hippocampus. In the case of visual learning, once the concepts of digits become available, neurons encoding individual retinal intensity values would cease to be used to create new associations (and possibly be physically disconnected). need for convergence As already suggested in <cit.>, having all neurons project their axons to a small region is the only method to realise a fast association between features encoded in neurons that can be dispersed across a vast (in brain's terms) cortical area. If the neurons are distributed on the surface of a sphere (as it is in the brain's case), the best solution is that their axons project to a central point. In this point, the tips of all axons are very close to each other and the establishment of connections between them can be done very quickly. And it must be done quickly, to keep up with the rapid flow of information fed from the perceptual apparatus. role of hippocampus Based on these considerations, we suggest that the function of the hippocampus is to register and store the brain extended dataset, structured as an array of records composed of features, encoded in cortical neurons, that co-occurred. From this records, the “algorithms” of the cortex extract high-level features, encoded in other cortical neurons through adjustments of their lateral connections. Based on the two-phase algorithmic structure described in section 2, we assume that the optimisation procedure requires the continuous access to the hippocampal formation by cortical neurons, in both reading and writing. The generation of new features and the addition of new records to the hippocampus are continuous processes that run in parallel.instantiation of cortical memories Cortical memories are instantiated from the beginning in immature form, and gradually develop to more mature forms, exactly as described in <cit.>. For some episodic memories, that are essentially random associations of features without any hidden rules (the car can be parked near a tree or in front of a building without any deep reason other than the availability of a parking place), the process of “cortical transfer” can take longer, and essentially consists in copying to the cortex some salient elements of the otherwise untouched hippocampal record. Figs. <ref> and <ref> show the interplay between cortex and hippocampus that results in learning and memory formation. Cortical neurons register their activations in the hippocampus, whose records keep track of which neurons' activations co-occurred. In parallel, high-level features are learned and encoded in cortical neurons. The subset of cortical neurons that have access to the hippocampus is continuously updated, adding new optimised neurons and removing obsolete neurons. § DISCUSSION hypothesis consistentOur model of the hippocampus as the extended dataset of the brain is consistent with the finding the hippocampal neurons do more than encoding the spatial context <cit.> and goes beyond: we think that hippocampal neurons not only include space and time features, but all features. The model also accounts for the involvement of the hippocampus in flexible cognition and social behaviour <cit.>, a task that requires the continuous updating of information and presupposes the continuous access to the hippocampus in both reading and writing.connectionist models /1 Hypotheses on the functioning of biological memory based on knowledge derived from connectionist models have already been proposed. <cit.> carry out a very thorough analysis of the subject and conclude that “interleaved learning” is the most plausible model for the kind on learning that takes place in the brain. The authors argue that “neural networks or connectionist models adhere to many aspects of the account of the mammalian memory system, but do not incorporate a special system for rapid acquisition of the contents of specific episodes and events”. connectionist models /2 Our counterargument to this observation is that neural networks do have a “special system for rapid acquisition of the contents of specific episodes and events” and this is nothing else than the computer memory section where the extended dataset is stored. The motivation for the failed recognition of this component is probably the notion that the data are strictly not part of the neural network model: they are however part of the system. procedural memory /1 H.M. had both hippocampal regions removed and, as a result, was unable to form new declarative memory: however, he was able to acquire new procedural memory. Different kinds of tasks fall under the definition of procedural memory: repetition priming, classical conditioning (Pavlov's experiments), emotional conditioning, various skills and habits such as mirror tracing, mirror reading or jigsaw puzzles. H.M.'s performance was good in many of these tasks <cit.>.procedural memory /2 A first explanation for the fact that procedural learning can take place without hippocampus is that procedural tasks involve other brain regions (e.g., sensory cortex, cerebellum, amygdala, striatum). A second explanation is that the presence of a stored dataset is not always required. Learning may still be possible by processing each new record (or a small number or records, depending on the storage capacity available without hippocampus) “on the fly”, changing the neurons' parameters and then testing the performance of the modified neurons on subsequent records. This would explain the ability of amnesiacs to learn often-repeated material gradually over time <cit.>.hippocampus in PTSD The model proposed can have implications for the field of psychology: structural alterations to the hippocampus are in fact present in post-traumatic stress disorder <cit.>, a condition caused by exposure to traumatic experiences that affects war veterans, victims of violence and abuse, etc. A common post-traumatic symptom is represented by dissociation, defined as the limitation or loss of the normal associative links between perceptions, thoughts and emotions. Dissociation can take the form of mental “black-out”, depersonalisation, derealisation, selective amnesia and emotional detachment <cit.>.interpretation: thin dataset Since the hippocampus is the device that provides the initial association among co-occurring features, it is natural to think that this structure is also involved in the absence of association. Dissociative symptoms can be modelled as a negative modulation of the links fed to the hippocampus (as well as to other brain regions), leading to a shrinkage of their target areas. Based on our model, such links are the carriers of as many cognitive features, and their disconnection translates to having an extended dataset “thinner” than normal (fewer rows, with reference to Fig. <ref>). hippocampus in schizophrenia Also the hippocampus of schizophrenic subjects is significantly reduced in size, and seems to be less susceptible to the phenomenon of habituation <cit.>: these observations are consistent with the hypothesis of psychosis as a “learning and memory problem” <cit.>. However, it is not clear whether such alterations are present also before the onset of symptoms or develop afterwards, as a result of disease progression. unreliable associations in psychosis One of the most characteristic symptoms of psychosis is represented by delusions, which are beliefs held with strong conviction despite evidence to the contrary. For instance, if a person with psychosis sees one day two red cars parked near home, he/she may think that these cars are part of a conspiracy plot organised by a foreign government to spy and secure important industrial secrets: a conclusion that sounds absurd to most (non-psychotic) people. However, what would a normal person think if he/she saw two red cars parked next to the person's car each day for one month? Also this person would probably develop the idea of being followed.unreliable associations due to short dataset Therefore, the problem seems to be that the psychotic person jumps to the conclusion after an insufficient number of observations: as a result, the conclusion lacks statistical robustness (and is wrong). This could be caused by a smaller hippocampus able to host fewer data records. Based on our model, this translates to having an extended dataset “shorter” than normal (fewer columns, with reference to Fig. <ref>): as a result, the cortical algorithms would be forced to learn from fewer records, leading to the establishment of unreliable associations. This, together with the phenomenon of aberrant salience <cit.>, could help to explain the nature of delusions in schizophrenia.§ CONCLUSIONS xxxx The objective of this work was to use the knowledge derived from the field of neural networks to gain insight into the functioning of biological memory and learning. Our central hypothesis is that the hippocampus is the biological device to store the brain extended dataset. Following this idea, memory and learning in neural networks appear fully compatible with the structure of the brain and the latest discoveries on biological memory. The model proposed can help to explain some psychological conditions, such as post-traumatic stress disorder and schizophrenia, in which the hippocampus presents structural alterations. § DISCLAIMER The author is an employee of the European Research Council Executive Agency. The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission.apalike | http://arxiv.org/abs/1706.05932v1 | {
"authors": [
"Alessandro Fontana"
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"published": "20170526152143",
"title": "A deep learning-inspired model of the hippocampus as storage device of the brain extended dataset"
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Sub-nanometre resolution of atomic motion during electronic excitation in phase-change materials Muneaki Hase December 30, 2023 ================================================================================================ Predicting human interaction is challenging as the on-going activity has to be inferred based on a partially observed video. Essentially, a good algorithm should effectively model the mutual influence between the two interacting subjects. Also, only a small region in the scene is discriminative for identifying the on-going interaction. In this work, we propose a relative attention model to explicitly address these difficulties. Built on a tri-coupled deep recurrent structure representing both interacting subjects and global interaction status, the proposed network collects spatio-temporal information from each subject, rectified with global interaction information, yielding effective interaction representation. Moreover, the proposed network also unifies an attention module to assign higher importance to the regions which are relevant to the on-going action. Extensive experiments have been conducted on two public datasets, and the results demonstrate that the proposed relative attention network successfully predicts informative regions between interacting subjects, which in turn yields superior human interaction prediction accuracy. § INTRODUCTION Action prediction is defined as the problem of recognizing on-going activities based on temporally incomplete observations. It is a challenging task as only a part of the video is available for observation. Compared to individual action prediction, human interaction prediction is even harder, because the activities are more complex and involve more actors in the scene. More importantly, the incoming action of a subject might depend on the intention of the other subject, and this intention has to be inferred based on certain movement of this subject. In other words, to predict interaction, a good model should understand one subject's current action and how it will affect the other's response to this action in the near future.Despite significant progress in the past few years, human interaction prediction is still challenging mainly due to the following two unanswered questions. The first one is how to model interaction or relative information. Second is how to discover the most discriminative regions and make use of them to make prediction. Solving these two difficulties will always bring performance gain over holistic or global feature learning methods.However, previous methods do not address these questions in a proper way. Recent methods mainly resort to: (1) holistic representation <cit.>; (2) individual representation <cit.> and (3) discriminative part based representation <cit.>. Despite their favorable performance on recent benchmark datasets <cit.>, we have the following observations on their limitations. First, holistic feature based methods <cit.> usually encode the whole scene into a global feature vector, the richer information contained in individual subjects is ignored. Second, although individual representation based method <cit.> models both interactive subjects, they are usually modeled separately. How to effectively model their relationship is not well exploded. Third, discriminative part based methods <cit.> try to select discriminative patches/parts to represent the actions. Such discriminative patch/part detectors usually apply to the video frame-by-frame, thus the detected patches/parts are not temporally consistent. Moreover, such methods are hard to distinguish similar movements, as the generated patches are also similar.To explicitly address the above issues, we propose a tri-coupled relative attention framework. On one hand, a tri-coupled interaction fusion network is proposed to model mutual influence between subjects involved in the interaction. This network is composed of three recurrent sub-structures, which accept three streams of information representing both interacting subjects and the global interaction region enclosing both subjects. To capture the dependency between subjects, at each time-step, information flows from all three streams are aggregated to the hidden node of the current time-step, and then output the new status information for both interacting subjects. We make two remarks. First, we denote it by coupled recurrent network because status information of one subject is linked to the other stream, in order to assist the prediction of the next status of the other subject. Second, information extracted from the global scene (which encloses both subjects) is also utilized to predict the interaction status of both subjects. In this way, both local motion information and global motion information are fully utilized, which are complements to each other. On the other hand, built on this tri-coupled recurrent infrastructure, we introduce a relative attention network. The motivation is that some local motion (attended small regions) might give very useful information to predict the other subject's response in the future. For example, if a person extends his arm or leg, another person is likely to dodge, a punching/kicking is more likely to happen. In this situation, the arm/leg region is crucial for predicting another person's response. Motivated by this observation, a visual attention module is embedded to the recurrent structure to predict the discriminative regions of each subject. At each time-step, the attention module receives information from both interactive subjects, as well as their hidden states of previous time-step, and then output the attended regions of both subjects. In this way, only the attended regions are input into the recurrent networks, providing discriminative local information.The proposed network is extensively compared with some popular methods for encoding human interaction on two popular datasets, the results of the proposed method show favorable performance against the state-of-the-art methods.§ RELATED WORKTraditional methods. For action prediction, many previous works focus on finding good feature representation (usually bag-of-words features or sparse coding) and training SVM-like classifiers. For example, Ryoo DBLP:conf/iccv/Ryoo11 proposes two BoW based representation, i.e., the integral bag-of-words (IBoW) and dynamic bag-of-words (DBoW). Cao et al. DBLP:conf/cvpr/CaoBBNYMLDSW13 apply sparse coding to derive the activity bases, and use the reconstruction error in the likelihood computation. Lan et al. DBLP:conf/eccv/LanCS14 propose a hierarchical representation and combine it with a max-margin learning framework for action prediction. Another two max-margin frameworks <cit.> are built upon structured SVM model, but extend it to accommodate sequential data.Kong and Fu DBLP:journals/pami/KongF16 further extend this framework using compositional kernels to model the relationship of partial observations. Xu et al. DBLP:conf/iccv/XuQM15 consider action prediction as a query auto-completion problem.These methods use hand-crafted features and encoding methods to represent the video. The difference of our work lies in the using CNN/LSTM features rather than hand-crafted features, which enables our model to be trained end-to-end.CNN based methods. Many CNN based methods have been focused on activity recognition and video classification. In <cit.>, a 3D CNN model is proposed for action recognition.Karpathy et al. DBLP:conf/cvpr/KarpathyTSLSF14 explore several approaches for fusing information over temporal dimension trough the CNN, but only achieving marginal improvement than the single frame baseline, which indicates that learning motion information is difficult for CNN. To address this issue, Simonyan and Zisserman DBLP:conf/nips/SimonyanZ14 propose a two-stream architecture which directly incorporate motion information from optical flows, achieving significant improvements compared to previous CNN based methods. However, such approaches are based on single frames, not able to represent long-term temporal clue.RNN based methods.Recurrent neural network (RNN) and Long Short Term Memory (LSTM) <cit.> are powerful tools to model sequential data. LSTMs have been applied to action classification in <cit.>.The work of <cit.> further improves the performance by building a hybrid model incorporating both spatial and temporal clue.Ibrahim et al. Ibrahim_2016_CVPR build a 2-stage deep temporal model for group activity recognition.Ma et al. Ma_2016_CVPR design novel ranking losses for training LSTM which enforce the score margin between the correct and incorrect categories to be monotonically non-decreasing.Visual attention model is also investigated for action recognition in <cit.>. Song et al DBLP:journals/corr/SongLXZL16 build a spatio-temporal attention model from skeleton data. These works mainly focus on recognizing action of a single object or group activity, they achieve promising result when the complete video is observed. In contrast, our framework is explicitly designed for person interaction involving a pair of persons in the scene, and it still achieves satisfactory results when only a small part of the video is observed.§ METHODOLOGYThe problem is formulated as follows. We denote a complete video of duration T as V[1:T], the task is to predict the action y with only partial observation V[1:t], t∈{1,...,T}, the observation ratio is t/T. The complete videos are only accessible for training, and the performance is evaluated by calculating the prediction accuracy with a fixed observation ratio for all the test videos. In this work, we assume the bounding box of each person and the global scene enclosing the two actors are located in each frame. In the rest of this section, we use 𝐗 to denote the CNN feature extracted from raw frames. 𝐋 denotes the attention weights corresponding to the attended region. The inputs and hidden states of LSTM network are denoted as 𝐱 and 𝐡 respectively. The weights and bias terms in our networks are denoted as 𝐖, 𝐔,𝐕 and 𝐛. §.§ Tri-coupled Interaction Fusion NetworkWith an LSTM network, information could be propagated from the first node to the last one, and the good nature of LSTM is very useful for our given task, i.e., to make full use of the observed information and make a prediction. Motivated by the success of recurrent neural networks in temporal sequence analysis, we employ LSTM network as our network prototype. The frame-level features are input into LSTMs to model the spatio-temporal information.In particular, each LSTM node includes three gates, (i.e., the input gate 𝐢, the output gate 𝐨 and the forget gate 𝐟) as well as a memory cell. At each time-step t, the input feature 𝐱_t and the previous hidden state 𝐡_t-1 are input into the LSTM, as illustrated in Figure <ref>. The LSTM network updates as follows:𝐢_t = σ(𝐖_i𝐱_t +𝐔_i𝐡_t-1+𝐕_i𝐜_t-1 + 𝐛_i)𝐟_t = σ(𝐖_f𝐱_t + 𝐔_f𝐡_t-1+𝐕_f𝐜_t-1 + 𝐛_f)𝐜_t = 𝐟_t·𝐜_t-1 + 𝐢_t·tanh(𝐖_c𝐱_t + 𝐔_c𝐡_t-1+ 𝐛_c)𝐨_t = σ(𝐖_o𝐱_t + 𝐔_o𝐡_t-1+𝐕_o𝐜_t + 𝐛_o)𝐡_t = 𝐨_t·tanh(𝐜_t)where σ is the sigmoid function and · denotes the element-wise multiplication operator. 𝐖_*, 𝐔_* and 𝐕_* are the weight matrices, and 𝐛_* are the bias vectors. The memory cell 𝐜_t is a weighted sum of the previous memory cell 𝐜_t-1 and a function of the current input. The weights are the activations of forget gate and input gate respectively.For the task of interaction prediction, the most straightforward idea is to model the global interaction regions enclosing both subjects with a single LSTM network, as other activity recognition system <cit.>. We denote it a global LSTM network, which takes the complete region of action as input and models the global information of the observed video. As shown in Figure <ref>, the frame-level features are extracted by a CNN extractor, and then input into the LSTM network for classification. Here, we use Alexnet <cit.> as CNN feature extractor. The good nature of this structure is that all the information is modeled by a global LSTM, which is simple and effective for action recognition. However, the interaction of individual subjects is not explicitly modeled in the structure, the performance might be limited for the task of interaction prediction.There are multiple options to model the mutual interactions of the interactive subjects. A naive approach is to model each subject with an individual LSTM model and then combine their predictions, which can be further enhanced by employing the prediction of the global LSTM network. We denote this structure as naive fusion network, see Figure <ref>. This structure employs both global and local interactive information, but it also suffers from a major limitation. Some subjects are likely to have very similar behaviours in different interactions, e.g., the dodge action in both kick and box. The prediction scores of these subjects can be very confusing, directly summing up their prediction scores may bring side effects to the overall results.To address this issue, we design a joint training scheme that simultaneously models the interactive state of the two subjects. In particular, each subject is also represented by an LSTM network, but the hidden states of the two LSTMs are shared at each time-step. In this case, the terms 𝐔_*𝐡_t-1 in Equation <ref> to Equation <ref> are further represented by:𝐔_*𝐡_t-1= 𝐔_*,s_1𝐡_t-1,s_1 + 𝐔_*,s_2𝐡_t-1,s_2,where 𝐡_t-1,s_1is the previous hidden state of the network and 𝐡_t-1,s_2 is the previous hidden state of the other subject. This enables the information communication between the subjects, i.e., the statues of one subject can be used to help predict the action of the other subject. Moreover, the outputs of the LSTMs are concatenated as a union feature for prediction, which is in contrast of combing the prediction scores of individual subject level LSTMs. This structure allows the two the LSTMs to be trained together, i.e., there is a single loss for the networks. We denote it a coupled network, see the top part of Figure <ref>.Although the coupled network explicitly models the spatio-temporal correlations of the two subjects, the global interactive information is not used in the structure. To integrate the global information into the network, we design a Tri-coupled interaction fusion network, as shown in the middle part of Figure <ref>. For the tri-couple structure, the LSTM representing the global interaction status is pre-trained as a vanilla LSTM, the other two LSTMs modeling the mutual interactions are modeled as:𝐔_*𝐡_t-1= 𝐔_*,s_1𝐡_t-1,s_1 + 𝐔_*,s_2𝐡_t-1,s_2+𝐔_*,g𝐡_t,g,where 𝐡_t,g is the hidden state of global LSTM.§.§ Relative Attention Network For the task of action prediction, usually only a certain region is crucial for identifying an action. Therefore, we would like our model to focus on these regions and to model the fine-grained details. Here, we embed our tri-coupled network with a relative attention module.Two kinds of attention model have been used to address this issue. Hard attention <cit.> samples attention location at each time stamp, which causes the system not differentiable. In contrast, soft attention <cit.> aims to learn a set of weights corresponding to each region, the model is differentiable and can be trained end-to-end using standard back-propagation. Therefore, we adopt the soft attention model in our work. Instead of extracting feature from the last fully connected layer, the soft attention model employs the last convolutional layer, resulting to K convolutional maps of size D*D, which can be denoted as:𝐗_t={𝐗_t,1, ...,𝐗_t,D^2}, X_t,i∈ℝ^K.Specially, each vector 𝐗_t,i corresponds to a specific receptive field in the original image.At each time-step t, we would like to assign weights to each location in the D*D feature map. The attended region should have higher weights compared to less important regions. As each location in the feature maps corresponds to a certain receptive field in the original image, attending to the feature map plays the same role as attending to the original image. The attention weights 𝐋_t={l_t,1,...,l_t,D^2} at time-step t is usually calculated using the following two features: the hidden state of the previous time-step 𝐡_t-1 and the CNN feature map of the current time-step 𝐗_t. See Figure <ref>. The weights are normalized after a softmax layer:l_t,i = exp(𝐖_h,i𝐡_t-1+ 𝐖_X,i𝐗_t)/∑_j=1^D*Dexp(𝐖_h,j𝐡_t-1+ 𝐖_X,j𝐗_t) ,where i∈1,...,D^2 and 𝐖_*,i are the weights for the inputs. l_t,i can be viewed as the probability of the i-th region to be important. For the tri-coupled network, we can also take advantage of the mutual information to help locate the interesting region, i.e., to use the hidden states of neighboring LSTMs. The 𝐖_h,i𝐡_t-1 term in Equation <ref> can be further decomposed into hidden state information from both subjects:𝐖_h,i𝐡_t-1 = 𝐖_h,i,s_1𝐡_t-1,s_1+ 𝐖_h,i,s_2𝐡_t-1,s_2.The final inputs for LSTM is a weighted summation of the attention vector 𝐋_t and the CNN features 𝐗_t:𝐱_t=∑_i=1^D^2l_t,i𝐗_t,i. §.§ Training the Network The proposed tri-coupled network and relative attention network can be jointly trained as a classification problem of N classes (N is the number of human interaction category). At each time-step, the hidden state 𝐡_t of each sub-LSTM is concatenated as the feature representation vector, which is further connected to a softmax layer. The output of the N-way softmax is the prediction of the probability distribution over N different actions:y_i = exp(y'_i)/∑_k=1^Nexp(y'_k),where y'_j=𝐰_j·𝐡_t +b_j linearly combines the LSTM outputs, and 𝐰 and b are the weight matrix and bias term of the softmax layer. The network is learned by minimizing -logy_k, where k is the index of the true label for a given input. Stochastic gradient descent is used with gradients calculated by back-propagation.§ EXPERIMENTSIn this section, we present extensive experimental evaluations and in-depth analysis of the proposed method on the following two human interaction prediction benchmarks:UT dataset. The UT-Interaction dataset (UTI) <cit.> contains videos of 6 classes of human-human interactions: shake-hands, point, hug, push, kick and punch. Except that point is a single action, all other activities are performed by a pair of actors. This dataset contains two subsets: UTI #1 and UTI #2. The backgrounds of UTI #1 are mostly static with little camera jitter, while the backgrounds of UTI #2 are moving slightly and containing more camera jitters. Both of the two subsets contain 10 videos of each interaction class. We adopt 10-folder leave-one-out cross validation setting to measure the performance of the two subsets.BIT dataset. The BIT dataset <cit.> contains 8 types of interactions: bend, box, handshake, hifive, hug, kick, pat and push, all the activities are performed by a pair of actors. Each activity contains 50 video sequences, i.e., totally 400 videos in the dataset. Following <cit.>, a random subset containing 272 videos is used for training, and the remaining 128 videos are used for testing. §.§ Implementation DetailsThe implementation of the proposed networks are based on Caffe <cit.>. The LSTM layer contains 512 hidden units, and a dropout layer is placed after it to avoid over-fitting. To increase training instances and to make our model applicable for sequences of variable length, we randomly extract subsequences of fixed length L (L=10 in our experiments) for training. To train the LSTM networks, the original learning rate is initialized as 0.001, and the learning rate is decreased to 1/10 of the original value after each 10 epochs. The whole training phase includes 30 epochs. The complete duration of training time is about 12 hours on a Titan X GPU. During testing, we extract the subsequences in the testing video with a stride of 5, and averaging their classification score as prediction. We test our network on top of both RGB frames and optical flows. The optical flow is computed using the approach of <cit.>. As point action in the UTI dataset is a single action, we duplicate the image as input for the networks that require both subjects, i.e., the naive fusion network, the coupled network and the tri-coupled network. §.§ Results on UTI DatasetThe proposed tri-coupled relative attention network is compared with some leading approaches on interaction prediction. (1) Bag-of-words based methods: DBow and IBoW <cit.>; (2) Sparse coding based method: MSSC <cit.>; (3) Max margin structure SVM based methods: MTSSVM <cit.> and MMAPM <cit.>; and (4) discriminative patch based method: AAC <cit.>. The comparative results on UTI #1 is shown in Figure <ref>, and the quantitative results with observation ratio 0.5 and 1 are shown in Table <ref>. We report our best performance with tri-coupled relative attention network on top of optical flow inputs. Our method achieves favorable performance compared to other methods. It's remarkable that our tri-coupled structure achieves 100% recognition accuracy when the observation ratio is larger than 0.6. This is better than the previous state-of-the-art method <cit.>, which also achieves remarkable performance on this dataset, i.e., 91.67% and 96.67% for half video and full video. We further notice that our results are significantly higher than DBow, IboW <cit.> and other encoding based models. This is mainly because that tri-coupled network explicitly employs the interactive information, while most other methods only rely on the global information.Comparative results on UTI #2 are displayed in Figure <ref>. We notice that other methods have significant lower prediction accuracies compared to the results on UTI #1, due to more complex backgrounds and more camera jitter. Even the discriminative patch based method AAC <cit.> suffers from about 10% decrease. Compared to these methods, our tri-coupled relative attention model achieves better performance, more than 90% prediction accuracies when observation ratio is larger than 0.5, which is higher than other methods. This well demonstrates the robustness of the proposed method in existence of noise, and it is mainly due to our relative attention network, which is able to attend to discriminative regions on each interactive subject.Component analysis. Our framework consists of two major components: the tri-coupled network and the relative attention network. To evaluate the effectiveness of each component, we compare our network with some baseline structures introduced in Section <ref>: (1) global LSTM network; (2) naive fusion network; (3) coupled network; and (4) tri-coupled network without relative attention. The results on UTI #2 dataset with both RGB inputs and optical flow inputs are shown in Table <ref>, we have three observations. First, our baseline networks with optical flows achieves much better performance than the baseline methods using RGB frames. This is mainly because that the motion information contained in optical flows is crucial for identifying the actions.Second, we note that the naive fusion method only achieves marginally increase to the performance compared to global LSTM network, for both optical flows and RGB frames. This is because that some motion patterns of individual subjects can be very similar though different interactions, which may even provide negative information for prediction. e.g., the dodge motion occurs in both kick and punch, it will be difficult to make a prediction when observing such pattern.Last but not least, the tri-coupled network brings significant performance gain to the above baseline methods, especially in the case of high observation ratios. When embedded with the relative attention network, the performance is further improved. This demonstrates the effectiveness of the proposed tri-coupled network as well as the relative attention network. §.§ Result on BIT DatasetThe results on BIT dataset are shown in Figure <ref>. All the other methods get worse results compared to the results on UTI datasets, due to the fact that BIT dataset contains more category of interactions, and the videos in this dataset are with more complex backgrounds and sometimes with heavy occlusion. Therefore, the compared methods <cit.> achieve less than 80% prediction accuracies even with full observation, which is far away from real-world applications. While our network achieves more than 90% accuracy when only only half the video is observed, which outperforms the compared methods by a large margin (more than 10% with any observation ratio). This is because of the effectiveness of the relative attention module, which is able to attend to the discriminative regions in the scene, thus make the proposed method more robust to occlusions. §.§ Qualitative ResultsFigure <ref> visualizes the attended regions generated by our relative attention model. As the best performance of our method is achieved upon optical flows, the visualization is based on optical flow images. The first example illustrates the interaction of bend. It's easy to notice that the major subject is on the right side, and the subject on the left nearly has no movement during the interaction. For the subject on the right, we can find very strong correlation between the attended regions and the movements of the upper part of the body. The second example depicts two people shaking their hands. Both subjects are involved during the interaction, and they share similar behaviours: stepping forward and reaching out their hands. Our model consistently attends to the arms of both subjects, which shares similar intuition of human cognition. In the last example, the subject on the left is kicking the right subject. The attended regions are focused on the extended leg of the left subject, and the upper body of the right subject is attended to as he/she falls down.§ CONCLUSIONIn this paper, we propose a tri-coupled relative attention network for human interaction prediction. Experimental results convincingly demonstrate that the proposed relative attention network successfully predicts informative regions between interacting subjects, which in turn yields superior human interaction prediction accuracy. Although this paper is explicitly designed to model two subject interaction, our method is easily extendable to model group people interaction. Here is a brief illustration for this generalization. For each subject, the relative attention could be calculated with his/her nearest neighbors. The computational complexity only increases linearly w.r.t. the number of neighboring subjects. Finally, we can aggregate all groups via a LSTM structure to achieve global group activity label prediction.§ ACKNOWLEDGMENTS The work was supported by State Key Research and Development Program (2016YFB1001003). This work was partly supported by National Natural Science Foundation of China (NSFC61502301, NSFC61521062), China's Thousand Youth Talents Plan, the 111 Project (B07022) and the Shanghai Key Laboratory of Digital Media Processing and Transmissions. named | http://arxiv.org/abs/1705.09467v1 | {
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thmTheorem[section] lem[thm]Lemma cor[thm]Corollary prop[thm]Proposition rem[thm]Remark deff[thm]Definition conj[thm]Conjecture key[thm]Keywords prob[thm]Problem probb | http://arxiv.org/abs/1705.09482v1 | {
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Three-dimensional hydrodynamicalmodel atmospheres of red giant stars Sven Wedemeyer1 Arūnas Kučinskas2 Jonas Klevas2 Hans-Günter Ludwig36 January 2017; accepted 30 May 2017 ============================================================================The batch exponentiated gradient (EG) method provides a principled approach to convex smooth minimization on the probability simplex or the space of quantum density matrices. However, it is not always guaranteed to converge. Existing convergence analyses of the EG method require certain quantitative smoothness conditions on the loss function, e.g., Lipschitz continuity of the loss function or its gradient, but those conditions may not hold in important applications. In this paper, we prove that the EG method with Armijo line search always converges for any convex loss function with a locally Lipschitz continuous gradient.Because of our convergence guarantee, the EG method with Armijo line search becomes the fastest guaranteed-to-converge algorithm for maximum-likelihood quantum state estimation, on the real datasets we have.§ INTRODUCTION §.§ Problem FormulationConsider the convex minimization problem∈ f ( ρ ) | ρ∈𝒟 ,where f is a continuously differentiable convex loss function, and 𝒟 is the set of (quantum) density matrices, i.e., for some d ∈ℕ, 𝒟 := ρ∈𝒞^d × d | ρ = ρ^H, ρ≥ 0,( ρ ) = 1.A density matrix is a non-commutative analog of a probability distribution—if ρ is diagonal, its diagonal elements define a probability distribution on 1, …, d.The (batch) exponentiated gradient (EG) method <cit.> provides a principled approach to solving such a convex program. Starting with some non-singular density matrix ρ_0, the EG method iterates asρ_k + 1 = c_k^-1exp[ log ( ρ_k ) - α_k f' ( ρ_k ) ] ,k ∈ℤ_+ ,for some given step size α_k, where c_k is a positive number normalizing the trace of ρ_k + 1. The EG method, in its formulation, is also a special case of mirror descent <cit.> and the interior gradient method <cit.>.We choose to call (<ref>) the EG method, as this name refers exactly to the expression we consider.Our goal is to show that if the step sizes are computed by Armijo line search, the EG method converges for almost all continuously differentiable convex loss functions. We will define precisely the class of loss functions we consider in Section <ref>.By considering only diagonal matrices, the convex program (<ref>) and the EG method (<ref>) are equivalent to their vector counterparts, respectively (see, e.g., Section 4.3 in <cit.> for the vector formulation).The theory in this paper hence automatically specializes to the vector case. §.§ MotivationTo derive a step size α_k that guarantees the convergence rate of the EG method, one needs to impose some quantitative smoothness condition on the loss function. The standard condition is L-Lipschitz continuity of the loss function or its gradient on 𝒟 <cit.>.L-Lipschitz continuity with respect to the relative entropy, instead of a norm, was considered in <cit.>. An L-Lipschitz-like condition was proposed in <cit.>, requiring L h - f to be convex for some L > 0, where h denotes the negative entropy function. The Lipschitz-like condition was later shown to be equivalent to L-Lipschitz continuity of the gradient with respect to the relative entropy in <cit.>.Once a condition is verified and the corresponding parameter L is explicitly computed, the step size α_k is then set as a function of L and the iteration counter k.However, the conditions may not hold, and verifying the conditions is usually non-trivial. For instance, consider minimizing the loss functionf_1 ( x, y ) := - log ( x ) - log ( y ) ,on the probability simplex 𝒫 :=( x, y ) ∈ℝ^2 | x ≥ 0, y ≥ 0, x + y = 1.Neither f_1 nor its gradient f_1' is Lipschitz continuous, due to the presence of the logarithmic function. The Lipschitz-like condition <cit.> requires the convexity of L h ( x, y ) - f_1 ( x, y ) for some L > 0 on 𝒟, where h ( x, y ) is the negative entropy function: h ( x, y ) := x log ( x ) + y log ( y ) .A necessary condition isL ∂^2 h ( x, y )/∂ x^2 - ∂^2 f_1 ( x, y )/∂ x ^ 2= L/x - 1/x^2≥ 0 , for all( x, y ) ∈𝒫,which cannot hold for any fixed L, because x can be arbitrarily close to zero.The loss function f_1 is not simply an artificial example. Consider a generalization of minimizing f_2 ( x ) := - 1/n∑_i = 1^n log⟨ b_i, x |⟩on the probability simplex for some n ∈ℕ, where b_1, …, b_n are vectors in the non-negative orthant, for which f_1 is a special case with b_1 = ( 1, 0 ) and b_2 = ( 0, 1 ). A minimizer corresponds to the best constant rebalanced strategy for log-optimal portfolio selection <cit.>.Consider a further generalization under the non-commutative setting: f_3 ( ρ ) := - 1/n∑_i = 1^n log ( M_i ρ ) , ρ∈𝒟 ,where M_1, …, M_n are given positive semi-definite matrices in ℂ^d × d. A minimizer of f_3 on 𝒟 is a maximum-likelihood (ML) estimate for quantum state estimation <cit.>, and also an ML estimate of the PhaseLifted signal for phase retrieval with Poisson noise <cit.>. As log-optimal portfolio selection by the EG method had been studied under the on-line setting (see, e.g., <cit.>), it is possible to extend existing results to the batch non-commutative formulation (i.e., minimizing f_2 on 𝒟). Such an extension, however, might not be able to address all other cases. For example, the hedged approach to ML quantum state estimation considers minimizing f_2 - λ_1 log ( ρ ) for some λ_1 > 0 <cit.>; the max-entropy approach considers minimizing f_2 + λ_2( ρlogρ ) for some λ_2 > 0 <cit.>; the approach to low-rank matrix estimation proposed in <cit.> considers minimizing ∑_i [ y_i -( M_i ρ ) ]^2 + λ_3( ρlogρ ) for some real numbers y_i, Hermitian matrices M_i, and λ_3 > 0; and a similar vector formulation of empirical risk minimization with Shannon entropy penalization was studied in <cit.>. In all examples, the loss functions are not Lipschitz continuous in function values nor their gradients. Why do we not use the projected gradient method?Indeed, it was shown in <cit.> that the projected gradient method with Armijo line search converges for minimizing any continuously differentiable loss function.We notice that, however, the projected gradient method may be not well-defined.Consider minimizing f_1 on the probability simplex as an example. As projection onto the probability simplex often results in a sparse output, it can happen that some iterate ( x_k, y_k ) is exactly sparse; then f_1 ( x_k, y_k ) and f_1' ( x_k, y_k ) are not defined, and the algorithm is forced to terminate. An explicit example is given by setting ( x_k - 1, y_k - 1 ) = ( 0.99999, 0.00001 ) and the step size (or the upper bound of it for Armijo line search) to be 1, for which ( x_k, y_k ) = ( 0, 1 ). §.§ Our ContributionUnlike existing results, we are interested in seeking for an universal approach to convex smooth minimization on 𝒟, which converges for minimizing almost all continuously differentiable convex functions. We consider finding the step sizes by the Armijo line search rule. The pseudo code is shown in Algorithm <ref>, in which we defineρ ( α ) := c^-1exp[ log ( ρ ) - α f' ( ρ ) ],for any non-singular density matrix ρ and α > 0, where the positive number c normalizes the trace of ρ ( α ). The outer for-loop in Algorithm <ref> implements the EG method; the inner while-loop applies the Armijo rule to find a proper step size.The EG method with Armijo line search had been studied in <cit.>, but the analyses therein assume Lipschitz continuity of f'.Our contribution lies in deriving a convergence guarantee under a very weak smoothness condition on the loss function. We say that f has a locally Lipschitz continuous gradient, if for every x ∈ ( f ), there exists a neighborhood in ( f ) on which f' is Lipschitz continuous.It is easily checked that if f is twice continuously differentiable on ( f ), then f has a locally Lipschitz continuous gradient. Therefore, for instance, the functions f_1,f_2, and f_3 all have locally Lipschitz continuous gradients. The main result of this paper is Theorem <ref>, which is proved in Section <ref>.Consider solving the convex program (<ref>) by Algorithm <ref>. Assume that f has a locally Lipschitz continuous gradient, and ( f ) contains all non-singular density matrices.The following statements hold. * The Armijo line search (Line 3–5) terminates in finite steps.* ρ_k ∈𝒟 for all k.* f ( ρ_k + 1 ) ≤ f ( ρ_k ) for all k.* The sequence ( ρ_k )_k ∈ℕ has at least one limit point.* Every limit point of ( ρ_k )_k ∈ℕ minimizes f on 𝒟.Notice that both Algorithm <ref> and Theorem <ref> do not assume the local Lipschitz constants of f' to be known nor uniformly bounded.Our problem formulation does not impose any quantitative smoothness condition on the loss function, so we do not have a guarantee on the convergence rate.Numerical experiments on ML quantum state estimation (Section <ref>), nevertheless, show that the empirical convergence rate of the EG method with Armijo line search can be competitive. In fact, the EG method with Armijo line search is the fastest among all existing guaranteed-to-converge algorithms for ML quantum state estimation, on the real experimental data we have. Recall that existing analyses for the EG method, with and without line search, do not directly apply to ML quantum state estimation, and the projected gradient method is, rigorously speaking, not applicable. §.§ Notations Let g be a convex function taking values in ℝ∪±∞.The (effective) domain of g, denoted by ( g ), is given by ( g ) =x | g ( x ) < + ∞. We denote the gradient of g by g', and the Hessian by g”. We will focus on the non-commutative formulation (<ref>) in the rest of this paper. To define the gradient of f properly is tricky, as a non-constant real-valued function of complex variables cannot be analytic. We define f' ( x ) at x ∈ ( f ) as the unique matrix such that f ( y ) ≥ f ( x ) +f' ( x ), y - x,for all y ∈ ( f ), where the inner product is the Hilbert-Schmidt inner product, i.e., for any matrices X, Y ∈ℂ^d × d, X, Y:=( X^H Y ).The definition of the EG method (cf. (<ref>)) presumes that f' is Hermitian.The inner products in the rest of this paper will be all Hilbert-Schmidt, unless otherwise specified. We denote by ·_F the Frobenius norm, and ·_ the trace norm.The functions exp ( · ) and log ( · ) in (<ref>) are matrix exponential and matrix logarithmic functions. Generally speaking, let X = ∑_j ∈𝒥λ_j P_j be the spectral decomposition of a Hermitian matrix X, where P_j is the projection onto the eigenspace corresponding to λ_j for all j ∈𝒥. Let g be a real-valued function whose domain contains λ_j : j ∈𝒥. Then g ( X ) is defined as ∑_j ∈𝒥 g ( λ_j ) P_j.The von Neumann entropy of a density matrix ρ is given byh ( ρ ) := -( ρlogρ ) ,where we adopt the convention that 0 log 0 = 0. The quantum relative entropy between two density matrices ρ and σ, denoted by D ( ρ, σ ), is given byD ( ρ, σ ) := {[ ( ρlogρ ) -( ρlogσ )if( ρ ) ⊇ ( σ ) ,;+ ∞otherwise . ].The relative entropy is always non-negative. Two non-singular density matrices ρ and σ are the same, if and only if D ( ρ, σ ) = 0.§ PROOF OF THEOREM <REF>Section <ref> provides some necessary background knowledge.Section <ref> presents a local Peierls-Bogoliubov inequality, which is key in establishing the convergence statement in Theorem <ref>. Section <ref> shows the complete proof of Theorem <ref>. §.§ PreliminariesWe defined ρ ( α ) explicitly in (<ref>).The following lemma shows that ρ ( α ) admits an equivalent definition. For any non-singular density matrix ρ and α > 0, one hasρ ( α ) =f' ( ρ ), σ - ρ + 1/α D ( σ, ρ ) | σ∈𝒟 . Combine the arguments in <cit.> and Section 4.3 of <cit.>, or directly solve the convex program as in <cit.>. Notice that ρ itself is a feasible point of the convex program (<ref>).One then hasf' ( ρ ), ρ ( α ) - ρ + 1/α D ( ρ ( α ), ρ ) ≤ 0 .This proves the following corollary.For any non-singular density matrix ρ and α > 0, one hasf' ( ρ ), ρ ( α ) - ρ≤ - D ( ρ ( α ), ρ )/α .Lemma <ref> implies a fixed-point characterization of a minimizer.A non-singular density matrix ρ minimizes f on 𝒟, if ρ = ρ ( α ) for some α > 0.On the other hand, if a non-singular density matrix ρ minimizes f on 𝒟, then ρ = ρ ( α ) for all α≥ 0.The first-order optimality condition (see, e.g., <cit.>) says that ρ is a minimizer, if and only iff' ( ρ ), σ - ρ≥ 0 ,for all σ∈𝒟. Equivalently, we writef' ( ρ ) + α^-1h̃' ( ρ ) - α^-1h̃' ( ρ ), σ - ρ≥ 0 ,where h̃ ( ρ ) :=( ρlogρ ) -( ρ ).It is easily checked that (<ref>) is the optimality condition ofρ =f' ( ρ ), σ - ρ + α^-1 D ( σ, ρ ) | σ∈𝒟 ,as D ( ·, · ) coincides with the Bregman divergence defined by h̃ on 𝒟×𝒟 (see, e.g., <cit.>). The lemma then follows from Lemma <ref>. The local Lipschitz continuity of f' allows us to bound the first-order approximation error locally. Let ρ be a non-singular density matrix. For α small enough, one has0 ≤ f ( ρ ( α ) ) - [ f ( ρ ) +f' ( ρ ), ρ ( α ) - ρ] ≤ L_ρ D ( ρ ( α ), ρ ) ,where L_α is the local Lipschitz continuity constant for f' in a neighborhood of ρ. Notice that ρ ( α ) is a continuous function wrt α. Following the proof of Lemma 1.2.3 in <cit.>), one has0 ≤ f ( ρ ( α ) ) - [ f ( ρ ) +f' ( ρ ), ρ ( α ) - ρ] ≤L_ρ/2ρ ( α ) - ρ_F^2 ,for small enough α, where L_ρ denotes the local Lipschitz constant of f'. By Pinsker's inequality <cit.>, one hasL_ρ/2ρ ( α ) - ρ_F^2 ≤L_ρ/2ρ ( α ) - ρ_^2 ≤ L_ρ D ( ρ ( α ), ρ ) ,which proves the lemma. §.§ A Local Peierls-Bogoliubov InequalityLet ρ be any non-singular density matrix.Defineφ ( α; ρ ) := logexp[ log ( ρ ) - α f' ( ρ ) ].The function φ plays a key role in the proof of Theorem <ref>. We will often omit ρ and write φ ( α ) for convenience, when the corresponding ρ is irrelevant, or clear from the context.The Peierls-Bogoliubov inequality says that φ is a convex function (see, e.g., <cit.>); equivalently, one has φ” ( α ) ≥ 0 for all α∈ℝ. In this paper, we need a slightly stronger version. One has φ” ( α ) ≥ 0 for all α∈ℝ.Moreover, φ” ( α ) = 0, if and only if f' ( ρ ) = κ I for some κ∈ℝ.The proof below is essentially a combination of the proofs in <cit.> and <cit.>.We show it to identify the condition for φ” = 0.Let A, B be two Hermitian matrices. Define H_t := A + t B, and Φ ( t ) := logexp ( H_t ) for t ∈ℝ.By the relation <cit.> ∂exp ( H_t )/∂ t= ∫_0^1 exp[ ( 1 - u ) H_t ] B exp( u H_t ) u ,one can obtainΦ” ( t ) =B, B _BKM I, I _BKM -I, B _BKM^2 /[ exp ( H_t ) ] ^ 2 ,where ⟨·, ·|_⟩BKM denotes the Bogoliubov-Kubo-Mori inner product with respect to H_t: X, Y _BKM := ∫_0^1 {exp[ ( 1 - u ) H_t ] X exp ( u H_t ) Y }u ,for any Hermitian matrices X, Y.Set A = log ( ρ ) and B = - f' ( ρ ).The theorem follows from the Cauchy-Schwarz inequality and its equality condition. The following lemma establishes the connection between φ and the EG method, which is easy to prove, but perhaps not obvious at first glance.For any non-singular density matrix ρ and α > 0, one hasD ( ρ ( α ), ρ )= φ ( 0 ) - [ φ ( α ) + φ' ( α ) ( 0 - α ) ], D ( ρ, ρ ( α ) )= φ ( α ) - [ φ ( 0 ) + φ' ( 0 ) ( α - 0 ) ] . By Theorem 3.23 in <cit.>, one can obtainφ' ( α ) = { - f' ( ρ ) exp[ log ( ρ ) - α f' ( ρ ) ] }/exp[ log ( ρ ) - α f' ( ρ ) ].The lemma is then verified by direct calculation.We now prove the main result of this sub-section, a local Peierls-Bogoliubov inequality. Its formulation was motivated by a result in <cit.>, which, in the context of this paper, says that the mappingα↦Π_𝒟 ( ρ - α f' ( ρ ) ) _F/αis non-increasing on ( 0, + ∞ ), where Π_𝒟 denotes the projection onto 𝒟 with respect to the Forbenius norm ·_F. For any non-singular density matrix ρ and α̅ > 0, there exists some γ≥ 2 such that Γ ( α ) :=D ( ρ ( α ), ρ ) /α^γis non-increasing on ( 0, α̅ ]. Moreover, γ depends continuously on ρ.We prove the proposition by verifying Γ' ( α ) ≤ 0 on ( 0, α̅ ]. Applying Lemma <ref>, a direct calculation givesΓ' ( α ) = φ ( α ) - φ' ( α ) α + φ” ( α )/γα ^ 2/γ^-1α^γ + 1 = φ ( α ) + φ' ( α ) ( 0 - α ) + φ” ( α )/γ ( 0 - α ) ^ 2/γ^-1α^γ + 1 .Notice that 0 = φ ( 0 ).Then one has Γ' ( α ) ≤ 0, if and only ifφ ( 0 ) - [ φ ( α ) + φ' ( α ) ( 0 - α ) ] ≥φ” ( α ) /γ ( 0 - α )^2 .The function φ” is continuous, so it takes its minimum μ≥ 0 and maximum L ≥ 0 on [ 0, α̅ ]. The Taylor formula with the integral remainder (see, e.g., <cit.>) givesφ ( 0 ) - [ φ ( α ) + φ' ( α ) ( 0 - α ) ]= α^2 ∫_0^1 ∫_0^t φ” ( α + τ ( 0 - α ) )τt ≥μ/2α^2 .Therefore, the inequality (<ref>) holds, if ( μ / 2 ) ≥ ( L / γ ). We consider two cases:* If μ = 0, Theorem <ref> implies that f' ( ρ ) = κ I for some κ∈ℝ.Then one can verify ρ ( α ) = ρ for all α.Therefore, Γ ( α ) = 0 for all α, and the proposition trivially holds with γ = 2. * If μ≠ 0, one can simply choose γ = ( 2 L / μ ) ≥ 2.Write L := maxφ” ( α ; ρ ) | α∈ [ 0, α̅ ].Notice that φ” ( α; ρ ) is continuous on [ 0, α̅ ] ×𝒟 as a function of the pair ( α, ρ ), and 𝒟 is a compact set.Therefore, L is continuously dependent on ρ <cit.>.Similarly, μ and hence γ are also continuously dependent on ρ. While the Peierls-Bogoliubov inequality requires φ” ( α ) ≥ 0 for all α, Proposition <ref> essentially requires φ” ( α ) to be strictly positive restricted on [ 0, α̂ ].This explains why we call Proposition <ref> a local Peierls-Bogoliubov inequality. §.§ Proof of Theorem <ref>We present the proofs of the five statements in Theorem <ref> one by one.The proofs of Statements 1–4 are simple; the difficulties lie in the proof of Statement 5. Proof of Statement 1Statement 1 follows from the following proposition. For any non-singular density matrix ρ in (f) and τ∈ ( 0, 1 ), there exists some α̃ > 0 such thatf ( ρ ( α ) ) ≤ f ( ρ ) + τ f' ( ρ ), ρ ( α ) - ρ ,for all α∈ ( 0, α̃ ).Equivalently, we have to verifyf ( ρ ( α ) ) - [ f ( ρ ) +f' ( ρ ), ρ ( α ) - ρ] ≤ - (1 - τ)f' ( ρ ), ρ ( α ) - ρ .By Corollary <ref> and Lemma <ref>, it suffices to prove L D ( ρ ( α ), ρ ) ≤( 1 - τ ) D ( ρ ( α ), ρ )/α ,in a neighborhood of ρ, where L denotes the local Lipschitz constant of f' in the neighborhood. If ρ is a minimizer of f on 𝒟, one has ρ ( α ) = ρ by Lemma <ref>; hence the proposition holds. If ρ is not a minimizer, (<ref>) is equivalent to L ≤ ( 1 - τ ) / α, which holds when α is small enough.Proof of Statement 2This is obvious by definition. Proof of Statement 3The Armijo rule ensures thatf ( ρ_k + 1 ) ≤ f ( ρ_k ) + τ f' ( ρ_k ), ρ_k + 1 - ρ_k,k ∈ℤ_+ .Notice that ρ_k + 1 = ρ_k ( α_k ).Statement 3 then follows from Corollary <ref>. Proof of Statement 4This statement follows from Statement 2 and the compactness of the constraint set 𝒟. Proof of Statement 5Equivalently, we will show that any convergent sub-sequence of ( ρ_k )_k ∈ℕ converges to a minimizer of f on 𝒟.We first check the feasibility of a limit point.All limit points of ( ρ_k )_k ∈ℕ lie in ( f ).Otherwise, Statement 3 in Theorem <ref> cannot hold by the continuity of f. Lemma <ref> allows one to talk about the local Lipschitz constant of f' around any limit point. Let ( ρ_k )_k ∈𝕂 be a convergent sub-sequence for some 𝒦⊆ℕ, converging to some ρ̅∈𝒟. Then there exists some constant β > 0, such that D ( ρ_k ( β ), ρ_k ) → 0 as k →∞ in 𝒦.If ρ_k' is a minimizer for some k' ∈𝒦, Lemma <ref> implies that ρ_k = ρ_k' for all k > k' in 𝒦, and the proposition trivially holds.In the rest of the proof, we assume that ρ_k is not a minimizer for all k ∈𝒦. We will denote by γ_k the value of γ in Proposition <ref> corresponding to ρ_k for all k.By continuity, γ_k converges to some γ̅≥ 2; hence one has ( 1 / 2 ) γ̅≤γ_k ≤ 2 γ̅ for large enough k ∈𝒦.Suppose that lim infα_k | k ∈𝒦≥ for some > 0. Let ( α_k )_k ∈𝒦' be a sub-sequence of ( α_k )_k ∈𝒦 converging to .By assumption, one has α_k ≤ 2 for large enough k ∈𝒦'. Then one can writef ( ρ_k ) - f ( ρ_k + 1 ) ≥ - τ f' ( ρ_k ), ρ_k + 1 - ρ_k ≥τα_k^-1 D ( ρ_k + 1, ρ_k )= τα_k^γ_k - 1α_k^- γ_k D ( ρ_k + 1, ρ_k ) ≥τα_k^γ_k - 1α̅^- γ_k D ( ρ_k ( α̅ ), ρ_k ) ≥ C D ( ρ_k ( α̅ ), ρ_k ),where C := ( 2)^2 γ̅ - 1 is independent of k.We have applied the definition of the Armijo rule in the first inequality, Corollary <ref> in the second inequality, and Proposition <ref> in the third inequality.The proposition follows from the continuity of f.Suppose that lim infα_k | k ∈𝒦 = 0.Let ( α_k )_k ∈𝒦' be a sub-sequence of ( α_k )_k ∈𝒦 converging to 0. Since then it is impossible to have α_k = α for all k ∈𝒦', one hasf ( ρ_k ( r^-1α_k ) ) > f ( ρ_k ) + τ f' ( ρ_k ), ρ_k ( r^-1α_k ).By Lemma <ref> and Lemma <ref> , one can writeL D ( ρ_k ( r^-1α_k ), ρ_k ) ≥ f ( ρ_k ( r^-1α_k ) ) - [ f ( ρ_k ) +f' ( ρ_k ), ρ_k ( r^-1α_k ) - ρ_k ]> - ( 1 - τ )f' ( ρ_k ), ρ_k ( r^-1α_k ) ≥ ( 1 - τ ) D ( ρ_k ( r^-1α_k ), ρ_k ) / r^-1α_k,for large enough k in 𝒦', where L is a local Lipschitz constant of f' in a neighborhood of ρ̅. Proposition <ref> then impliesD ( ρ_k ( r^-1α_k ), ρ_k ) ≥C̃[ D ( ρ_k ( r^-1α ), ρ_k ) ]^1 / γ_k[ D ( ρ_k ( γ^-1α_k ), ρ_k ) ]^1 - 1 / γ_k,where C̃ := ( 1 - τ ) / ( r^-1α L ) is independent of k. Since we assume ρ_k is not a minimizer for all k, D ( ρ_k ( r^-1α_k ), ρ_k ) ≠ 0 for all k. Then one obtains[ D ( ρ_k ( r^-1α ), ρ_k ) ]^1 / γ_k≤C̃^-1[ D ( ρ_k ( r^-1α_k ), ρ_k ) ]^1 / γ_k .The dependence of γ_k on k can be removed by writing[ D ( ρ_k ( r^-1α ), ρ_k ) ]^1 / 2 γ̅≤C̃^-1[ D ( ρ_k ( r^-1α_k ), ρ_k ) ]^2 / γ̅ ,for large enough k ∈𝒦'. It remains to show that D ( ρ_k ( γ^-1α_k ), ρ_k ) → 0 as k →∞ in 𝒦'. This can be verified by Lemma <ref> and the assumption that α_k → 0 as k →∞ in 𝒦': D ( ρ_k ( r^-1α_k ), ρ_k ) = φ_k ( 0 ) - [ φ_k ( r^-1α_k ) ( 0 - r^-1α_k ) ] ≤L_k/2( α_k/r) ^ 2 ≤L̅( α_k/r) ^ 2,for large enough k ∈𝒦', where φ_k ( t ) := φ ( t ; ρ_k ) for t ∈ℝ, L_k denotes the supremum of φ_k” on [ 0, γ^-1α ], and L̅ denotes the supremum of φ” ( ·; ρ̅ ) on the same interval. We used the fact that L_k ≤ 2 L̅ for k large enough in the second inequality; notice that L_k converges to L̅, as shown at the end of the proof of Proposition <ref>. If ρ̅ is non-singular, Proposition <ref> implies D ( ρ̅ ( β ), ρ̅ ) = 0 for some β > 0; therefore, ρ̅ ( β ) = ρ̅, so ρ̅ is a minimizer by Lemma <ref>.However, if ρ̅ is singular, ρ̅ ( β ) is not well-defined in (<ref>).Although the equivalent definition of ρ̅ ( β ) given by Lemma <ref> is still valid when ρ̅ is singular, it is unclear whether the limiting argument goes through.We show explicitly that Proposition <ref> implies the optimality of ρ̅ in the rest of this sub-section.The idea is to consider the first-order optimality condition—although ρ̅ ( β ) might be not well-defined when ρ̅ is non-singular, the first-order optimality condition is always well-defined. For any ρ∈ ( f ), defineψ ( ρ ) := inf f' ( ρ ), σ - ρ | σ∈𝒟 .The first-order optimality condition says that a density matrixminimizes f on 𝒟, if and only if ψ () = 0 (see, e.g., <cit.>). Notice that ψ is a continuous function well-defined on ( f ). Our goal is to show thatψ ( ρ̅ ) = lim_k →∞∈𝒦ψ ( ρ_k ) = 0,for any convergent sub-sequence ( ρ_k )_k ∈𝒦.For any non-singular density matrix ρ and β > 0, it holds that- β^-1 D ( ρ ( β ), ρ ) ≤ψ ( ρ ) ≤ 0 . The upper bound on ψ is obvious, as one can choose σ = ρ in (<ref>). It is easily verified thatψ ( ρ ) = λ_min ( f' ( ρ ) ) -f' ( ρ ), ρ ,for any ρ∈𝒟, where λ_min ( · ) denotes the minimum eigenvalue. A direct calculation givesD ( ρ ( β ), ρ ) = - β f' ( ρ ), ρ ( β )- logexp[ log ( ρ ) - β f' ( ρ ) ] .We bound the two terms at the right-hand side separately. Noticing that D ( ρ ( β ), ρ ) ≥ 0, Corollary <ref> implies -f' ( ρ ), ρ ( β ) ≥ -f' ( ρ ), ρ.As f' ( ρ ) - λ_min ( f' ( ρ ) ) I is positive semi-definite, one haslogexp{log ( ρ ) - β[ f' ( ρ ) - λ_min ( f' ( ρ ) ) I ] }≤logexplog ( ρ ) = 0,i.e., - logexp[ log ( ρ ) - β f' ( ρ ) ] ≥βλ_min ( f' ( ρ ) ) .The lemma follows. Consider any convergent sub-sequence ( ρ_k )_k ∈𝒦 converging to a limit point ρ̅. We have proved that there exists some constant β > 0 such that D ( ρ_k ( β ), ρ_k ) → 0 as k →∞ in 𝒦. Lemma <ref> and the continuity of ψ then implylim_k →∞ in 𝒦ψ ( ρ_k ) = ψ( ρ̅ ) = 0,which establishes the optimality of ρ̅.§ NUMERICAL EXPERIMENT: ML QUANTUM STATE TOMOGRAPHYQuantum state tomography is the problem of estimating an unknown density matrix ρ∈𝒞^d × d, by measuring multiple independent and identically prepared copies of it (for details, see, e.g., <cit.>). It is essential in quantum information applications; for example, researchers estimate the density matrix of a prepared quantum gate for calibration.A measurement setting is mathematically described by a probability operator-valued measure (POVM), a set of Hermitian positive semi-definite matrices summing up to the identity. Let ℳ :=M_j : j ∈𝒥 be a POVM. The corresponding measurement outcome of ρ is a random variable ξ, taking values in 𝒥 and satisfying ℙξ = j=( M_j ρ ) for all j ∈𝒥. Given n independent measurement outcomes on n copies, the normalized negative log-likelihood function is then given by f_3 (cf. (<ref>)), where each M_i is an element in the POVM applied to the i-th copy of ρ.The experimental data we have was generated following the setting in <cit.>, in which Pauli-based measurements are used to measure the W-state (a specific single-rank density matrix).Under this setting, each M_i is a single-rank matrix of the form v v^H, v being a tensor product of eigenvectors of Pauli matrices.As discussed in Section <ref>, f_3 is not Lispchitz continuous in its function value nor its gradient; hence there are few guaranteed-to-converge existing algorithms. To the best of our knowledge, the diluted R ρ R algorithm <cit.>, SCOPT <cit.>, and the modified Frank-Wolfe algorithm <cit.> are the only existing algorithms that are guaranteed to converge. We will also consider the R ρ R algorithm <cit.>, which does not converge in some cases <cit.>, but is much faster than its diluted version, the diluted R ρ R algorithm.We compare the convergence speeds for the 6-qubit (d = 2^6) and 8-qubit (d = 2^8) cases, in Fig. <ref> and <ref>, respectively. The corresponding “sample sizes” (i.e., number of summands in f_3) are n = 60640 and n = 460938, respectively. The experiments were done in MATLAB R2015b, on a MacBook Pro with an Intel Core i7 2.8GHz processor and 16GB DDR3 memory. We set α = 10, and γ = τ = 0.5 in Algorithm <ref> for both cases. In both figures, f^⋆ denotes the minimum value of f_3 found by the five algorithms in 120 iterations.One can observe that the EG method with Armijo line search has the fastest empirical convergence speed, in terms of the actual elapsed time. The numerical results can be explained by theory. * The diluted R ρ R algorithm, using the notation of this paper, iterates as ρ_k + 1 = c_k^-1[ I + λ_k f' ( ρ_k ) ]^H ρ_k [ I + λ_k f' ( ρ_k ) ],where c_k normalizes the trace of ρ_k + 1, and to guarantee convergence, the step size λ_k is computed by exact line search. The exact line search procedure renders the algorithm slow. * SCOPT is a projected gradient method for minimizing self-concordant functions <cit.>, which chooses the step size such that each iterate lies in the Dikin ellipsoid centered at the previous iterate.It is easily checked that f_3 is a self-concordant function of parameter 2 √( n ). Following the theory in <cit.>, the radius of the Dikin ellipsoid shrinks at the rate O( n^-1/2 ), so SCOPT becomes slow when n is large. * The modified Frank-Wolfe algorithm is essentially the same as the standard Frank-Wolfe algorithm, with a novel step size to guarantee convergence for minimizing f_3.Like the standard Frank-Wolfe algorithm, the modified version suffers for a sub-linear convergence rate due to the zig-zagging phenomenon (see, e.g., <cit.> for an illustration). We notice that the empirical convergence rate of the EG method with Armijo line search is linear, despite that f_3 is not globally strongly convex.§ A HISTORICAL REMARK We have discussed existing analyses of the EG method in Section <ref>.As for Armijo line search, there are few existing convergence results as general as Theorem <ref>. The Armijo rule was originally proposed for unconstrained convex minimization <cit.>, assuming that the loss function has a Lipschitz continuous gradient.Bertsekas extended the formulation of Armijo line search for continuously differentiable convex functions, and showed that the projected gradient method with Armijo line search (henceforth abbreviated as PGA) always converges for the box and positive orthant constraints in <cit.>.According to <cit.> and <cit.>, Goldstein proved the convergence of PGA for a class of constraint sets in a conference paper in 1974. A general convergence result for PGA, which is valid for any continuously differentiable convex function and any convex constraint set, appeared first in <cit.>, and was then summarized in <cit.> (what we cited is the last edition of the book).To the best of our knowledge, there was no such general convergence result for the EG method.Our Theorem <ref> fills this gap. acm | http://arxiv.org/abs/1705.09628v1 | {
"authors": [
"Yen-Huan Li",
"Volkan Cevher"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170526155559",
"title": "A General Convergence Result for the Exponentiated Gradient Method"
} |
𝐱 ŁL Konda Reddy Mopuri, Vishal B. Athreyaand R. Venkatesh BabuVideo Analytics Lab, Computational and Data Sciences Indian Institute of Science, [email protected], [email protected], [email protected] image representations using caption generators D. L. Kovrizhin December 30, 2023 =================================================== Deep learning exploits large volumes of labeled data to learn powerful models. When the target dataset is small, it is a common practice to perform transfer learning using pre-trained models to learn new task specific representations. However, pre-trained CNNs for image recognition are provided with limited information about the image during training, which is label alone. Tasks such as scene retrieval suffer from features learned from this weak supervision and require stronger supervision to better understand the contents of the image. In this paper, we exploit the features learned from caption generating models to learn novel task specific image representations. In particular, we consider the state-of-the art captioning system Show and Tell <cit.> and the dense region description model DenseCap <cit.>. We demonstrate that, owing to richer supervision provided during the process of training, the features learned by the captioning system perform better than those of CNNs. Further, we train a siamese network with a modified pair-wise loss to fuse the features learned by <cit.> and <cit.> and learn image representations suitable for retrieval. Experiments show that the proposed fusion exploits the complementary nature of the individual features and yields state-of-the art retrieval results on benchmark datasets.strong supervision, image representations, image retrieval, feature fusion, transfer learning, caption generators, region descriptors § INTRODUCTION Deep learning has enabled us to learn various sophisticated models using large amounts of labeled data. Computer vision tasks such as image recognition, segmentation, face recognition, etc. require large volumes of labeled data to build reliable models. However, when the training data is not sufficient, in order to avoid over-fitting, it is a common practice to use pre-trained models rather than training from scratch. This enables us to utilize the large volumes of data (eg: <cit.>) on which the pre-trained models are learned and transfer that knowledge to the new target task. Hierarchical nature of the learned representations and task specific optimization makes it easy to reuse them. This process of reusingpre-training and learning new task specific representations is referred to as transfer learning or fine-tuning the pre-trained models. There exist many successful instances of transfer learning (e.g. <cit.>) in computer vision using Convolution Neural Networks (CNNs). Large body of these adaptations are fine-tuned architectures of the well-known recognition models <cit.> trained on the IMAGENET <cit.> and Places <cit.> datasets. However, these models perform object or scene classification and have very limited information about the image. All that these models are provided with during training is the category label. No other information about the scene is provided. For example, the image shown in Figure <ref> has dog and person as labels. Useful information such as indoor or outdoor, interaction between the objects, presence of other objects in the scene is missing. Tasks such as image search (similar image retrieval) suffer from the features learned by this weak supervision. Image retrieval requires the models to understand the image contents in a better manner (eg: <cit.>) to be able to retrieve similar images. Specifically, when the images have multiple objects and graded relevance scores (multiple similarity levels, eg: on a scale from 1 to 5) instead of binary relevance (similar or dissimilar), the problem becomes more severe.On the other hand, automatic caption generation models (e.g. <cit.>) are trained with human given descriptions of the images. These models are trained with stronger supervision compared to the recognition models. For example, Figure <ref> shows pair of images form MSCOCO <cit.> dataset along with their captions. Richer information is available to these models about the scene than mere labels. In this paper, we exploit the features learned via strong supervision by these models and learn task specific image representations for retrieval via pairwise constraints. In case of CNNs, the learning acquired from training for a specific task (e.g. recognition on IMAGENET) is transferred to other vision tasks. Transfer learning followed by task specific fine-tuning has proven to be efficient to tackle less data scenarios. However, similar transfer learning is left unexplored in the case of caption generators. For the best of our knowledge, this is the first attempt to explore that knowledge via fine-tuning the representations learned by them to a retrieval task.The major contributions of our work can be listed as: * We show that the features learned by the image captioning systems represent image contents better than those of CNNs via image retrieval experiments. We attempt to exploit the strong supervision observed during their training via transfer learning. * We train a siamese network using a modified pair-wise loss suitable for non-binary relevance scores to fuse the complementary features learned by <cit.> and <cit.>. We demonstrate that the task specific image representations learned via our proposed fusion achieve state-of-the-art performance on benchmark retrieval datasets.The paper is organised as follows: Section <ref> provides a short summary of <cit.> and <cit.> before presenting details about the proposed approach to perform transfer learning. This section also discusses the proposed fusion architecture. Section <ref> details the experiments performed on benchmark datasets and discusses various aspects along with the results.Finally, Section <ref> concludes the paper. § ALTERNATE IMAGE REPRESENTATIONS Transfer learning followed by task specific fine-tuning is a well known technique in deep learning. In this section we present an approach to exploit the fine supervision employed by the captioning models and the resulting features. Especially, we target the task of similar image retrieval and learn suitable features. Throughout the paper, we consider the state-of-the art captioning model Show and Tell by Vinyals et al. <cit.>. Their model is an encoder-decoder framework containing a simple cascading of a CNN to an LSTM. The CNN encodes visual information from the input image and feeds via a learnable transformation W_I to the LSTM. This is called image encoding, which is shown in Figure <ref> in green color. The LSTM's task is to predict the caption word by word conditioned on the image and previous words. Image encoding is the output of a transformation (W_I) learned from the final layer of the CNN (Inception V3 <cit.>) before it is fed to the LSTM. The system is trained end-to-end with image-caption pairs to update the image and word embeddings along with the LSTM parameters. Note that the Inception V3 layers (prior to image encoding) are frozen (not updated) during the first phase of training and they are updated during the later phase.The features at the image encoding layer W_I (green arrow in Figure <ref>) are learned from scratch. Note that these are the features input to the text generating part and fed only once.These features are very effective to summarize all the important visual content in the image to be described in the caption. These features need to be more expressive than the deep fully connected layers of the typical CNNs trained with weak supervision (labels). Therefore, we consider transferring these features to learn task specific features for image retrieval. We refer to these features as Full Image Caption (FIC) features since the generated caption gives a visual summary of the whole image.On the other hand Johnson et al. <cit.> proposed an approach to densely describe the regions in the image, called dense captioning task. Their model contains a fully convolutional CNN for object localization followed by an RNN to provide the description. Both the modules are linked via a non-linear projection (layer), similar to <cit.>. The objective is to generalize the task of object detection and image captioning. Their model is trained end-to-end over the Visual genome <cit.> dataset which provides object level annotations and corresponding descriptions. They fine-tune the later (from fifth) layers of the CNN module (VGG <cit.> architecture) along with training the image encodings and RNN parameters. Similar to FIC features we consider the image encodings to transfer the ability of this model to describe regions in the image. This model provides encodings for each of the described image regions and associated priorities. Figure <ref> (right panel) shows an example image and the region descriptions predicted by DenseCap model. Note that the detected regions and corresponding descriptions are dense and reliable. In order to have a summary of the image contents, we perform mean pooling on the representations (features) belonging to top-K (according to the predicted priorities) regions. We refer to the pooled encodings as Densecap features.Especially for tasks such as image retrieval, models trained with strong object and attribute level supervision can provide better pre-trained features than those of weak label level supervision. Therefore, we propose an approach to exploit the Densecap features along with the FIC features and learn task specific image representations. §.§ Complementary features and Fusion Figure <ref> shows descriptions predicted by <cit.> and <cit.> for a sample image. Note that the predictions are complementary in nature. FIC provides the summary of the scene: a boy is standing next to a dog. Where as, Densecap provides more details about the scene and objects: presence of green grass, metal fence, brick wall and attributes of objects such as black dog, white shirt,etc. In the proposed approach, we attempt to learn image representations that exploit the strong supervision available from the training process of <cit.> and <cit.>. Further, we take advantage of the complementary nature of these two features and fuse them to learn task specific features for image retrieval. We train a siamese network to fuse both the features. The overview of the architecture is presented in Figure <ref>. The proposed siamese architecture has two wings. A pair of images is presented to the network along with their relevance score (high for similar images, low for dissimilar ones). In the first layer of the architecture, FIC and Densecap features are late fused (concatenated) and presented to the network. A sequence of layers is added on both the wings to learn discriminative embeddings. Note that these layers on both the wings have tied weights (identical transformations in the both the paths). In the final layer, the features are compared to find the similarity and the loss is computed with respect to the ground truth relevance. The error gets back-propagated to update the network parameters. Our network accepts the complementary information provided by both the features and learns a metric via representations suitable for image retrieval. More details about the training are presented in section <ref>.§ EXPERIMENTS§.§ Datasets We begin by explaining the retrieval datasets[The datasets are available at <http://val.serc.iisc.ernet.in/attribute-graph/Databases.zip>] considered for our experiments. In order to have more natural scenario, we consider retrieval datasets that have graded relevance scores instead of binary relevance (similar or dissimilar). We require the relevance to be assigned based on overall visual similarity as opposed to any one particular aspect of the image (e.g. objects). To match these requirements, we consider two datsets rPascal (ranking Pascal) and rImagenet (ranking Imagenet) composed by Prabhu et al. <cit.>. These datasets are subsets of aPascal <cit.> and Imagenet <cit.> respectively. Each of the datasets contains 50 query images and a set of corresponding relevant images. They are composed by 12 annotators participating to assign relevance scores. The scores have 4 grades, ranging from 0 (irrelevant) to 3 (excellent match).* rPascal: This dataset is composed from the test set of aPascal <cit.>. The queries comprise of 18 indoor and 32 outdoor scenes. The dataset consists of a total of 1835 images with an average of 180 reference images per query.* rImagenet: It is composed from the validation set of ILSVRC 2013 detection challenge. Images containing at least 4 objects are chosen. The queries contain 14 indoor scenes and 36 outdoor scenes. The dataset consists of a total of 3354 images with an average of 305 reference images per query. Figure <ref> shows sample images from the two datasets. Note that the first image in each row is query and the following images are reference images with relevance scores displayed at top right corner. §.§ Evaluation metricWe followed the evaluation procedure presented in <cit.>. For quantitative evaluation of the performance, we compute normalized Discounted Cumulative Gain (nDCG) of the retrieved list. nDCG is a standard evaluation metric used for ranking algorithms (e.g. <cit.> and <cit.>). For all the queries in each dataset, we find the nDCG value and report the mean nDCG per dataset evaluated at different ranks (K). §.§ Features learned by the caption generator (FIC) In this subsection we demonstrate the effectiveness of the features obtained from the caption generation model <cit.>. For each image we extract the 512D FIC features to encode it's contents. Note that these are the features learned by the caption generation model via the strong supervision provided during the training. Retrieval is performed by computing distance between the query and the reference images' features and arranging in the increasing order of the distances. Figure <ref> and <ref> show the plots of nDCG evaluted at different ranks (K) on the two datasets. As a baseline comparison, we have compared the performance of FIC features with that of the non-finetuned visual features of inception v3 model (blue color plot in Figure <ref> and <ref>). Note that these are 2048D features that are extracted from the last fully connected layer of the inception v3 model <cit.>. The FIC features outperform the non-finetuned visual features by a large margin emphasizing the effectiveness of the strong supervision.We have considered another baseline using the natural language descriptors. For a given image, we have predicted the text caption using the Show and Tell <cit.> model. After pre-processing (stop word removal and lemmatizing), we encode each of the remaining words using word2vec <cit.> embeddings and mean pool them to form an image representation. Note that the FIC features perform better than this baseline also.We also compare the performance of FIC features against the state-of-the art Attribute graph approach <cit.>. The FIC features clearly outperform the Attribute Graph approach in case of both the benchmark datasets. §.§ Task specific fine-tuning through fusionThe proposed fusion architecture[Project codes can be found at <https://github.com/mopurikreddy/strong-supervision>] is trained with pairs of images and corresponding relevance scores (y). The typical pairwise training consists of binary relevance scores: simila r(1) or dissimilar (0). The objective is to reduce the distance between the projections of the images if they are similar and separate them if dissimilar. Equation (<ref>) shows the contrastive loss <cit.> typically used to train siamese networks.E = 1/2N∑_n=1^N (y) d + (1-y) max(∇-d, 0)where, E is the prediction error, N is the mini-batch size, y is the relevance score (0 or 1), d is the distance between the projections of the pair of images and ∇ is the margin to separate the projections corresponding to dissimilar pair of images.However, in practice images can have non-binary relevance scores. To handle more fine grained relevances, we modified the contrastive loss function to include non-binary scores as shown in equation (<ref>) E = 1/2N∑_n=1^N (y^2) d +1(y=0) max (∇-d^2, 0) where 1(.) is indicator function. Note that the modified loss function favours the nDCG measure by strongly punishing (due to the square term) the distances between images with higher relevance scores.We train a siamese network with 5 fully connected layers on both the wings, with tied weights. The number of units in each wing are 1024-2048-1024-512-512. The representations learned at the last layer are normalized and euclidean distance is minimized according to Equation (<ref>). We divide the queries into 5 splits to perform 5 fold validation and report the mean nDCG. That is, each fold contains image pairs of 40 queries and corresponding reference images for training. The remaining 10 queries form the evaluation set. On an average, each fold contains 11300 training pairs for rPascal and 14600 pairs for rImagenet.Figures <ref> and <ref> show the performance of the task specific image representations learned via the proposed fusion. For evaluating the performance of the Densecap <cit.> method, we have mean pooled the encodings corresponding to top-5 image regions resulting a 512D feature. FIC feature is also 512D vector, therefore forming an input of 1024D to the network. Note that the transfer learning and fine-tuning through fusion improves the retrieval performance on both the datasets.§ CONCLUSION In this paper, we have presented an approach to exploit the strong supervision observed in the training of caption generation systems. We have demonstrated that image understanding tasks such as retrieval can benefit from this strong supervision compared to weak label level supervision. Transfer learning followed by task specific fine-tuning is commonly observed in CNN based vision systems. However similar practice is relatively unexplored in the case of these captioning systems. Our approach can potentially open new directions for exploring other sources for stronger supervision and better learning. It can also motivate to tap the juncture of vision and language in order to build more intelligent systems. § ACKNOWLEDGEMENTThis work was supported by Defence Research and Development Organization (DRDO), Government of India.IEEEbib | http://arxiv.org/abs/1705.09142v1 | {
"authors": [
"Konda Reddy Mopuri",
"Vishal B. Athreya",
"R. Venkatesh Babu"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170525121327",
"title": "Deep image representations using caption generators"
} |
⟨#|1#1 |#⟩1#1 #1#1 ⟨#|1⟩#1 #1‖#1‖ #1#2#2#1 #1#[email protected] Physics Department, Technion-Institute of Technology, Haifa 32000, [email protected] Physics Department, Technion-Institute of Technology, Haifa 32000, [email protected] Physics Department, Technion-Institute of Technology, Haifa 32000, [email protected] Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA We suggest that the exclusive Higgs + light (or b)-jet production at the LHC, pp → h+j(j_b), is a rather sensitive probe of the light-quarks Yukawa couplings and of other forms of new physics (NP) in the Higgs-gluon hgg and quark-gluon qqg interactions. We study the Higgs p_T-distribution in pp → h+j(j_b) →γγ + j(j_b), i.e., in h+j(j_b) production followed by the Higgs decay h →γγ, employing the (p_T-dependent) signal strength formalism to probe various types of NP which are relevant to these processes and which we parameterize either as scaled Standard Model (SM) couplings (the kappa-framework) and/orthrough new higher dimensional effective operators (the SMEFT framework). We find that the exclusive h+j(j_b) production at the 13 TeV LHC is sensitive to various NP scenarios, with typical scales ranging from a few TeV to O(10) TeV, depending on the flavor, chirality and Lorentz structure of the underlying physics.Light-quarks Yukawa couplings and new physics in exclusive high-p_T Higgs + jet and Higgs + b-jet events Amarjit Soni December 30, 2023 ======================================================================================================== § INTRODUCTIONThe next runs of the LHC will be dedicated to two primary tasks: the search for new physics (NP) and the detailed scrutiny of the Higgs properties, which might shed light on NP specifically related to the origin of mass and flavor and to the observed hierarchy between the two disparate Planck and ElectroWeak (EW) scales. Indeed, the study of Higgs systems is in particular challenging, since it requires precision examination of some of its weakest couplings (within the SM) and measurements of highly non-trivial processes involving high jet multiplicities, large backgrounds and low detection efficiencies.The s-channel Higgs production and its subsequent decays, pp → h → ff, which led to its discovery, are relatively inefficient for NP searches. In particular, if the NP scale, Λ, is of O( TeV) and larger, then its effect in these processes is expected to be suppressed by at least ∼ m_h^2/Λ^2, since most of these events come from the dominant gluon fusion s-channel production mechanism and are, therefore, clustered around √(s)≃ m_h. However, in some fraction of the events, the Higgs recoils against one or more hard jets and, thus, carries a large p_T, which may play a key role in the hunt for NP and/or for background rejection in Higgs studies. Indeed, a key observable for Higgs boson events is the number of jets produced in the event. For that reason, and since the Higgs p_T distribution is sensitive to the production mechanism, there has recently been a growing interest, both experimentally <cit.> and theoretically <cit.>, in the behavior of the Higgs p_T distribution in inclusive and exclusive Higgs production, where the Higgs carries a substantial fraction of transverse momentum (for earlier work see <cit.>). In particular, the Higgs p_T distribution in the exclusive Higgs + jets production, pp → h + nj, was one of the prime targets of the measurements performed recently by ATLAS and CMS <cit.>.In this paper we will thus focus on the exclusive Higgs + 1-jet production, pp → h + j, where j stands for either a “light-jet" defined as any non-flavor tagged jet originating from a gluon or light-quarks j = g,u,d,c,s (i.e., assuming them to be indistinguishable from the observational point of view) or a b-quark jet (j_b). It is interesting to note that there has been some hints in the LHC 8 TeV data for an excess in the h+j channel <cit.>, although the statistics are still limited and the theoretical uncertainties are relatively large. Indeed, a significant effort has been dedicated in recent years, from the theory side, towards understanding and reducing the uncertainties pertaining to the Higgs+jet production cross-section at the LHC <cit.>, with special attention given to higher transverse momentum of the Higgs, where NP effects are expected to become more apparent. In particular, the high-p_T Higgs spectrum in pp → h+j(j_b) can be sensitive to various well motivated NP scenarios, such as supersymmetry <cit.>, heavy top-partners <cit.>, higher dimensional effective operators <cit.> and NP in Higgs-top quark and Higgs-gluon interactions in the so-called “kappa-framework", where one assumes that the hgg and htt interactions are scaled by some factor with respect to the SM <cit.>.In general, there is a tree-level contribution to pp → h+j(j_b) in the SM from the hard processes gq → q h, g q̅→q̅ h and q q̅→ g h (q = u,d,c,s,b). The corresponding SM tree-level diagrams, which are depicted in Fig. <ref>, are proportional to the light-quarks Yukawa couplings, y_q, so that the SM tree-level contribution to the overall pp → h+j(j_b) cross-section is small (e.g., in the case of pp → h+c, it is at the percent level). In particular, the squared matrix elements, summed and averaged over spins and colors, for these tree-level hard processes are: ∑|M_SM^q q̅→ gh|^2 = 2 g_s^2 y_q^2/ C_qqm_h^4+ŝ^2/t̂û ,∑|M_SM^q g → q h|^2 =-C_qq/ C_qg∑|M_SM^q q̅→ gh|^2 (ŝ↔t̂) , ∑|M_SM^q̅ g →q̅ h|^2 =-C_qq/ C_qg∑|M_SM^q q̅→ gh|^2 (ŝ↔û) , where ŝ=(p_1+p_2)^2, t̂=(p_1+p_3)^2 and û=(p_2+p_3)^2, defined for the process q(-p_1)+ q̅(-p_2) → h + g(p_3). Also, g_s is the strong coupling constant and C_qq=N^2, C_qg=NV are the color average factors, where V=N^2-1=8 corresponds to the number of gluons in the adjoint representation of the SU(N) color group.Thus, in the limit y_q → 0, the dominant and leading order (LO) SM contribution to the Higgs + light-jet cross-section, σ(pp → h+j), arises from the 1-loop process gg → gh, which is generated by 1-loop top-quark exchanges (and the subdominant b-quark loops <cit.>), and can be parameterized by an effective Higgs-gluon ggh interaction Lagrangian: L_eff^ggh = C_g h G_μν^a G^μν, a , where C_g is the Higgs-gluon point-like effective coupling, which at lowest order in the SM is <cit.>: C_g=α_s/(12π v), where v=246 GeV is the Higgs vacuum expectation value (VEV). In what follows we will use the point-like ggh effective coupling of Eq. <ref> with C_g given as an asymptotic expansion in 1/m_t up to m_t^-6, as implemented in MADGRAPH5 for the Higgs effective field theory (HEFT) model <cit.>. We will neglect throughout this work the 1-loop effects of the b-quark and of the lighter quarks with enhanced Yukawa couplings (i.e., as large as the b-quark Yukawa), which are expected to yield a correction at the level of a few percent compared to the dominant top-quark loops, when the Higgs transverse momentum is larger than ∼ m_h/2 <cit.>.This prescription for the Higgs-gluon coupling is a good approximation for a Higgs produced with a p_T(h)200 GeV, see e.g., <cit.>, whereas, as will be shown in this work, the harder p_T(h)200 GeV regime is important for probing NP in Higgs +jet production. However, since the exact form of the loop induced ggh interaction (i.e., using a finite top-quark mass) is currently unknown beyond LO (1-loop), we choose to work with the effective ggh point-like interaction (as described above) in order to simplify the calculation and the presentation of our analysis. Given the exploratory nature of this work and the type of study presented, this approximation is not expected to have an effect on our results at a level which changes the main outcome and conclusions of this work. In particular, in order to give an estimate of the sensitivity of our results to the calculation scheme, we will also study and analyse some samples of our results using the exact LO calculation of the 1-loop diagrams (mass dependent top-quark exchanges) which involve the ggh interaction vertex. Indeed, since this LO 1-loop calculation is the only currently available exact (mass dependent) calculational setup for pp → h+j(j_b), a comparison between the NP effects calculated with the point-like ggh approximation and with the mass dependent 1-loop diagrams can serve as a yardstick for the uncertainty and sensitivity of our results to the calculational setup. The subprocesses gq → q h, g q̅→q̅ h and q q̅→ g h (which, as can be seen from Eqs. <ref>-<ref>, are proportional to y_q^2 at tree-level) also receive a 1-loop contribution from the above ggh effective vertex (i.e., from the top-quark loops), which is, however, small compared to the gg → gh <cit.>. In particular, the gg → gh contribution to σ(pp → h+j) at the LHC is about an order of magnitude larger than the one from gq → q h and more than two orders of magnitude larger than the two other channels g q̅→q̅ h and q q̅→ g h. The 1-loop (and LO for y_q =0) SM differential hard cross-sections for gg → gh, gq → q h, g q̅→q̅ h and q q̅→ g h (the corresponding SM diagrams for all channels are shown in Fig. <ref>), expressed in terms of the above effective ggh interaction and neglecting the light-quark masses, are given by <cit.>: ∑|M_SM^gg → gh|^2 ≃ 96 g_s^2 C_g^2/ C_ggm_h^8+ŝ^4+t̂^2+û^2/ŝt̂û ,∑|M_SM^q q̅→ gh|^2 ≃ 16 g_s^2 C_g^2/ C_qqt̂^2+û^2/ŝ ,∑|M_SM^q g → q h|^2 ≃-C_qq/ C_qg∑|M_SM^q q̅→ gh|^2 (ŝ↔t̂) , ∑|M_SM^q̅ g →q̅ h|^2 ≃-C_qq/ C_qg∑|M_SM^q q̅→ gh|^2 (ŝ↔û) , where C_gg=V^2=64 and ŝ=(p_1+p_2)^2, t̂=(p_1+p_3)^2, û=(p_2+p_3)^2, with momenta defined via h → g(p_1)+g(p_2)+g(p_3) for gg → gh and via q(-p_1)+ q̅(-p_2) → h + g(p_3) for q q̅→ gh.Turning now to the possible manifestation of NP in Higgs + jet production at the LHC, there are, in principle, two ways in which pp → h+j(j_b) can be modified: * when the NP generates new interactions that are absent in the SM and that can potentially change the SM kinematic distributions in this process.* when the NP comes in the form of scaled SM couplings, corresponding to the previously mentioned kappa-framework.We will explore both types of NP effects in pp → h+j and pp → h+j_band, in particular, focus on NP that modifies the light and b-quarks Yukawa couplings and/or the light and b-quarks interactions with the gluon, as well as the Higgs-gluon effective vertex in Eq. <ref>. Indeed, the Higgs mechanism of the SM implies that the fermion's Yukawa couplings are proportional to the ratio between their masses and the EW VEV, i.e., y_f ∝ m_f/v. Thus, at least for the light fermions of the 1st and 2nd generations [where m_f/v ∼ O(10^-5) and m_f/v ∼ O(10^-4-10^-3), respectively], any signal which can be associated with their Yukawa couplings would stand out as clear evidence for NP beyond the SM. The current experimental bounds on the Yukawa couplings of light-quark's of the 1st and 2nd generations, y_u,y_d,y_s,y_c, coming from fits to the measured Higgs data, allow them to be as large as the b-quark Yukawa y_b <cit.>. From the phenomenological point of view, it is, therefore, important to explore the possibility that the light-quark Yukawa couplings and/or their interactions with the gauge boson's are significantly enhanced or modified with respect to the SM. Indeed, there has recently been a growing interest in the study of light-quark's Yukawa couplings, see e.g., <cit.>. For example, in <cit.>, the Higgs p_T distributions in inclusive Higgs production, pp → h+X, was used to study the sensitivity to y_q, where it was shown that the measurements from the 8 TeV LHC run constrain the Yukawa couplings of the 1st generation quarks and the c-quark to be y_u,y_d0.5 y_b <cit.> and y_c5 y_b <cit.>, respectively. Slightly improved bounds are expected in the inclusive channel at the future LHC Runs: y_u,y_d0.3 y_b <cit.> and y_cy_b <cit.>. As we will see below, a p_T-dependent ratio between the NP and SM cross-sections (the signal strength) for the exclusive Higgs + jet production cross-section, σ(pp → h +j), followed by the Higgs decays to e.g., γγ and WW^⋆, may be used to put comparable and, in some cases, stronger constraints on y_q. In particular, we will show that, if the ggh effective coupling also deviates from its SM value, then significantly stronger bounds on y_q are expected.We also explore exclusive Higgs + jet production in the SMEFT, defined as the expansion of the SM Lagrangian with an infinite series of higher dimensional effective operators. We find that the exclusive pp → h+j(j_b) signal can probe the NP scenarios portrayed by the SMEFT with typical scales ranging from a few to O(10) TeV, depending on the details of underlying physics.The paper is organized as follows: in section <ref> we outline our notation and define our observables for the study of NP in pp → h+j and pp → h+j_b. In sections <ref> and <ref> we discuss the NP effects in pp → h+j(j_b) within the kappa and the SMEFT frameworks, respectively, and in section <ref> we summarize. § NOTATION AND OBSERVABLESWe define the signal strength for pp → h+j (and similarly for pp → h+j_b), followed by the Higgs decay h → ff, where f can be any of the SM Higgs decay products (e.g.,f=b, τ, γ, W, Z), as the ratio of the number of pp → h+j → ff + j events in some NP scenario relative to the corresponding number of Higgs events in the SM:μ_hj^f =N(pp → h+j → ff+j)/ N_SM(pp → h+j → ff+j) . In particular, N is the event yield N= Lσ Aϵ, where L is the luminosity, A is the acceptance in the signal analysis (i.e., the fraction of events that ”survive" the cuts) and ϵ is the efficiency which represents the probability that the fraction of events that pass the set of cuts are correctly identified. Clearly, the luminosity and efficiency factors, L and ϵ, cancel by definition in μ_hj^f of Eq. <ref>, whereas the acceptance factors, A and A_SM, do not in general, unless the NP in the numerator of μ_hj^f does not change the kinematics of the events. Given the exploratory nature of this work, we will assume, for simplicity, that A≃ A_SM in Eq. <ref>, in which case one obtains:^[1][1]The effect of A≠ A_SM can be estimated by simulating the detector acceptance in the actual analysis, and scaling our results below (for the signal strength μ_hj^f) by the factor A/ A_SM.μ_hj^f ≃σ(pp → h+j)/σ_SM(pp → h+j)·BR(h → ff)/BR_SM(h → ff) . We further assume that there is no NP in the Higgs decay h → ff and, for definiteness, we will occasionally consider the decay channel h →γγ (i.e., with a SM rate), at the LHC with a luminosity of 300 fb^-1 and/or 3000 fb^-1 (corresponding to the high-luminosity LHC, HL-LHC), representing the lower and higher statistics cases for the Higgs + jet signal pp → h + j →γγ +j.We will henceforward use the p_T-dependent “cumulative cross-section", satisfying a given lower Higgs p_T cut, as follows: σ(p_T^cut) ≡σ( p_T(h) > p_T^cut) = ∫_p_T(h) ≥ p_T^cut dp_T dσ/dp_T , which turns out to be useful for minimizing the ratio between the higher-order and LO pp → h+j cross-sections (i.e., the K-factor) for values of p_T^cut 150 GeV <cit.>. Furthermore, as was mentioned earlier and will be shown below, the p_T-distribution of the Higgs may be sensitive to the specific type of the underlying NP, so that the cumulative cross-section of Eq. <ref> gives an extra handle for extracting the NP effects in pp → h+j, without having to analyze fully differential quantities associated with pp → h+j.All cross-sections are calculated using MadGraph5 <cit.> at LO parton-level, where a dedicated universal FeynRules output (UFO) model was produced for the MadGraph5 sessions using FeynRules <cit.>, for both the kappa and SMEFT frameworks. The analytical results were cross-checked with Formcalc <cit.>, while intermediate steps were validated using FeynCalc <cit.>. We use the LO MSTW 2008 PDF set <cit.>, in the 4 flavor and 5 flavor schemes MSTW2008lo68cl_nf4 and MSTW2008lo68cl, respectively, with a dynamical scale choice for the central value of the factorization (μ_F) and renormalization (μ_R) scales, corresponding to the sum of the transverse mass in the hard-process level: μ_F = μ_R = μ_T ≡∑_i √(m_i^2+p_T^2(i)) = √(m_h^2+p_T^2(h)) + p_T(j). The uncertainty in μ_Fand μ_R is evaluated by varying them in the range 1/2μ_T ≤μ_F, μ_R ≤ 2 μ_T. As mentioned above, all cross-sections were calculated with a lower p_T(h) cut and, in some instances, an overall invariant mass cut was imposed using Mad-Analysis5 <cit.>. To study the sensitivity of μ_hj^f to NP we define our NP signal to be (recall that μ_hj^f(SM) = 1): Δμ_hj^f ≡|μ_hj^f - 1| , and assume that μ_hj^f will be measured to a given accuracy δμ_hj,exp^f(1σ), with a central value μ̂_hj,exp^f: μ_hj,exp^f=μ̂_hj,exp^f ±δμ_hj,exp^f(1σ) .Thus, taking μ̂_hj,exp^f = μ_hj^f (μ_hj^f being our prediction for the measured value μ̂_hj,exp^f), the statistical significance of the NP signal is: N_SD = Δμ_hj^f/δμ_hj^f , which we will use in the following analysis, where δμ_hj^f represents the combined experimental and theoretical 1σ error, e.g., δμ_hj^f = √((δμ_hj,theory^f)^2 + (δμ_hj,exp^f)^2). In particular, in the spirit of the ultimate goal of the Higgs physics program, which is to reach a percent level accuracy in the measurements and calculations of Higgs production and decay modes <cit.>, we will assume throughout this work that the signal strength defined above, for Higgs+jet production followed by the Higgs decay, will be measured and known to a 5%(1σ) accuracy. That is, that the combined experimental and theoretical uncertainties will be pushed down to δμ_hj^f = 0.05(1σ). Indeed, achieving such an accuracy is both a theoretical and experimental challenge, which, however, seems to be feasible in the LHC era with the large statistics expected in the future runs and in light of the recent progress made in higher-order calculations.Finally, we wish to briefly address the uncertainty associated with the effective point-like ggh approximation which we use for the calculation of all the SM-like diagrams for pp → h+j(j_b) that involve the ggh interaction (i.e., all diagrams in Fig. <ref> in the pp → h+j case and diagram (e) in Fig. <ref> for the pp → h+j_b case). As mentioned earlier, for the differential p_T(h) distribution, d σ/dp_T(h), this approximation is accurate up to p_T(h)200 GeV. As a result, the p_T-dependent cumulative cross-section defined in Eq. <ref> accrues an error which depends on the p_T^cut used. To estimate the corresponding uncertainty in σ_SM(p_T^cut), we plot in Fig. <ref> the ratio: r_ggh≡σ_SM^point-like(p_T^cut)/σ_SM^exact-LO(p_T^cut) , as a function of p_T^cut for both pp → h+j and pp → h+j_b, where σ_SM^point-like(p_T^cut) and σ_SM^exact-LO(p_T^cut) are the cumulative cross-sections which are calculated for a given p_T^cut, using the point-like ggh approximation and the full LO 1-loop set of diagrams (i.e., top-quark loops with a finite top-quark mass), respectively. The loop-induced SM cross-sections were calculated using the loopSM model of MadGraph5.We see that the point-like ggh approximation overestimates the cumulative cross-sections for exclusive Higgs + jet production, in particular at large p_T(h), and that the effect is more pronounced in the Higgs + b-jet case. In particular, for p_T^cut = 100, 200, 400 GeV, we find r_ggh∼ 1, 1.4, 2.9 for pp → h+j and r_ggh∼ 1.3, 1.8, 3.6 for pp → h+j_b. Thus, by using the effective point-like ggh vertex we are overestimating the Higgs + jet cross-sections (which are dominated by the SM diagrams involving the ggh interaction) and, therefore, the corresponding expected number of Higgs + jet events, roughly by a factor of r_ggh. On the other hand, as will be shown later, the statistical significance of the signals (N_SD defined in Eq. <ref> above) only mildly depend on the calculation scheme (i.e., on r_ggh). We will address these issues in a more quantitative manner below. § HIGGS + JET PRODUCTION IN THE KAPPA-FRAMEWORKThe kappa-framework is defined by multiplying the SM couplings g_i by a scaling factor κ_i, which parameterizes the effects of NP when it has the same Lorentz structure as the corresponding SM interactions <cit.>. In the case of pp → h+j(j_b), the relevant scaling factors apply to the effective (1-loop) Higgs-gluon interaction of Eq. <ref> and to the light and/or b-quark Yukawa couplings. In particular, the effective interaction Lagrangian for pp → h+j(j_b) in the kappa-framework, takes the form: L_eff^h+j = - ∑_q=u,d,s,c,bκ_q m_b/v h q̅ q + κ_g C_g h G_μν^a G^μν, a , where we have scaled the light-quark Yukawa coupling, y_q, with the SM b-quark Yukawa: κ_q ≡y_q/y_b^SM , and y_b^SM= √(2)m_b/v. In particular, κ_g=1, κ_b=1, κ_c∼ 0.3, κ_s∼ O(10^-2) and κ_u,d∼ O(10^-3) are the SM strengths for the corresponding couplings. In what follows, we will refer to the SM case by κ_u,d,c,s = 0, since the effect of the small SM values for κ_u,d,c,s in pp → h+j are negligible. §.§ The case of Higgs + light-jet productionAs mentioned earlier, in the case of pp → h+j, where j=g,u,d,s,c is a non-flavor tagged light-jet originating from a gluon or any quark of the 1st and 2nd generations, the SM tree-level diagrams involving the light-quarks Yukawa couplings are vanishingly small (see Eqs. <ref>-<ref>). Therefore, the dominant SM contribution to σ(pp → h+j) arises at 1-loop via the sub-processes gg → gh, gq → q h, g q̅→q̅ h and q q̅→ g h (the corresponding diagrams are depicted in Fig. <ref>, where the loops are represented by an effective ggh vertex). In particular, using the Higgs-gluon effective Lagrangian of Eq. <ref>, the corresponding total SM cross-section for pp → h+j can be written as: σ_SM^hj = C_g^2 (σ^gg_SM + σ^gq_SM + σ^g q̅_SM + σ^q q̅_SM) , where σ^ij_SM, for ij=gg,gq,g q̅,q q̅, can be obtained from the corresponding squared amplitudes given in Eqs. <ref>-<ref>. For example, σ^gg_SM is part of the SM cross-section coming from gg → gh, which is the dominant sub-process in the SM.On the other hand, turning on the light-quark qqh Yukawa couplings and allowing for deviations also in the Higgs-gluon ggh interaction, within the kappa-framework of Eq. <ref>, we obtain the total NP cross-section for pp → h+j: σ^hj = κ_g^2 σ_SM^hj + κ_q^2 σ_qqh^hj , where σ_SM^hj≃σ^hj(κ_g=1,κ_q=0) is given in Eq. <ref> and σ_qqh^hj=σ^hj(κ_g=0,κ_q=1) arises from the the s-channel and t-channel tree-level gq → qh diagrams, depicted in Fig. <ref>, where only the (scaled) light-quarks qqh Yukawa couplings contribute. The interference term between the diagrams involving the ggh and qqh couplings is proportional to the light-quark mass and is, therefore, neglected in Eq. <ref>. In particular, σ^hj is practically insensitive to the signs of κ_g and κ_q.Furthermore, in the hgg - h q̅ q kappa-framework of Eq. <ref>, the ratio of branching ratios in Eq. <ref> is given by: μ_h → ff ≡ BR(h → ff)/BR_SM(h → ff)= 1/1 + (κ_g^2 -1 ) BR_SM^gg + κ_q^2 BR_SM^bb , where BR_SM^gg,bb = BR_SM(h → gg,bb) and we will assume no NP in the Higgs decay h → ff. In particular, as mentioned above, we assume that the Higgs decays via h →γγ with a SM decay rate.Collecting the expressions from Eqs. <ref>, <ref> and <ref>, we obtain the signal strength in the kappa-framework:μ_hj^f = ( κ_g^2 + κ_q^2 R^hj) ·μ_h → ff , where R^hj≡σ_qqh^hj/σ_SM^hj , is the NP contribution scaled with the SM cross-section and calculated using cumulative cross-sections, as defined in Eq. <ref>, i.e., for a given p_T^cut in both numerator and denominator: R^hj = R^hj(p_T^cut) = σ_qqh^hj(p_T^cut)/σ_SM^hj(p_T^cut). The ratio R^hj contains all the dependence of μ_hj^f on the Higgs p_T and, as will be further discussed below, is where all the uncertainties reside, i.e., the higher order corrections (K-factor), the theoretical uncertainty of the PDF due to variations of the renormalization and factorization scales and the acceptance factors. In Fig. <ref> we show the dependence of R^hj and the signal strength, μ_hj^f, on p_T^cut, assuming no NP in the hgg interaction (κ_g=1) and for the cases in which either a single or all light-quark Yukawa couplings are modified, i.e., κ_q=1 for any one of the light-quarks q=u,d,s,c or κ_q=1 for all q=u,d,s,c. We find that the effect of κ_q ≠ 0 is to change the softer p_T(h) spectrum, so that R^hj drops when p_T^cut is increased. As a result, the contribution of κ_q to pp → h+j sharply drops in the harder p_T(h) region, p_T(h)300 GeV, where R^hj O(0.1), see Fig. <ref>.Note, however, that the signal strength approaches an asymptotic value as p_T^cut is further increased, which corresponds to the region where the κ_q dependence of μ_hj^f is dominated by the decay factor μ_h → ff in Eq. <ref>. In particular, μ_hj^f → 0.6-0.7 in the single κ_q =1 case and μ_hj^f → 0.3 when κ_q =1 for all light-quarks. Thus, in the high Higgs p_T regime, the difference between the effects of a single κ_q ≠ 0 is small, i.e., for either of the quark flavors q=u,d,c,s. The advantage of monitoring the high p_T(h) spectrum, where R^hj is suppressed is, therefore, reducing the theoretical and experimental uncertainties which, as mentioned above, reside only in R^hj. Indeed, this will be illustrated in Table <ref> below, where we show the sensitivity of the signal to the theoretical uncertainty obtained by scale variations. In Fig. <ref> we plot the expected statistical significance, N_SD defined in Eq. <ref>, assuming a 5% relative error (δμ_hj^f=0.05), as a function of κ_q for two cases: (i) κ_q ≠ 0 for all q=u,d,s,c and (ii) only κ_u ≠ 0. In both cases we assume no NP in the Higgs-gluon coupling (κ_g=1) and we use two different p_T^cut values p_T^cut=100, 400 GeV. We see that, in the single κ_u ≠ 0 case, there is a 3σ sensitivity to values of κ_u0.6, for κ_g =1 and using p_T^cut=400 GeV. In the case where the NP modifies κ_q for all q=u,d,c,s, one can expect a deviation of more than 3 σ for values of κ_q0.3. We also show in Fig. <ref> the corresponding expected number of pp → h + j →γγ +j events, as a function of κ_q for cases (i) and (ii) considered above, with p_T^cut=100 and 400 GeV and an integrated luminosity of 300 and 3000 fb^-1, respectively, assuming a signal acceptance of 50%. We can see that around 1000(100) pp → h + j →γγ +j events with p_T(h) > 100(400) GeV are expected at the LHC(HL-LHC), i.e., with L =300(3000) fb^-1. Thus, in both cases it should be possible to probe the NP effects when the Higgs decays via h →γγ. The signal strength μ_hj^f is more sensitive to NP in the Higgs-gluon coupling, i.e., to κ_g. We find, for example, that if μ_hj^f is known to a 5%(1σ) accuracy, then a deviation of more than 3σ is expected for κ_g0.9 for any value of κ_q and for any p_T^cut 500 GeV. This is illustrated in Fig. <ref> where we plot the 68%, 95% and 99% confidence level (CL) allowed ranges in the κ_q-κ_g plane, for p_T^cut=400 GeV and assuming that the signal strength has been measured to be μ_hj^f ∼ 1 ± 0.05(1σ), i.e., with a SM central value and to an accuracy of δμ_hj^f=5%(1σ). Here also, we consider both the single κ_u case where κ_u ≠ 0 and κ_d=κ_s=κ_c=0 and the case where κ_q ≠ 0 for all q=u,d,s,c. In particular, values of {κ_q,κ_g} outside the shaded 99% contour will be excluded at more than 3σ, if the signal strength will be measured to lie within 0.85 < μ_hj^f < 1.15. In Table <ref> we list the statistical significance of the NP signal, N_SD=Δμ_h_b^f/δμ_h_b^f, as defined in Eq. <ref>, again assuming 5% error (δμ_hj^f=0.05(1σ)), for p_T^cut=400 GeV and some discrete values of the scaled couplings: κ_q=0,0.25,0.5 and κ_g=0.8,0.9,1,1.1,1.2. Here also, results are given in the single κ_u case and in the case where κ_q ≠ 0 for all q=u,d,s,c. We include the theoretical uncertainty obtained by scale variations and (although of little use) write N_SD up to the 2nd digit to illustrate the small uncertainty due the scale variation. Note that for κ_q=0 there is no dependence on the scale of the PDF since, in this case, it is cancelled in the ratio of cross-sections as defined in the signal strength μ_hj^f. We see that indeed the effect of the variation of scale with which the PDF is evaluated is negligible due to the smallness of R^hj in the harder p_T spectrum, in particular for p_T^cut=400 GeV used in the Table <ref> (see also discussion above).All the results presented in this section were obtained using the effective point-like ggh approximation, which as was shown in section <ref> (see Fig. <ref>), overestimates the contribution of the SM-like diagrams involving the 1-loop ggh vertex when compared to the 1-loop induced (top-mass dependent) terms. In particular, this approximation effects the denominator of the scaled NP ratio R^hj in Eq. <ref>, i.e., the SM cumulative cross-section σ_SM^hj(p_T^cut). To give a feeling for the sensitivity of our results to the underlying calculation setup at the high p_T(h) regime, where the point-like ggh approximation shows O(1) deviations, we recalculate the statistical significance N_SD in Table <ref> using the top-mass dependent 1-loop result for σ_SM^hj(p_T^cut) in Eq. <ref>. In this case, the scaled NP ratio R^hj changes to: R^hj→R̃^hj = r_ggh R^hj , where r_ggh, which is defined in Eq. <ref>, is the ratio between the point-like and the LO loop-induced (mass dependent) SM cross-sections. Thus, replacing R^hj→R̃^hj in the expression of Eq. <ref> for the signal strength and using the definition for N_SD in Eq. <ref>, we obtained the statistical significance in the exact 1-loop case: Ñ_SD = r_ggh N_SD - (r_ggh-1 ) ( κ̃_g^2 μ_h → ff - 1 ) /δμ_hj^f , where μ_h → ff is the scaled Higgs decay branching ratio defined in Eq. <ref> and δμ_hj^f is the assumed 1 σ error (see Eq. <ref>). Note that in Eq. <ref> above we have denoted the the modified ggh interaction by κ̃_g (rather than κ_g), since caution has to be taken when interpreting the NP associated with the ggh vertex in the exact top-quark 1-loop case. In particular, in the calculation of σ^hj = σ(pp → h+j) using the effective point-like ggh interaction, κ_g simply corresponds to the scaling of the effective ggh SM vertex (see Eq. <ref>) and, therefore, to the ratio κ_g = √(σ^hj/σ_SM^hj) (see Eq. <ref> for κ_q=0). On the other hand, in the exact LO (1-loop) calculation, the diagrams in Fig. <ref> involving NP in the effective ggh interaction should be added at the amplitude level to the SM 1-loop diagrams (i.e., with the top-quark loops). Thus, in this case, generic NP effects associated with the ggh vertex in σ^hj can be parameterized as follows <cit.>: σ^hj(κ_q=0) = ( κ_t^2 + A κ_t κ_g + B κ_g^2 ) σ_SM^hj≡κ̃_g^2 σ_SM^hj , where κ_t ≡ y_t/y_t^SM is the tth coupling modifier (which parameterizes potential NP in the SM top-quark loop diagrams) and A,B are phase-space coefficients which depend on the lower Higgs p_T cut (p_T^cut), see <cit.>. Thus, when considering NP in pp → h+j within the exact 1-loop calculation, the ggh coupling modifier κ̃_g (defined in Eq. <ref>), which appears in Eq. <ref> and in Table <ref> should be interpreted as the overall NP effect in the ggh interaction, where κ̃_g = κ_t corresponds to NP which modifies only the tth Yukawa coupling while κ̃_g = √(1 + A κ_g + Bκ_g^2 ) applies to the case where κ_t=1 and the NP arises from some other underlying heavy physics which is integrated out and generates the ggh effective interaction of Eq. <ref>. This interpretation of κ̃_g applies to all instances below where we discuss our results for the NP effect in pp → h+j(j_b) within the exact LO 1-loop case.In Table <ref> we list the statistical significance Ñ_SD calculated according to Eq. <ref>, again taking a 5% error δμ_hj^f=0.05(1σ), p_T^cut=400 GeV and the same values of the scaled couplings as in Table <ref>, where here only the single κ_u ≠ 0 case is considered. We also list in Table <ref> the values of N_SD of Table <ref> (i.e., corresponding to the case where the diagrams involving the ggh interaction are calculated with the point-like ggh interaction). We see that the expected significance of the NP signal in pp → h+j is mildly sensitive to the calculation scheme. In particular, variations at the level of 0.1σ -1 σ are observed in N_SD depending on the values of the scaled NP couplings κ_q and κ_g (note that Ñ_SD = N_SD for κ_u=0), so that the point-like ggh approximation is indeed useful for estimating the NP effect in pp → h+j even for events with p_T(h) > 400 GeV.§.§ The case of Higgs + b-jet productionWe next turn to Higgs + b-jet production, which can be described in the five flavor scheme (5FS), where one treats the b-quark as a massless parton while keeping its Yukawa coupling finite <cit.>, see also <cit.>. In particular, the LO contribution to pp → h + j_b arises at tree-level by the same diagrams that drive the subprocess qg → hq (and the charged conjugate one g b̅→b̅ h), shown in Fig. <ref> with q=b. The cross-section for these diagrams is proportional to the bbh Yukawa coupling (squared) and can be obtained from the corresponding squared amplitudes which are given in Eqs. <ref>-<ref>. The 1-loop contribution to gb → b h, which, in the infinite top-quark mass limit, can be described by the effective ggh vertex (see Fig. <ref>), is given in Eqs. <ref>-<ref>. It is comparable to the LO tree-level one at low p_T(h)100 GeV, while it dominates at the higher p_T(h) spectrum (see below).^[2][2]Note that the Higgs + light-jet processes (in particular, the dominant gluon-fusion process gg → hg) may ”contaminate" the Higgs + b-jet signal, when the light jet is mistagged as a b-jet. The probability for that is, however, expected to be at the sub-percent level for a b-tagging efficiency of ϵ_b ∼ 60-70% and is, therefore, neglected.Let us denote the corresponding tree-level and 1-loop cumulative cross-sections (following Eq. <ref>) for pp → h + j_b as σ_bbh^hj_b≡σ_bbh^hj_b(p_T^cut) and σ_ggh^hj_b≡σ_ggh^hj_b(p_T^cut), respectively. Thus, in the kappa-framework where κ_b and κ_g are the only NP scaled couplings, the total Higgs + b-jet cross-section is (again there is negligible interference between the diagrams involving the bbh and ggh interactions): σ^hj_b = κ_g^2 σ_ggh^hj_b + κ_b^2 σ_bbh^hj_b , so that the SM cross-section is obtained for κ_g=κ_b=1, i.e., σ_SM^hj_b = σ_ggh^hj_b + σ_bbh^hj_b.The signal strength for pp→ h + j_b → ff +j_b is then given by:μ_hj_b^f=N(pp → h+j_b → ff+j_b)/ N_SM(pp → h+j_b → ff+j_b)≃ ( κ_g^2/1+R^hj_b + κ_b^2/1+(R^hj_b)^-1) ·μ_h → ff^b ,where R^hj_b≡σ_bbh^hj_b/σ_ggh^hj_b , and μ_h → ff^b≡ BR(h → ff)/BR_SM(h → ff)=1/1 + (κ_g^2 -1 ) BR_SM^gg + (κ_b^2 -1 ) BR_SM^bb . Once again, all the uncertainties associated with the measurement of μ_hj_b^f reside in the ratio of cross-sections R^hj_b and in the limit R^hj_b≪ 1, we get an expression for μ_h j_b^f which is similar to the one obtained for the Higgs + light-jet case in Eq. <ref>, with the replacement κ_q →κ_b: μ_hj_b^f (R^hj_b≪ 1) ≃( κ_g^2 + κ_b^2 R^hj_b) ·μ_h → ff^b .In particular, we find that, as in the Higgs + light-jet case, the κ_b term is important for softer p_T(h)for which R^hj_b∼ O(1), while the κ_g contribution is dominant at the harder p_T(h) regime, where R^hj_b≪ 1. For example, we obtain R^hj_b∼ 2 for p_T^cut∼ 35 GeV, dropping to R^hj_b∼ 1 at p_T^cut∼ 90 GeV (i.e., the point where σ_bbh^hj_b is comparable to σ_ggh^hj_b), then to R^hj_b∼ 0.4 for p_T^cut∼ 200 GeV and further to R^hj_b∼ 0.15 at p_T^cut∼ 400 GeV. Thus, here also, the effects of higher-order corrections and variation of scales, as well as the acceptance factors, become insignificant when the signal strength is evaluated for a high p_T^cut∼ 400 GeV, for which R^hj_b∼ O(0.1).In Fig. <ref> we show the dependence of the signal strength μ_hj_b^f on p_T^cut, assuming no NP in the Higgs-gluon ggh interaction (κ_g=1) and for values of κ_b within 0 < κ_b < 1.5, which are consistent with the current measurements of the 125 GeV Higgs production and decay processes <cit.>. We see that, once again, the signal strength approaches an asymptotic value (for a given κ_b value) as p_T^cut is increased, which is where the κ_g term dominates and the κ_b dependence arises mostly from the decay factor μ_h → ff^b in Eq. <ref>.We also show in Fig. <ref> the expected number of pp → h +j_b →γγ + j_b events, N(pp → h + j_b →γγ + j_b) =L·σ(pp → h + j_b ) · BR(h →γγ) · A·ϵ_b, as a function of p_T^cut at the HL-LHC with L=3000 fb^-1, an acceptance of A = 0.5 and a b-jet tagging efficiency of 70%, i.e., ϵ_b=0.7. We see that, under these conditions and for the values of κ_g and κ_b considered, a p_T^cut 100 GeV is required to ensure O(100) pp → h +j_b →γγ + j_b events. In particular, N(pp → h +j_b →γγ + j_b) ∼ O(1000) for p_T^cut∼ 30 GeVand N(pp → h +j_b →γγ + j_b) ∼ O(10) for p_T^cut∼ 200 GeV, respectively. In the following, we will therefore use p_T^cut = 30 GeV and 200 GeV as two representative extreme cases, where the former can be detected in the pp → h +j_b →γγ + j_b channel, while the latter is more suited for a higher statistics channel, such as pp → h +j_b → WW^⋆ + j_b followed by the leptonic W-decays WW^⋆→ 2ℓ 2ν, which has a rate about five times larger than pp → h +j_b →γγ + j_b. In Fig. <ref> we plot the statistical significance of the signals, N_SD=Δμ_hj_b^f/δμ_hj_b^f, for p_T^cut = 30 and 200 GeV, as a function of κ_b, assuming κ_g=1 and a 5%(1σ) error δμ_hj_b^f = 0.05. We see that, for p_T^cut = 200 GeV a 3 σ effect is expected if κ_b0.8 and/or κ_b1.3, while for p_T^cut = 30 GeV a larger deviation from the SM is required, i.e., κ_b0.5 and/or κ_b2.2, for a statistically significant signal of NP in pp → h +j_b →γγ + j_b. In Fig. <ref> we plot the 68%, 95% and 99% CL sensitivity ranges of NP in the κ_b-κ_g plane, for pp → h+ j_b with p_T^cut=30 GeV and p_T^cut=200 GeV, assuming again that μ_hj^f ∼ 1 ± 0.05(1σ), i.e., around the SM value with a 5%(1σ) accuracy. We see that the two p_T^cut cases probe different regimes in the κ_g - κ_b plane and are, therefore, complementary.Finally, in Table <ref> we list the statistical significance of NP in pp → h +j_b, for δμ_hj_b^f=0.05(1σ), p_T^cut=200 GeV and for several discrete values of the scaled couplings: κ_b=0.5,0.75,1,1.25,1.5 and κ_g=0.8,0.9,1,1.1,1.2. We include again the theoretical uncertainty obtained by scale variations, which we find to be somewhat higher than in the case of pp → h +j.Here also we can estimate the sensitivity of the signal to the calculational setup, using the prescription described in the previous section. In particular, we find that calculating R^hj_b in Eq. <ref> with the exact 1-loop finite top-quark mass effect in σ_ggh^hj_b, the statistical significance values quoted in Table <ref> can vary by up to a few standard deviations depending on the values of the scaled couplings κ̃_g and κ_b. For example, for (κ_b,κ̃_g)=(0.5,0.8),(0.5,1.0),(0.75,1.1),(1.0,1.2),(1.25,0.8) (see the definition of κ̃_g in Eq. <ref> and discussion therein), the expected statistical significance changes from N_SD=0.4,7.5,6.7,5.3,6.0 in the point-like ggh approximation to Ñ_SD=2.3,4.0,4.4,4.1,4.0 in the loop-induced (top-quark mass dependent) case. § HIGGS + JET PRODUCTION IN THE SMEFTThe SMEFT is defined by expanding the SM Lagrangian with an infinite series of higher dimensional operators, O_i^(n) (using only the SM fields), as <cit.>: L_SMEFT = L_SM + ∑_n=5^∞1/Λ^(n-4)∑_i f_i^(n) O_i^(n) , where Λ is the scale of the NP that underlies the SM, n denotes the dimension and i all other distinguishing labels. Considering the expansion up to operators of dimension 6 (for a complete list of dimension 6 operators in the SMEFT, see e.g. <cit.>), we will study here the following subset of operators that can potentially modify the Higgs + jet production processes: O_u ϕ = ( ϕ^†ϕ) (Q̅_L ϕ̃u_R ) + h.c. , O_d ϕ = ( ϕ^†ϕ) (Q̅_L ϕ d_R ) + h.c. , O_u g = (Q̅_L σ^μν T^a u_R ) ϕ̃G_μν^a + h.c. , O_d g = (Q̅_L σ^μν T^a d_R ) ϕ G_μν^a +h.c. , O_ϕ g = (ϕ^†ϕ) G_μν^a G^a, μν , where ϕ is the SM Higgs doublet (with ϕ̃≡ i σ_2 ϕ^⋆), G^a, μν denotes the QCD gauge-field strength and Q_L and u_R(d_R) are the SU(2)_L quark doublet and charge 2/3(-1/3) singlets, respectively. In particular, we assume that the physics which underlies Higgs+jet production is contained within (dropping the dimension index n=6): L_SMEFT = L_SM + ∑_i=u ϕ,d ϕ,u g,dg,ϕ gf_i/Λ_i^2 O_i , and, to be as general as possible, we allow different scales of the NP which underly the different operators. For example, Λ_u ϕ corresponds to the typical scale of O_u ϕ, where by “typical scale" we mean that the corresponding Wilson coefficient is f_u ϕ∼ O(1).The effects of the operators O_u ϕ, O_d ϕ and O_ϕ g can be “mapped" into the kappa-framework, satisfying: κ_q ≃y_q^SM/y_b^SM - f_q ϕ/y_b^SMv^2/Λ_q ϕ^2 , κ_g = 1+ 12 π f_ϕ g/α_sv^2/Λ_ϕ g^2 ,where y_q^SM/y_b^SM→ 0 for e.g., q=u or d, while y_q^SM/y_b^SM =1 for the b-quark. Thus, the sensitivity of the signal strength μ_hj^f for pp → h+j (defined in Eqs. <ref> and <ref>) to the effective Lagrangian containing the operators O_u ϕ, O_d ϕ and O_ϕ g can be obtained from the analysis that has been performed for the kappa-framework in the previous section. For example, it follows from Eq. <ref> that, for f_u ϕ, f_ϕ g∼ O(1), one expects |κ_u|0.5 and Δκ_g = | κ_g -1|0.1, if the corresponding scales of NP are Λ_u ϕ 3 TeV and Λ_ϕ g 15 TeV, respectively.On the other hand, the (flavor diagonal) operators O_u g and O_d g induce new chromo-magnetic dipole moment (CMDM) type, qqg and contact qqgh interactions, which have a new Lorentz structure and, therefore, cannot be described by scaling the SM couplings. In particular, these new CMDM-like operators give rise to different Higgs + jet kinematics with respect to the SM. The effects of the light-quarks and b-quark CMDM-like effective operators, O_q g (q=u,d,c,s,b), in Higgs production at the LHC was studied in <cit.>, where it was found that the inclusive Higgs production, pp → h+X, and Higgs + b-jets events can be used to probe the CMDM-like interactions if its typical scale is Λ_qg∼ few TeV. Here we will show that a better sensitivity to the scale of the effective quark CMDM-like operators, Λ_qg, can be achieved by analysing the exclusive pp → h + j(j_b) →γγ + j(j_b) Higgs production and decay channels and using the signal strength formalism with the cumulative cross-sections for a high p_T^cut∼ 200-300 GeV.Note that, in the general case where the Wilson coefficients f_u ϕ, f_d ϕ, f_u g and f_d g are arbitrary 3 × 3 matrices in flavor space, the operators O_u ϕ, O_d ϕ, O_u g and O_d g will generate tree-level flavor-violating u_i → u_j and d_i → d_j transitions (i,j =1-3 are flavor indices). One way to avoid that is to assume proportionality of these Wilson coefficients to the corresponding 3 × 3 Yukawa coupling matrices (Y_u and Y_d), in which case the field redefinitions which diagonalize the quark matrices also diagonalize these operators and the effective theory is automatically minimally-flavor-violating (MFV). That is, f/Λ^2 O_u g→Y_q ·f_MFV/Λ^2_MFV O_u g , so that the relation between generic NP parameters (f,Λ) and the corresponding parameters in the MFV effective theory is (for a single flavor q): Λ_MFV^2/Λ^2 = y_q f_MFV/f .Thus, if f_MFV∼ f, then Λ_MFV∼√(y_q)·Λ, in which case Λ_MFV≪Λ for q ≠ t. On the other hand, for √(y_q f_MFV/f)∼ O(1) we have Λ_MFV∼Λ. In what follows we would like to keep our discussion as general as possible, not restricting to any assumption about the possible flavor structure of the Wilson coefficients. In particular, we will focus below on a single flavor (diagonal element) of these operators and assume that flavor violation is controlled by some underlying mechanism in the high-energy theory (not necessarily MFV), thereby suppressing the non-diagonal elements of these operators to an acceptable level.§.§ The case of Higgs + light-jet productionLet us consider first the operators O_u ϕ and O_ϕ g, which, as seen from Eq. <ref>, modify the SM uuh and ggh couplings in a way that is equivalent to the kappa-framework (we will focus below only on the case of the 1st generation u-quark operator O_u ϕ).^[3][3]The effects of O_ϕ g and the top and bottom quarks operators O_t ϕ and O_b ϕ on the subprocess gg → hg were considered in <cit.>, in the context of Higgs-p_T distribution in Higgs + jet production at the LHC. In particular, using Eq. <ref> and the analysis performed in the previous section for NP in the kappa-framework, we plot in Fig. <ref> the 68%, 95% and 99% CL sensitivity ranges in the Λ_u ϕ-Λ_ϕ g plane, for p_T^cut=400 GeV, assuming that μ_hj^f ∼ 1 ± 0.05(1σ). The sensitivity ranges are shown for the two cases f_ϕ g= ± 1, where in both cases we set |f_u ϕ|=1, since the cross-section is ∝κ_q^2 (see Eq. <ref>) so that there is no dependence on the sign of f_u ϕ for y_u^SM/y_b^SM→ 0 (see Eq. <ref>).We see that a measured value of μ_hj^f which is consistent with the SM at 3σ (i.e, with 0.85 ≤μ_hj^f ≤ 1.15) will exclude NP with typical scales of Λ_ϕ g 15 TeV (equivalent to κ_u0.6) and Λ_u ϕ 2 TeV (equivalent to κ_g1.1), for f_ϕ g=-1. In the case of f_ϕ g=1, there is an allowed narrow band in the Λ_u ϕ-Λ_ϕ g plane, stretching down to NP scales of Λ_ϕ g∼ 5 TeV and Λ_u ϕ∼ 1 TeV, which are consistent with 0.85 ≤μ_hj^f ≤ 1.15. We note that, as in the kappa-framework analysis, these sensitivity ranges in the Λ_u ϕ-Λ_ϕ g plane mildly depend on the calculation scheme of the SM-like diagrams involving the ggh interaction, i.e., on the difference between the point-like ggh approximation and the exact 1-loop results.We study next the effect of the CMDM-like operator O_u g on pp → h+j (again focusing only on the u-quark operator). The tree-level diagrams corresponding to the contribution of O_u g to pp → h+j are depicted in Fig. <ref>. They contain the momentum dependent CMDM-like uug vertex and uugh contact interaction, which do not interfere with the SM diagrams in the limit of m_u → 0. In particular, in the presence of O_u g, the total pp → h+j cross-section can be written as: σ^hj = σ_SM^hj + (f_u g/Λ_u g^2)^2 σ_ug^hj , where the squared amplitudes for σ_SM^hj are given in Eqs. <ref>-<ref> (see also Eq. <ref>) and σ_ug^hj is the NP cross-section corresponding to the square of the CMDM-like amplitude, which is generated by the tree-level diagrams for q q̅→ gh, q g → q h and q̅ g →q̅ h shown in Fig. <ref>, with an insertion of the effective CMDM-like uug and uugh vertices. In particular, σ_ug^hj is composed of σ_ug^hj = σ_ug^hj(q q̅→ gh ) + σ_ug^hj(q g → q h ) + σ_ug^hj(q̅ g →q̅ h ), where the corresponding amplitude squared (summed and averaged over spins and colors) are given by: ∑|M_ug^q q̅→ gh|^2 = 8/ C_qqût̂[ 1- 4 v C_g + 8 v^2 C_g^2 ] ,∑|M_ug^q g → q h|^2 =-C_qq/ C_qg∑|M_ug^q q̅→ gh|^2 (ŝ↔t̂) , ∑|M_ug^q̅ g →q̅ h|^2 =-C_qq/ C_qg∑|M_ug^q q̅→ gh|^2 (ŝ↔û) , with ŝ=(p_1+p_2)^2, t̂=(p_1+p_3)^2 and û=(p_2+p_3)^2, defined for q(-p_1)+ q̅(-p_2) → h + g(p_3).As illustrated in Fig. <ref>, the momentum dependent contribution from O_u g drastically changes the p_T(h)-dependence of the cross-section with respect to the SM and also with respect to the case where the NP is in the form of scaled couplings (i.e., in the kappa-framework). Indeed, the effect of O_u g (or any other NP with a similar p_T(h) behaviour) are better isolated in the harder Higgs p_T regime. This can be obtained by using a relatively high p_T^cut for the cumulative cross-section (see below).Assuming no additional NP in the decay (the effects of O_u g in the Higgs decay is ∝ (m_h/Λ_u g)^4 and is, therefore, negligible for Λ∼ few TeV), the corresponding signal strength is: μ_hj^f ( O_u g) = 1 + (f_u g/Λ_u g^2)^2 R_ug^hj , R_ug^hj≡σ_ug^hj/σ_SM^hj , so that the NP signal, as defined in Eq. <ref>, is: Δμ_hj^f ( O_u g) = |μ_hj^f( O_u g) - 1 | = (f_u g/Λ_u g^2)^2 R_ug^hj .In Fig. <ref> we plot the NP signal, Δμ_hj^f ( O_u g), as a function of Λ_ug with f_ug=1, for p_T^cut values of 100, 250 and 400 GeV and an invariant mass cut m_h+j≤ 2 TeV. As expected (see Fig. <ref>), the sensitivity to Λ_ug is significantly improved the higher the p_T^cut is. In particular, while Δμ_hj^f/μ_hj^f5% for p_T^cut=100 GeV and Λ_u g 4 TeV, for p_T^cut=400 GeV we obtain Δμ_hj^f/μ_hj^f5% for Λ_u g 8.5 TeV. In Fig. <ref> we plot the statistical significance of the signal, N_SD=μ_hj^f/δμ_hj^f, for δμ_hj^f = 0.05(1σ), and the expected number of events, again assuming that the Higgs decays via h →γγ, i.e., N(pp → h + j →γγ + j), as a function of p_T^cut and for Λ_ug=2, 4, 6 and 8 TeV with f_ug=1 and an invariant mass cut m_h+j≤ 2 TeV. N(pp → h + j →γγ + j) is shown for an integrated luminosity of 300 fb^-1 and a signal acceptance of 50%. We see, for example, that if Λ_ug=6 TeV, then a high p_T^cut∼ 350 GeV is required in order to obtain a 3 σ effect, for which N(pp → h + j →γγ + j) ∼ O(10) and O(100) is expected at the LHC with L=300 fb^-1 and the HL-LHC with L=3000 fb^-1, respectively.Note that the effect of changing the calculation scheme of the SM cross-section from the point-like ggh interaction to the exact mass dependent 1-loop one is to change R_ug^hj→ r_ggh R_ug^hj in Eq. <ref> (r_ggh is defined in Eq. <ref>) and therefore it also increases the statistical significance N_SD by a factor of r_ggh which depends on the p_T^cut used (see Fig. <ref>). Thus, the statistical significance values reported in the upper plot of Fig. <ref> are on the conservative side. §.§ The case of Higgs + b-jet productionAs mentioned above, the effects of the NP operators O_b ϕ and O_ϕ g in pp → h+ j_b, can be described using the kappa-framework formalism of Eq. <ref>, with the NP factors multiplying the SM bbh Yukawa coupling (κ_b) and ggh coupling (κ_g) as prescribed in Eq. <ref>.In Figs. <ref> and <ref> we plot the 68%, 95% and 99% CL sensitivity ranges in the Λ_b ϕ-Λ_ϕ g plane, for (f_b ϕ,f_ϕ g)=(1,1),(1,-1),(-1,1),(-1,-1) and p_T^cut=30 GeV and 200 GeV, assuming again that the signal strength had been measured to a 5%(1σ) accuracy with a SM central value, i.e., μ_hj^f ∼ 1 ± 0.05(1σ). As in the kappa-framework analysis of the previous section, we use the two p_T^cut values, p_T^cut = 30 GeV and p_T^cut = 200 GeV, as two representative examples of a high and low statistics pp → h +j_b→γγ +j_b signal at the HL-LHC (see also Fig. <ref>). As expected, a better sensitivity to the NP is obtained for the higher p_T^cut = 200 GeV, where Λ_b ϕ 3 TeV and Λ_ϕ g O(10) TeV can be excluded at 3 σ if μ_hj_b^f is found to be consistent with the SM within15% (3 σ). Here also, similar to the kappa-framework analysis for pp → h+j_b, the sensitivity ranges in the Λ_b ϕ-Λ_ϕ g plane for the p_T^cut = 200 GeV case mildly depend on whether the SM cross-section is calculated with the point-like ggh approximation or at 1-loop with a finite top-quark mass.Finally, we consider the case where the NP in pp → h+j_b is due only to the b-quark CMDM-like operator O_b g. The corresponding tree-level diagrams with the new momentum dependent CMDM-like bbg vertex and bbgh contact interaction are shown in Fig. <ref>, where, as opposed to the pp → h+j case, here there is an interference (though small - see below) between the CMDM-like diagrams and the tree-level SM ones (depicted in Fig. <ref>). In particular, in the presence of O_b g, the total pp → h+j_b cross-section can be written as: σ^hj_b = σ_SM^hj_b + f_b g/Λ_b g^2σ_bg^1,h j_b + (f_b g/Λ_b g^2)^2 σ_bg^2, hj_b , where σ_SM^hj_b is the SM cross-section (the relevant SM squared amplitude terms are given in Eqs. <ref>,<ref>,<ref>,<ref>) and the NP terms σ_bg^1,2 can be obtained from the following CMDM-like NP squared amplitudes (summed and averaged over spins and colors): ∑|M_bg^1,b g → bh|^2 = 8 g_s y_b/ C_qg( 4 v C_g t̂ -m_h^2 ) ,∑|M_bg^2,b g → bh|^2 =- 8/ C_qg[ŝû( 1- 4 v C_g + 8 v^2 C_g^2 ) .. + y_b^2 v^2 t̂] ,∑|M_bg^1,b̅ g →b̅h|^2 = ∑|M_bg^1,b g → bh|^2 (û↔t̂) , ∑|M_bg^2,b̅ g →b̅h|^2 = ∑|M_bg^2,b g → bh|^2 (û↔t̂) , where again ŝ=(p_1+p_2)^2, t̂=(p_1+p_3)^2 and û=(p_2+p_3)^2, defined for b(-p_1)+ b̅(-p_2) → h + g(p_3).We see from Eqs. <ref> and <ref> above that the interference terms M_bg^1,b g → bh and M_bg^1,b̅ g →b̅h (corresponding to σ_bg^1,hj_b in Eq. <ref>) are proportional to y_b ∼ O(m_b/v) and are therefore sub-leading, so that the dependence of the pp → h+j_b cross-section on the sign of the CMDM-like Wilson coefficient, f_bg, is tenuous. As a result, σ^hj_b has a very similar p_T-behaviour as the one depicted in Fig. <ref> for the pp → h+j case. In particular, here also, the Higgs p_T spectrum becomes appreciably harder with respect to the SM and also with respect to the case of the NP operators O_b ϕ and O_g ϕ, due to the momentum-dependent σ_bg^2, hj_b term, which corresponds to the square of the b-quark CMDM-like diagrams, generated by the operator O_b g and depicted in Fig. <ref>.In Fig. <ref> we plot the statistical significance of the O_b g signal for δμ_hj^f = 0.05(1σ), as a function of p_T^cut for f_bg=1 and Λ_bg=2, 3, 4 and 6 TeV, imposing an invariant mass cut of m_h+j_b≤ 2 TeV. The results for f_bg=-1 are very similar due to the small interference between the CMDM-like and SM amplitudes (see discussion above). We see that, as expected, the sensitivity to the scale of the CMDM-like operator, Λ_bg, is higher the higher the p_T^cut is. We find, for example, that the effect of O_bg with a typical scale of Λ_bg∼ 4 TeV can be probed in pp → h+j_b →γγ +j_b to the level of N_SD∼ O(10σ) with p_T^cut = 200 GeV. The expected number of pp → h +j_b →γγ + j_b events in this case (i.e., for Λ_bg∼ 4 TeV, p_T^cut = 200 GeV and an invariant mass cut of m_h+j_b≤ 2 TeV), assuming an integrated luminosity of 3000 fb^-1, a signal acceptance of A = 0.5 and a b-jet tagging efficiency of 70%, ϵ_b=0.7, is N(pp → h + j_b →γγ + j_b) ∼ 30 (see also Fig. <ref>).As for the sensitivity of the above results to the calculational scheme: due to the smallness of the interference term it is similar to that of the u-quark CMDM-like case in pp → h+j. In particular, the statistical significance N_SD shown in Fig. <ref> should also be considered conservative with respect to the values which would have been obtained using the exact 1-loop induced SM cross-section, i.e., N_SD is naively larger by a factor of r_ggh in the exact 1-loop calculation case. § SUMMARYWe have examined the effects of various NP scenarios, which entail new forms of effective qqh and qqg interactions in conjunction with beyond the SM Higgs-gluon effective coupling, in exclusive Higgs + light-jet (pp → h+j) and Higgs + b-jet (pp → h+j_b) production at the LHC. We have defined the signal strength for pp → h+j (j_b) followed by the Higgs decay h → ff, as the ratio of the corresponding NP and SM rates, and studied its dependence on the Higgs p_T spectrum. We specifically focused on h →γγ and assumed that there is no NP in this decay channel.We first analyse NP in pp → h+j (j_b) →γγ + j(j_b) within the kappa-framework, in which the SM Higgs couplings to the light-quarks (qqh) and to the gluons (ggh) are assumed to be scaled by a factor of κ_q and κ_g, respectively. In particular, in our notation the scale factors κ_q for all light-quark's Yukawa couplings (q=u,d,c,s,b) are normalized with respect to the b-quark Yukawa, κ_q = y_q/y_b^SM, so that in the SM we have e.g., κ_b=1 and κ_u ∼ O(10^-3). This NP setup does not introduce any new Lorentz structure in the underlying hard processes (i.e., gg → gh, qg → q h, q̅ g →q̅ h, q q̅→ g h in the case of pp → h+j and bg → b h, b̅ g →b̅ h in the case of pp → h+j_b), thus retaining the SM pp → h+j (j_b) kinematics. In particular, we find that strong bounds can be obtained in the κ_g - κ_q plane at the LHC, by measuring a p_T-dependent signal strength for Higgs + jet events at relatively high Higgs p_T. For example, the combination of κ_g < 0.8 with κ_u > 0.25 (κ_g < 0.8 with κ_b > 1.5) can be excluded at more than 7 σ at the HL-LHC with a luminosity of 3000 fb^-1, if the signal strength in the pp → h+j(j_b) →γγ +j(j_b) channels will be measured and known to an accuracy of 5%(1σ), for high p_T(h) events with p_T(h) ≥ 400(200) GeV. Recall that in our notation the corresponding SM strengths of these couplings are κ_b=κ_g=1 and κ_u ∼ O(10^-3).We also considered NP effects in pp → h+j(j_b) in the SMEFT framework, where higher dimensional effective operators modify the SM qqh Yukawa couplings and the Higgs-gluon ggh interaction by a scaling factor, similar to the case of the kappa-framework for NP. We thus utilize an interesting “mapping" between the SMEFT and kappa-frameworks to derive new bounds on the typical scale of NP that underlies the SMEFT lagrangian. We find, for example, that pp → h+j(j_b) →γγ +j(j_b) events with high p_T(h) > 400(200) GeV at the HL-LHC, are sensitive to the new effective operators that modify the qqh (Yukawa) and ggh couplings, if their typical scale (i.e., with O(1) dimensionless Wilson coefficients) is a few TeV and O(10) TeV, respectively.Finally, as a counter example, we study the effects of NP in the form of dimension six u-quark and b-quark chromo magnetic dipole moment (CMDM)-like effective operators, which induce new derivative and new contact interactions that significantly distort the pp → h+j(j_b) SM kinematics and, therefore, cannot be described in terms of scaled couplings. In particular, in this case, the high-p_T Higgs spectrum becomes significantly harder with respect to the SM. We thus show that pp → h+j(j_b) →γγ + j(j_b) events at the HL-LHC, with a high Higgs p_T of p_T(h)400(200) GeV, can probe the higher dimensional CMDM-like u-quark and b-quark effective operators, if their typical scale is around Λ∼ 5 TeV.Our main results were obtained using an effective point-like ggh interaction approximation. To estimate the sensitivity to this approximation, we also compared samples of our results to the case where the ggh vertex is calculated explicitly at leading order, which, for Higgs + jet, corresponds to a 1-loop mass dependent calculation using a finite top-quark mass.Acknowledgments: The work of AS was supported in part by the US DOE contract #DE-SC0012704.99 ATLAS1 G. Aad et al., the ATLAS collaboration, JHEP 1409 (2014), 112, arXiv:1407.4222 [hep-ex].ATLAS2 G. Aad et al., the ATLAS collaboration, Phys.Lett. B738 (2014), 234, arXiv:1408.3226 [hep-ex].ATLAS3 G. 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Aad et al.), JHEP 1608 (2016), 045, arXiv:1606.02266 [hep-ex].EFTpapers W. Buchmiiller, D. Wyler, Nucl. Phys. B268 (1986), 621; C. Arzt, M.B. Einhorn, J. Wudka, Nucl. Phys. B433 (1995), 41; M. B Einhorn and J. Wudka, Nucl. Phys. B876 (2013), 556.warsha B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, JHEP 1010 (2010), 085.1411.0035A. Hayreter, G. Valencia, Phys.Rev. D88 (2013), 034033, e-Print: arXiv:1304.6976 [hep-ph]; ibid.,arXiv:1411.0035 [hep-ph]. | http://arxiv.org/abs/1705.09295v4 | {
"authors": [
"Jonathan Cohen",
"Shaouly Bar-Shalom",
"Gad Eilam",
"Amarjit Soni"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170525180002",
"title": "Light-quarks Yukawa couplings and new physics in exclusive high-$p_T$ Higgs + jet and Higgs + $b$-jet events"
} |
Energy-Efficient Transponder Configuration for FMF-based Elastic Optical Networks Mohammad Hadi, Member, IEEE, and Mohammad Reza Pakravan, Member, IEEEDecember 30, 2023 ==================================================================================== We propose an energy-efficient procedure for transponder configuration in FMF-based elastic optical networks in which quality of service and physical constraints are guaranteed and joint optimization of transmit optical power, temporal, spatial and spectral variables are addressed. We use geometric convexification techniques to provide convex representations for quality of service, transponder power consumption and transponder configuration problem. Simulation results show that our convex formulation is considerably faster than its mixed-integer nonlinear counterpart and its ability to optimize transmit optical power reduces total transponder power consumption up to 32%. We also analyze the effect of mode coupling and number of available modes on power consumption of different network elements. Convex optimization, Green communication, Elastic optical networks, Few-mode fibers, Mode coupling.§ INTRODUCTION3D temporally, spectrally and spatially Elastic Optical Network (EON) has been widely acknowledged as the next generation high capacity transport system and the optical society has focused on its architecture and network resource allocation techniques <cit.>. EONs can provide an energy-efficient network configuration by adaptive 3D resource allocation according to the communication demands and physical conditions. Higher energy efficiency of Orthogonal Frequency Division Multiplex (OFDM) signaling has been reported in <cit.> which nominates OFDM as the main technology for resource provisioning over 2D resources of time and spectrum. On the other hand, enabling technologies such as Few-Mode Fibers (FMFs) and Multi-Core Fibers (MCFs) have been used to increase network capacity and efficiency through resource allocation over spatial dimension <cit.>. Although many variants of algorithms have been proposed for resource allocation in 1D/2D EONs <cit.>, joint assignment of temporal, spectral and spatial resources in 3D EONs needs much research and study. Among the available works on 3D EONs, a few of them have focused on energy-efficiency which is a fundamental requirement of the future optical networks <cit.>. Moreover, the available energy-efficient 3D approaches do not consider transmit optical power as an optimization variable which results in inefficient network provisioning <cit.>.Flexible resource allocation is an NP-hard problem and it is usually decomposed into several sub-problems with lower complexity <cit.>. Following this approach, we decompose the resource allocation problem into 1) Routing and Ordering Sub-problem (ROS) and 2) Transponder Configuration Sub-problem (TCS) and mainly focus on TCS which is more complex and time-consuming <cit.>. We consider FMF because it has simple amplifier structure, easier fusion process, lower nonlinear effects and lower manufacturing cost compared to other Space-Division Multiplexed (SDM) optical fibers <cit.>. In TCS, we optimally configure transponder parameters such asmodulation level, number of sub-carriers, coding rate, transmit optical power, number of active modes and central frequency such that total transponders power consumption is minimized while Quality of Service (QoS) and physical constraints are met. Unlike the conventional approach, we provide convex expressions for transponder power consumption and Optical Signal to Noise Ratio (OSNR), as an indicator of QoS. We then use the results to formulate TCS as a convex optimization problem which can efficiently be solved using fast convex optimization algorithms. We consider transmit optical power as an optimization variable and show that it has an important impact on total transponder power consumption. Simulation results show that our convex formulation can be solved almost 20 times faster than its Mixed-Integer NonLinear Program (MINLP) counterpart. Optimizing transmit optical power also improves total transponder power consumption by a factor of 32% for European Cost239 optical network with aggregate traffic 60 Tbps. We analyze the effect of mode coupling on power consumption of the different network elements. As simulation results show, total network power consumption can be reduced more than 50% using strongly-coupled FMFs rather than weakly-coupled ones. Numerical outcomes also demonstrate that increasing the number of available modes in FMFs provides a trade-off between FFT and DSP power consumption such that the overall transponder power consumption is a descending function of the number of available modes.§ SYSTEM MODELConsider a coherent optical communication network characterized by topology graph G(𝐕, 𝐋) where 𝐕 and 𝐋 are the sets of optical nodes and directional optical strongly-coupled FMF links, respectively. The optical FMFs have ℳ modes and gridless bandwidth ℬ. 𝐐 is the set of connection requests and 𝐐_l shows the set of requests sharing FMF l on their routes. Each request q is assigned a contiguous bandwidth Δ_q around carrier frequency ω_q and modulates m_q modes of its available ℳ modes. The assigned contiguous bandwidth includes 2^b_q OFDM sub-carriers with sub-carrier space of ℱ so, Δ_q=2^b_qℱ. To have a feasible MIMO processing, the remaining unused modes of a request cannot be shared among others <cit.>. We assume that the assigned bandwidths are continuous over their routes to remove the high cost of spectrum/mode conversion <cit.>. Request q passes 𝒩_q fiber spans along its path and has 𝒩_q,i shared spans with request i≠ q. Each FMF span has fixed length of ℒ_spn and an optical amplifier to compensate for its attenuation.There are pre-defined modulation levels c and coding rates r where each pair of (c, r) requires minimum OSNR Θ(c, r) to get a pre-FEC BER value of 1× 10^-4 <cit.>. Each transponder is given modulation level c_q, coding rate r_q and injects optical power p_q/m_q to each active mode of each polarization. Chromatic dispersion and mode coupling signal broadenings are respectively proportional to 𝒩_q2^b_q and m_q^-0.78√(𝒩_q) with coefficients σ=2πβ_2ℱℒ_spn and ϱ=5Δβ_1 √(L_secℒ_spn) where β_2 is chromatic dispersion factor and Δβ_1 √(L_sec) is the product of rms uncoupled group delay spread and section length <cit.>. Transponders add a sufficient cyclic prefix to each OFMD symbol to resolve the signal broadening induced by mode coupling and chromatic dispersion. Transponders have maximum information bit rate 𝒞. There is also a guard band 𝒢 between any two adjacent requests on a link. Considering the architecture of Fig. <ref>, the power consumption of each pair of transmit and receive transponders P_q can be calculated as follows:P_q = 𝒫_trb+2𝒫_edcm_qr_q^-1+2m_q2^b_qb_q𝒫_fft+2m_q^22^b_q𝒫_dspwhere 𝒫_trb is transmit and receive transponder bias term, 𝒫_edc is the scaling coefficient of encoder and decoder power consumptions, 𝒫_fft denotes the power consumption for a two point FFT operation and 𝒫_dsp is the power consumption scaling coefficient of the receiver DSP and MIMO operations <cit.>.To have a green EON, we need a resource allocation algorithm to determine the values of system model variables such that the transponders consume the minimum power while physical constrains are satisfied and desired levels of OSNR are guaranteed. In general, such a problem is modeled as an NP-hard MINLP optimization problem <cit.>. To simplify the problem and provide a fast-achievable near-optimum solution, the resource allocation problem is usually decomposed into two sub-problems: ROS, where the routing and ordering of requests on each link are defined, and 2) TCS, where transponders are configured.Usually the search for a near optimal solution involves iterations between these two sub-problems. To save this iteration time, it is of great interest to hold the running time of each sub-problem at its minimum value. In this work, we mainly focus on TCS which is the most time-consuming sub-problem and formulate it as a convex problem to benefit from fast convex optimization algorithms. For a complete study of ROS, one can refer to <cit.>. § TRANSPONDER CONFIGUARTION PROBLEMA MINLP formulation for TCS is as follows:min_𝐜, 𝐛, 𝐫, 𝐩,𝐦, ω∑_q ∈𝐐P_qs.t.Ψ_q ⩾Θ_q, ∀ q ∈𝐐ω_Υ_l, j+Δ_Υ_l, j+Δ_Υ_l, j+1/2 + 𝒢⩽ω_Υ_l, j+1, ∀ l ∈𝐋, ∀ j ∈𝐌_1^𝐐_lΔ_q/2⩽ω_q ⩽ℬ- Δ_q/2, ∀ q ∈𝐐ℛ_q ⩽2ℱ^-1m_qr_q c_qΔ_q/ℱ^-1+σ𝒩_q2^b_q+ϱ m_q^-0.78√(𝒩_q), ∀ q ∈𝐐where 𝐜, 𝐛, 𝐫, 𝐩, 𝐦 and ω are variable vectors of transponder configuration parameters modulation level, number of sub-carriers, coding rate, transmit optical power, number of active modes and central frequency. 𝐌_a^b shows the set of integer numbers {a, a+1, ⋯, b-1}. The goal is to minimize the total transponder power consumption where P_q is obtained using (<ref>). Constraint (<ref>) is the QoS constraint that forces OSNR Ψ_q to be greater than its required minimum threshold Θ_q. Ψ_q is a nonlinear function of 𝐛, 𝐩, 𝐦 and ω while the value of Θ_q is related to r_q and c_q <cit.>. Constraint (<ref>) is nonoverlapping-guard constraint that prevents two requests from sharing the same frequency spectrum. Υ_l, j is a function that shows which request occupies j-th assigned spectrum bandwidth on link l and its values are determined by solving ROS <cit.>. Constraint (<ref>) holds all assigned central frequencies within the acceptable range of the fiber spectrum. The last constraint guarantees that the transponder can convey the input traffic rate ℛ_q in which wasted cyclic prefix times are considered. Generally, this problem is a complex MINLP which is NP-hard and cannot easily be solved in a reasonable time. Therefore, we use geometric convexification techniques to convert this MINLP to a mixed-integer convex optimization problem and then use relaxation method to solve it. To have a convex problem, we first provide a generalized posynomial expression <cit.> for the optimization and then define a variable change to convexify the problem. A posynomial expression for OSNR of a request in 2D EONs has been proposed in <cit.>. We simply consider each active mode as an independent source of nonlinearity and incoherently add all the interferences <cit.>. Therefore the extended version of the posynomial OSNR expression is:Ψ_q=p_q/m_q/ζ𝒩_qΔ_q+κ_1ςp_q/m_q∑_i ∈𝐐, q ≠ im_i(p_i/m_i)^2𝒩_q,i/Δ_i/d_q,i, ∀ q ∈𝐐where κ_1=0.4343, ζ=(e^αℒ_spn-1)hν n_sp and ς=3γ^2/2απβ_2. n_sp is the spontaneous emission factor, ν is the light frequency, h is Planck’s constant, α is attenuation coefficient, β_2 is dispersion factor and γ is nonlinear constant. Furthermore,d_q,i is the distance between carrier frequencies ω_q and ω_i and equals to d_q,i = ω_q-ω_i. We use Θ_q ≈ r_q^κ_2(1+κ_3 c_q)^κ_4 for posynomial curve fitting of OSNR threshold values where κ_2=3.37, κ_3=0.21, κ_4=5.73 <cit.>. Following the same approach as <cit.>, we arrive at this new representation of the optimization problem:min_𝐜, 𝐛, 𝐫, 𝐩, 𝐦, ω, 𝐭, 𝐝∑_q ∈𝐐P_q+𝒦∑_q,i ∈𝐐 q ≠ i, 𝒩_q,i≠ 0d_q,i^-1s.t.r_q^κ_2 t_q^κ_4[ζℱ𝒩_qm_qp_q^-12^b_q+ κ_1ςℱ^-1∑_i ∈𝐐, i ≠ q𝒩_q,ip_i^2m_i^-12^-b_id_q,i^-1]⩽ 1 , ∀ q ∈𝐐ω_Υ_l, j+0.5ℱ2^b_Υ_l, j +𝒢+ 0.5ℱ2^b_Υ_l, j+1⩽ω_Υ_l, j+1, ∀ l ∈𝐋 ,∀ j ∈𝐌_1^𝐐_l 0.5ℱ2^b_q + ω_q⩽ℬ, ∀ q ∈𝐐 0.5ℱ2^b_q⩽ω_q, ∀ q ∈𝐐0.5ℛ_qℱ^-1 r_q^-1c_q^-1m_q^-1 2^-b_q+ 0.5 σ𝒩_q ℛ_q m_q^-1r_q^-1c_q^-1 +0.5 ϱ√(𝒩_q)ℛ_qr_q^-1c_q^-1m_q^-1.78 2^-b_q⩽ 1,∀ q ∈𝐐 1+κ_3c_q⩽ t_q, ∀ q ∈𝐐 d_Υ_l, i,Υ_l, j + ω_Υ_l, j⩽ω_Υ_l, i, ∀ l ∈𝐋,∀ j ∈𝐌_1^𝐐_l,∀ i ∈𝐌_j+1^𝐐_l+1 d_Υ_l, i,Υ_l, j+ω_Υ_l, i⩽ω_Υ_l, j , ∀ l ∈𝐋,∀ j ∈𝐌_2^𝐐_l+1, ∀ i ∈𝐌_1^jIgnoring constraints (<ref>), (<ref>), (<ref>) and the penalty term of the goal function (<ref>), the above formulation is equivalent geometric program of the previous MINLP in which expressions (<ref>) and the mentioned posynomial curve fitting have been used for QoS constraint (<ref>). Constraints (<ref>) and (<ref>) and the penalty term are added to guarantee the implicit equality of d_q,i = ω_q-ω_i <cit.>. Constraint (<ref>) is also needed to convert the generalized posynomial QoS constraint to a valid geometric expression, as explained in <cit.>. Now, consider the following variable change:x=e^X:x ∈ℝ_>0⟶ X ∈ℝ, ∀ x ∉𝐛Applying this variable change to the goal function (which is the most difficult part of the variable change), we have:∑_q ∈𝐐 [𝒫_trb+2𝒫_edce^m_q-r_q+5.36e^0.82b_q+m_q𝒫_fft+ 2e^2m_q2^b_q𝒫_dsp]+𝒦∑_q,i∈𝐐, q ≠ i, 𝒩_q,i≠ 0e^-d_q,iClearly, e^-d_q,i, e^m_q-r_q and e^2m_q2^b_q are convex over variable domain.We use expression 5.36e^0.82b_q+m_q to provide a convex approximation for the remaining term 2e^m_qb_q2^b_q. The approximation relative error is less than 3% for practical values of m_q ⩾ 1 and 4 ⩽ b_q ⩽ 11.Consequently, function (<ref>) which is a nonnegative weighted sum of convex functions is also convex. The same statement (without any approximation) can be applied to show the convexity of the constraints under variable change of (<ref>) (for some constraints, we need to apply an extra log to both sides of the inequality). To solve this problem, a relaxed continuous version of the proposed mixed-integer convex formulation is iteratively optimized in a loop <cit.>. At each epoch, the continuous convex optimization is solved and obtained values for relaxed integer variables are rounded by a given precision. Then, we fix the acceptable rounded variables and solve the relaxed continuous convex problem again. The loop continues untill all the integer variables have valid values. The number of iterations is at most equal to (in practice, is usually less than) the number of integer variables. Furthermore, a simpler problem should be solved as the number of iteration increases because some of the integer variables are fixed during each loop.§ NUMERICAL RESULTSIn this section, we use simulation results to demonstrate the performance of the convex formulation for TCS. The European Cost239 optical network is considered with the topology and traffic matrix given in <cit.>. Simulation constant parameters are β_2=20393 fs^2/m , α=0.22 dB/km, ℒ_spn=80 km, ν=193.55 THz, n_sp=1.58, γ=1.3 1/W/km, ℱ=80 MHz, ϱ=113 ps, σ=14 fs, 𝒢=20 GHz, ℬ=2 THz, 𝒫_trb=36 W, 𝒫_edc=3.2 W, 𝒫_fft=4 mW, 𝒫_dsp=3 mW <cit.>. We use MATLAB, YALMIP and CVX software packages for programming, modeling and optimization.The total power consumption of different network elements in terms of aggregate traffic with and without adaptive transmit optical power assignment has been reported in Fig. <ref>. We have used the proposed approach of <cit.> for fixed assignment of transmit optical power. Clearly, for all the elements, the total power consumption is approximately a linear function of aggregate traffic but the slope of the lines are lower when transmit optical powers are adaptively assigned. As an example, adaptive transmit optical power assignment improves total transponder power consumption by a factor of 32% for aggregate traffic of 60 Tbps. Fig. <ref> shows total power consumption of different network elements versus number of available modes ℳ in FMFs. The power consumption values are normalized to their corresponding values for the scenario with single mode fibers ℳ=1. As ℳ increases, the amount of transponder power consumption decreases but there is no considerable gain for ℳ > 5. Moreover, there is a tradeoff between DSP and FFT power consumption such that the overall transponder power consumption is a decreasing function of the number of available modes. Fig. <ref> shows power consumption of different network elements in terms of aggregate traffic for strongly- and weakly-coupled FMFs. Obviously, total transponder power consumption is considerably reduced for strongly-coupled FMFs (in which group delay spread is proportional to square root of path lengths) in comparison to weakly-coupled FMFs (in which group delay spread is proportional to path lengths). This is the same as the results published in <cit.>. As an example, improvement can be more than 50% for aggregate traffic of 60 Tbps. Numerical outcomes also show that our convex formulation can be more than 20 times faster than its mixed-integer nonlinear counterpart which is compatible with the results reported in <cit.>. § CONCLUSIONEnergy-efficient resource allocation and quality of service provisioning is the fundamental problem of green 3D FMF-based elastic optical networks. In this paper, we decompose the resource allocation problem into two sub-problems for routing and traffic ordering, and transponder configuration. We mainly focus on transponder configuration sub-problem and provide a convex formulation in which joint optimization of temporal, spectral and spatial resources along with optical transmit power are considered. Simulation results show that our formulation is considerably faster than its mixed-integer nonlinear counterpart and its ability to optimize transmit optical power can improve total transponder power consumption up to 32%. We demonstrate that there is a tradeoff between DSP and FFT power consumptions as the number of modes in FMFs increases but the overall transponder power consumption is a descending function of the number of available modes. We also calculate the power consumption of different network elements and show thatstrongly-coupled FMFs reduce the power consumption of these elements. IEEEtran | http://arxiv.org/abs/1705.10198v1 | {
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]Tuning of Fermi Contour Anisotropy in GaAs (001) 2D Holes via Strain Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USADepartment of Physics, Northern Illinois University, DeKalb, IL 60115 USAPhysical Sciences Department, Rhode Island College, Providence, RI 02908 USADepartment of Electrical Engineering, Princeton University, Princeton, NJ 08544 USAWe demonstrate tuning of the Fermi contour anisotropy of two-dimensional (2D) holes in a symmetric GaAs (001) quantum well via the application of in-plane strain. The ballistic transport of high-mobility hole carriers allows us to measure the Fermi wavevector of 2D holes via commensurability oscillations as a function of strain. Our results show that a small amount of in-plane strain, on the order of 10^-4, can induce significant Fermi wavevector anisotropy as large as 3.3, equivalent to a mass anisotropy of 11 in a parabolic band. Our method to tune the anisotropy in situ provides a platform to study the role of anisotropy on phenomenasuch as the fractional quantum Hall effect and composite fermions in interacting 2D systems. [ M. Shayegan December 30, 2023 =====================High quality two-dimensional (2D) holes in GaAs exhibit various quantum mechanical phenomena, rendering them an attractive platform for fundamental resarch as well as applications for novel electronic and spintronic devices. For example, the strong spin-orbit interaction in GaAs leads to the intrinsic spin Hall effect, <cit.> and 2D holes in narrow quantum wells (QWs) have shown long spin coherence times, making them promising cadidates for spin qubits. <cit.> Also, high-mobility hole carriers and their zero-field spin-splitting, tunable by external electric field, <cit.> are essential components for mesoscopic spin devices. <cit.> In addition, recent magnetotransport measurements reveal that the inverted (“Mexican hat”) dispersion of the hole excited subband hosts an exotic annular Fermi sea. <cit.>The band-structure modifications resulting from the application of in-plane strain to GaAs 2D holes also provide exciting opportunities for fundamental studies and device applications. Previous studies demonstrated the tuning of the spin-splitting and the Fermi contour shape by applying in-plane strain to the GaAs (311)A 2D holes. <cit.> However, in such samples, despite the nearly isotropic Fermi contour, low symmetries in the wafer orientation and the interface corrugations cause severe anisotropic mobilities along the [011̅] and [2̅33] directions. <cit.> These complications can be eliminated by growing the 2D holes on a GaAs (001) substrate. There have indeed been magnetotransport studies on GaAs (001) 2D holes in the presence of strain, and these have reported an effective mass change for one of the spin-split subbands <cit.> as well as a control of the interaction-induced anisotropic phase in high magnetic fields. <cit.> Yet, quantitative measurements of strain-induced Fermi contour anisotropy have not been reported.In this Letter, we demonstrate tunable Fermi contour anisotropy via the application of in-plane strain to GaAs (001) 2D holes, and measure the Fermi wavevector anisotropy as a function of strain using commensurability oscillations.Figure 1 highlights our study to induce anisotropic Fermi contours in 2D holes confined to a symmetric GaAs QW grown on a GaAs (001) substrate. In Fig. 1(a) we show the calculated Fermi contours at a density of p=1.3 × 10^11 cm^-2 under various strain values applied along the [1̅10] direction. <cit.> The self-consistent numerical calculations are performed based on the 8×8 Kane model augmented by the Bir-Pikus strain Hamiltonian. <cit.> Owing to the bulk inversion asymmetry of the zinc-blende structure of GaAs, a finite spin-splitting is expected in our symmetric QW, as indicated by two Fermi contours (solid and dashed curves). <cit.> Without strain (ϵ=0), the Fermi contours show a four-fold symmetry in reciprocal space; the smaller contour is close to circular and the larger contour is slightly warped. When tensile strain (ϵ >0) is applied along [1̅10], the Fermi contours become elongated along [1̅10]. For compressive strain (ϵ <0), the distortion of the Fermi contours occurs in the opposite direction. Remarkably, the induced Fermi contour anisotropy is quite large even for a small amount of strain, of the order of 10^-4. This pronounced distortion is attributed to the large hole effective mass and the strong heavy-hole and light-hole mixing in the band structure of GaAs 2D holes; <cit.> the strain effect in GaAs 2D electrons is negligible. <cit.>Figures 1(b) and 1(c) show the schematic of our experimental setup. Our sample contains 2D holes confined to a 175-Å-wide, modulation-doped, GaAs QW grown on a GaAs (001) substrate by molecular beam epitaxy. The symmetric QW is flanked by a 960-Å-thick Al_0.24Ga_0.76As spacer layer and a C δ-layer on each side, resulting in a density p ≃1.3 × 10^11 cm^-2 and mobility 2 × 10^6 cm^2/Vs at 0.3 K. A 4 × 4 mm^2 piece of the wafer is thinned to ∼ 120 μm using mechanical lapping followed by chemical-mechanical polishing. <cit.> A Ti/Au gate is deposited on the back-side of the wafer to tune the density and also to shield any spurious external field from the piezo-actuator. The thinned sample, etched into the Hall bar geometry, is glued on one side of the stacked piezo-actuator using two-component epoxy, and cooled to 0.3 K for the magnetotransport measurements. A strain gauge is also glued on the other face of the piezo to monitor the relative change of the strain as a voltage bias V_P is applied to the piezo. <cit.> After sample cool-down, a finite built-in strain develops due to the different thermal contractions of the sample and the piezo. We determine the ϵ =0 condition when the measured Fermi wavevector equals k_0=√(2 π p). (see Ref. 29.)In order to measure the Fermi wavevector, we fabricate a grating of negative electron-beam resist with period a= 200 nm on the surface of the wafer. The grating induces a periodic strain on the GaAs surface which in turn results in a small periodic modulation of the 2DHS density via the piezoelectric effect . <cit.> When a small perpendicular magnetic field B is applied, the holes move along the cyclotron orbits, whose shapes resemble the Fermi contours, but rotated by 90^∘. When the cyclotron diameter along the current direction becomes commensurate with a, the magnetoresistance exhibits commensurability oscillations, also known as Weiss oscillations. <cit.> The minima in the trace are directly related to the 2D holes' Fermi wavevector along [1̅10], k_[1̅10], through 2R_c /a = i-1/4 (i=1,2,3 ...) where 2R_c = 2ħ k_[1̅10] /eB is the cyclotron diameter along [110], e is the electron charge, and ħ is the Planck constant. <cit.> [As an example, the shapes of the cyclotron orbits for ϵ >0 are depicted by black curves in Fig. 1(c).] Figure 2(a) shows the measured magnetoresistance traces at different ϵ. The green trace represents the ϵ =0 case, satisfying k_[1̅10]=k_0. The blue traces are for tensile strain (ϵ >0) while the red ones are for compressive strain (ϵ <0). The commensurability features appear symmetrically with respect to B=0, and the minima positions corresponding to i=2 and 3 are used to determine k_[1̅10]. The i=1 minimum is expected at higher field (≃ 0.8 T for ϵ=0), but it is masked by the Shubnikov-de Haas oscillations. The evolution of the mimima positions as a function of strain is clearly seen in Fig. 2(a) following the dashed lines; with increasing ϵ the minima move away from B=0, implying an increasing k_[1̅10]. In Fig. 2(b), black solid and dashed curves represent real-space cyclotron orbits of holes in different ϵ regimes; ϵ >0 (top), ϵ=0 (middle), and ϵ<0 (bottom). In principle, there are two cyclotron orbits corresponding to the two spin-subbands. However, we measure a single k_[1̅10] from the commensurability minima, implying that our experiments do not resolve the finite but small spin-splitting. The measured k_[1̅10] normalized to k_0 is shown in Fig. 2(c) as a function of ϵ. The red and blue circles are obtained from two different cool-downs. Although the built-in strains are different between the two cool-downs, both sets of data contain k_[1̅10]=k_0, i.e., ϵ=0. This enables us to determine the built-in strain, and thereby the absolute values of ϵ. It is clear that the overlapping k_[1̅10] values from two measurements agree very well each other. When ϵ is increased from -1.8 to 1.7 × 10^-4, k_[1̅10]/k_0 changes from 0.85 to 1.38, i.e., there is a ≃ 60% increase of the Fermi wavevector along the [1̅10] direction.Figure 3(a) shows the measured k_[1̅10]/k_0 and its comparison with the calculations as a function of ϵ. The circles, squares, and triangles represent the measured k_[1̅10]/k_0 in three different cool-downs where different built-in strains were attained. The black solid and dashed lines represent the calculated k_[1̅10]/k_0 for the two spin-subbands, and the thick orange curve shows their averaged values. The data shown by circles and squares are slightly lower than the orange curve. The small discrepancy indicates that the measured k_[1̅10] deduced from the commensurability features may not reflect exactly the averaged k_[1̅10] of the two spin-subbands, or the calculated Fermi contours are slightly different from those in our sample. Compared to the circles and squares, the measured data plotted by triangles show very large k_[1̅10]/k_0 values, implying that a large built-in strain develops for this cool-down. <cit.> Because k_[1̅10] cannot be tuned over a sufficiently large range to reach k_0 by applying V_P to the piezo, limited within ± 300 V, we are unable to determine the absolute magnitude of ϵ for this cool-down directly from the measured data. If we assume a built-in strain of ≃ 4.7 × 10^-4, however, the measured data match very well with the orange curve. Overall, the experimental data in Fig. 3(a) agree well with the averaged Fermi wavevector of the calculations to within 6%. Figure 3(b) shows the Fermi wavevector anisotropy, defined by k_[1̅10]/k_[110] where we use the average values of the two spin-subbands for each direction, based on calculations results for p=1.3 × 10^11 cm^-2. A remarkably large Fermi wavevector anisotropy, as high as 3.3, is achieved when ϵ= 5.5 × 10^-4 is applied. Note that this is equivalent to an effective mass anisotropy of 11 in a parabolic band, much larger than the intrinsic mass anisotropy (≃ 5) of semiconductors such as Si and AlAs. <cit.> Moreover, the tunability of the anisotropy in 2D holes allows for systematic studies of anisotropy effect on transport properties, which is not viable with the fixed, anisotropic 2D electrons in Si and AlAs. Before closing, we note that recent experiments have demonstrated that parallel magnetic fields can induce anisotropic Fermi contours in high-quality GaAs 2D hole (and electron) systems and that the induced anisotropy can be determined via commensurability oscillations measurements. <cit.> However, the induced anisotropy is primarily caused by the coupling between the in-plane and out-of-plane motions of the carriers, making the theoretical understanding of the data challenging. In contrast, the strain-induced anisotropy is originated from the band structure modifications at zero magnetic field. We also emphasize several different aspects between the two methods. First, the in-plane strain can both expand and contract the Fermi contour along a particular direction, while parallel-field can only elongate the Fermi contour in the direction perpendicular to the applied field direction. Second, the out-of-plane motions of the carriers driven by the in-plane field is affected by the QW width, thus the induced anisotropy is strongly depedent on the QW width as well as the out-of-plane effective mass, <cit.> while the strain does not bring in such complexity to the 2D holes. Third, a large parallel field inevitably leads to a large Zeeman spin-splitting of 2D holes. 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"published": "20170526044001",
"title": "Tuning of Fermi Contour Anisotropy in GaAs (001) 2D Holes via Strain"
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1Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, as part of the SPHERE GTO observations. 2ETH Zurich, Institute for Astronomy, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland 3Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago, Chile 4Millennium Nucleus "Protoplanetary Disk", Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago, Chile 5Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands 6Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands 7Universidad Autónoma de Madrid, Dpto. Física Teórica, Módulo 15, Facultad de Ciencias, Campus de Cantoblanco, E-28049 Madrid, Spain 8Núcleo de Astronomía, Facultad de Ingeniera, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile 9Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 10Max-Planck-Institut fur Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany 11Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109, USA 12INAFOsservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy 13Université Grenoble Alpes, IPAG, 38000 Grenoble, France 14CNRS, IPAG, 38000 Grenoble, France 15Aix Marseille Université, CNRS, LAM, UMR 7326, 13388, Marseille, France 16European Southern Observatory, Alonso de Cordova 3107, Casilla 19001 Vitacura, Santiago 19, Chile 17Kiepenheuer-Institut für Sonnenphysik, Schneckstr. 6, D-79104 Freiburg, Germany 18European Southern Observatory, Karl Schwarzschild St, 2, 85748 Garching, Germany 19NOVA Optical Infrared Group, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands We have observed the protoplanetary disk of the well-known young Herbig star HD 142527 using ZIMPOL Polarimetric Differential Imaging with the VBB (Very Broad Band, ∼ 600-900nm) filter. We obtained two datasets in May 2015 and March 2016. Our data allow us to explore dust scattering around the star down to a radius of ∼0.025 (∼ 4au). The well-known outer disk is clearly detected, at higher resolution than before, and shows previously unknown sub-structures, including spirals going inwards into the cavity. Close to the star, dust scattering is detected at high signal-to-noise ratio, but it is unclear whether the signal represents the inner disk, which has been linked to the two prominent local minima in the scattering of the outer disk, interpreted as shadows. An interpretation of an inclined inner disk combined with a dust halo is compatible with both our and previous observations, but other arrangements of the dust cannot be ruled out. Dust scattering is also present within the large gap between ∼30 and ∼140au. The comparison of the two datasets suggests rapid evolution of the inner regions of the disk, potentially driven by the interaction with the close-in M-dwarf companion, around which no polarimetric signal is detected.HD142527§ INTRODUCTIONPlanet formation cannot be understood from theory and first principles alone. Besides looking at the specific case of the solar system , the most promising way to achieve better comprehension is to study the environments planets form in - before, during, and after their formation. Protoplanetary and specifically transition disks, i.e. disks in which the inner region has already undergone some clearing <cit.>, are interesting targets when trying to examine and understand the intricate processes that occur in the circumstellar environment in its early phases. We know that planets form during these early phases in circumstellar disks. It is suspected that the transition disk phase is directly related to the formation of planets, because often, planets are a possible explanation for the features seen in the disk <cit.>. In a few cases, the connection between the disk and the (forming) planets has already been made. In HD 169142, a companion candidate was detected just inside the inner rim of the outer disk <cit.>, and an outer clump was seen at mm wavelengths <cit.>. In LkCa15, two companion candidates have been detected by means of non-redundant masking and/or direct Hα imaging <cit.>. In HD 100546, a companion candidatepotentially in its accretion phase was detected at ∼53±2 au, and a second one inside the cleared region is suspected based on the orbital notion seen in CO ro-vibrational line spectra and the shape of the inner rim of the outer disk <cit.>. All these disks are transition disks, and in all cases except for the outer HD 100546 companion the physical connection between the companion and the disk seems reasonably clear, strengthening the interpretation of transition disks as an important evolutionary phase related to the birth of at least certain types of planetary systems. Structures in protoplanetary disks are generally often interpreted in terms of planets or companions, which can induce gaps or spiral arms <cit.>. However, it is important to keep in mind that structures in disks can also be produced by processes not necessarily involving planets or planet formation, such as grain growth, photo-evaporation, magneto-rotational instabilities, shadow-driven spirals, or vortices <cit.>.Many transition disk systems are still accreting <cit.>, though perhaps at lower than expected levels <cit.>. This means that they must have an inner accretion disk, and given the typical clearing times of these inner disks, it also means that in most cases, material must be able to cross the gap to feed the accretion. In several cases, these inner disks have been detected directly with near-IR interferometry / imaging in the mid-IR or inferred from their SEDs <cit.>. In the case of LkCa15, an inner disk has been directly detected in scattered light <cit.>, but this is generally a challenging undertaking because of the proximity to the star and the contrasts needed. The dust emission is often too faint to be seen in (sub-)mm even with ALMA, though an unresolved inner disk has been inferred for the nearest protoplanetary transition disk, TW Hya, which could be responsible for a rotating azimuthal asymmetry seen in scattered light<cit.>.§.§ The HD142527 system HD 142527 is an example of a Herbig star surrounded by an optically thick protoplanetary disk and well-studied due to its proximity <cit.>, the brightness of its disk <cit.> and the size of its gap (∼140 au given the new Gaia distance). It is only moderately inclined <cit.>. These factors together have allowed for high-resolution, high-SNR images of the disk in optical/near-IR scattered light <cit.>, the mid-infrared <cit.> and at sub-mm wavelengths <cit.>. The disk has also been studied with respect to its sub-mm polarization <cit.>. These studies have revealed the gap to be highly depleted in micron-sized grains, though optically thick CO gas is present. They have also led to the interpretation of the two prominent local minima in scattered light as shadows cast from an inner, tilted disk, in agreement with the ALMA studies of gas close to the star <cit.>. More recently, <cit.> have produced an updated model of the inner and outer disk based on Herschel data, focusing on the water ice within the disk. Spirals in the outer disk extend outward from its inner rim and are detected both in sub-mm and scattered light <cit.>, though they are displaced w.r.t. each other at the two wavelengths. The inner disk has so far been revealed through mid-IR imaging and SED modeling, but has not been imaged directly in the near-IR at high resolution (sub-0.1). The disk still retains significant mass <cit.> and besides the fact that there is no good agreement on the extinction towards or the exact luminosity of the star, the star is still accreting at a significant rate <cit.>, which means that material must be transported in some way from the outer to the inner regions of the disk.So far, the inner disk surrounding the star has escaped detections in scattered light. Dust close to the star (up to to ∼30 au) is consistent with SED modelling and mid-IR imaging <cit.>, but using NACO, <cit.> reached an inner working angle of ∼0.1 (15 au) without detecting traces of the inner disk. This is interesting in light of the fact that HD 142527 has a still accreting M-dwarf companion, which has been detected with NACO/SAM and subsequently been followed up with NACO, GPI, MagAO, and most recently with SPHERE at close separation (77-90 mas). While it does not seem to orbit in the same plane as the outer disk, its orbit could be in agreement with the orientation of an inner disk casting shadows, though it is still not very well-determined <cit.>. This companion must necessarily be in causal contact and thus shape the dust and gas present within the inner 10-30 au around the star. Recent observations by <cit.> performed with the Gemini Planet Imager (Y band, 0.95-1.14μm) detect the companion in total intensity and find a point source in polarized light slightly offset from the location of the secondary, suggesting the presence of dust close to the location of the M-dwarf companion, possibly in a circumsecondary disk.Here, we present new SPHERE/ZIMPOL observations of HD 142527 specifically designed to study the smallest possible separations at the highest possible detail and SNR, being able to detect and resolve circumstellar dust down to an inner working angle (IWA) of ∼25 mas (4 au). In Section 2, we describe the data and data reduction procedures. In Section 3, we present results and analysis of these data, which are then discussed in more detail in Section 4. We conclude in Section 5.§ OBSERVATIONS AND DATA REDUCTIONThe observations were performed at the Very Large Telescope on the night of May 2nd, 2015, as part of the SPHERE GTO campaign. A second set of observations was obtained on March 31st, 2016. The ZIMPOL sub-instrument of SPHERE <cit.> was used in polarimetric differential imaging (PDI) mode, with both instrument arms set up with the Very Broad Band (590-881nm) filter, covering a wide wavelength range from the R to the I band. The read-out mode was set to FastPol with an integration time of 3s per frame, very slightly saturating the PSF core (2s and no saturation for the 2016 data). The data were taken using the P2 polarimetric mode (field stabilized) of ZIMPOL in 2015, and using the P1 (not field stabilized, i.e. the field rotates) and P2 modes for equal amounts of time in 2016 (this was done in order to compare the performance of the P1 and P2 modes for ZIMPOL). The instrument was set up to maximise the flux on the detector while enabling the study of the very smallest separations (down to ∼25 mas) and the gap of the disk as deeply as possible. The setup is not optimised for a high-SNR detection of the outer disk.In order to minimize suspected systematic effects close to the star (see section <ref>), images were taken in four blocks with different de-rotator angles in 2015 (0, 35, 80, 120 degrees). In 2016, the two polarimetric modes used naturally gave two different field rotations, with the field slightly rotating (∼4 degrees) in the P1 mode. The P1 and P2 mode observations were interleaved, with 3 blocks for each. For each of these blocks, the number of integrations (NDIT) was set to 14 (18 for the 2016 data), with the number of polarimetric cycles (NPOL) set to 6 (5 and 4 for the two last rotations after frame selection, 6 for the 2016 data). Using the QU cycle (full cycle of all four half-wave plate rotations), this adds up to a total on-source integration time of 3528s (3s * 14 (NDIT) * (6+6+5+4 (NPOL)) * 4 (HWP rotations)) in 2015 and 5184s (2s * 18 (NDIT) * 6 (NPOL) * 4 (HWP rotations) * 3 (blocks) * 2 (P1+P2)) in 2016, for a grand total of 8712s (2h 25.2min) of on-source integration time in both epochs combined.The most critical step in PDI is the centering of the individual frames. ZIMPOL data are special because of the way the detectors work <cit.>. The pixels cover an on-sky area of 7.2mas x 3.6mas each. The stellar position is determined before re-scaling the images by fitting a skewed (i.e. elliptical) 2-dimensional Gaussian to the peak. The data are then re-mapped onto a square grid, accounting for the difference in pixel scale along the x- and y-axis, and corrected for True North. The columns affected by the read-out in FastPol mode have been mapped out manually <cit.>. Besides these points specific to the ZIMPOL detectors, the data reduction follows the steps described in <cit.> with one important difference: Instead of performing the correction for instrumental (or interstellar) polarization for each pair of ordinary and extraordinary beams, the correction is done at the end by subtracting scaled versions of the total intensity I from the Stokes Q and U vectors, minimizing the absolute value of U_ϕ. This method has been used successfully before by the SEEDS team <cit.> and is better at suppressing very low surface brightness residuals. We note, however, that any such method that does not rely on separate calibration sources intrinsically assumes the star to be unpolarized. Any intrinsic polarization of the central source will diminish the resultant data quality. However, given the typical polarizations of Herbig stars (fractions of percent to few percent) compared to the polarization induced by dust scattering <cit.>, this would be a second-order effect. A correction for the efficiency in Stokes U vs. Stokes Q is not required for ZIMPOL thanks to the to the better controlled instrumental polarization compared to NACO <cit.>.The local Stokes vectors, now called Q_ϕ and U_ϕ by most authors <cit.> are calculated as: Q_ϕ=+Qcos(2ϕ)+Usin(2ϕ) U_ϕ=-Qsin(2ϕ)+Ucos(2ϕ) ϕ=arctanx-x_0/y-y_0+θ Here, θ is used to correct for the fine-alignment of the half-wave plate (HWP) rotation and is determined from the data by assuming that U_ϕ should on average be zero. We note that in cases of highly inclined optically thick disks, the reality can deviate from this assumption strongly due to multiple scattering <cit.>. HD142527 is only moderately inclined <cit.>, but the inner disk is suspected to be inclined by about 70 degrees <cit.>. However, in an optically thick, but symmetric disk of any inclination, the average of U_ϕ will still be zero for reasons of symmetry. In the case of single-scattering (optically thin disks) and non-aligned grains, the assumption of no signal in U_ϕ (polarization signal perpendicular to the incident light) holds for any inclination.The disk of HD142527 is neither optically thin nor symmetric. We use the region between 0.2and 0.6to measure U_ϕ for correction. This region has very little flux in either Q_ϕ or U_ϕ, resulting in a good correction for instrumental or interstellar polarization effects. Given the inherent problems with measuring flux in PDI images <cit.>, we do not attempt to perform an absolute flux calibration of our images. We roughly estimate the Strehl ratio of our data by comparing the flux within the first airy minimum to the total flux (measured within 1.5") and dividing this by the expected ratio for a perfect, diffraction-limited system of 0.838. Using this method, we arrive at an estimate of around 34% for all our datasets. The resolution achieved (as measured by the FWHM) is around 34 mas, again for all our datasets. § RESULTS AND ANALYSIS In this section, we first discuss the results for the combined data of both epochs, before investigating possible differences between both epochs in section <ref>.Figure <ref> shows the resulting combined (2015+2016) Q_ϕ and U_ϕ images obtained in the same color stretch. To first order, Q_ϕ contains polarimetric signal and noise, while U_ϕ contains no signal, but noise on the same level. As can be seen, the signal in Q_ϕ is - as expected - much stronger than the signal in U_ϕ, and polarized flux is seen close to the star. However, there remains a significant pattern close to the star in U_ϕ within the innermost ∼200 mas. This also affects the Q_ϕ data. This kind of noise is seen for other sources as well <cit.>. Taking the data in different orientations reduces this problem, but does not eliminate it. It is worth noting that this pattern noise, which is overlaid over the actual data in both Q_ϕ and U_ϕ, has both positive and negative components. There is no indication in either this or the HD 135344B dataset that the noise deviates from zero on average, and thus the noise mostly cancels itself out once taking azimuthal averages. The dust scattering close to the star remains a clear detection and it is evident in each of our 6 independent datasets individually. Its significance is further emphasized by statistical analysis (see below). Besides the noise pattern, we do not detect any significant astrophysical signal in U_ϕ. §.§ Geometrical appearance of the dust The Q_ϕ images reveal the well-known outer disk at high SNR and at higher resolution than previously available NACO data <cit.>. They furthermore are able to detect dust scattering close to the star. This structure is elongated in the ESE-WNW (position angle ∼120^∘ east of north) direction. However, it does notresemble a uniform disk of any inclination, but rather has extensions both on the south-eastern and western / north-western sides (these two lobes are seen in each of the six independent sub-datasets described before). The more prominent extension is seen on the north-western side. The western side is also special because there is a prominent dip in brightness towards the west, very close to the star (∼25-50mas). The dust structure as a whole has no apparent relation to the shadows seen in the outer disk <cit.>, because an inclined disk explaining those shadows would be elongated in the north-south direction.Furthermore, the gap that was first revealed to be largely devoid of dust down to ∼15au (Avenhaus et al. 2014) can now be seen to be asymmetric. The gap clearly deviates from an elliptic shape in the southwest. The spiral arms in this region seem to cross the inner wall of the outer disk, extending inwards into the gap region.§.§ Inner dust structure and dust within the gapIn order to further determine the reliability of the detected signal, we calculate azimuthally-averaged surface brightness profiles for the disk. The results can be seen in Figure <ref>. The polarization signal close to the star is detected at more than 20σ, with σ calculated from the U_ϕ data as described in <cit.>. This does take into account the (systematic) errors close to the star discussed above. We can thus clearly state that the inner dust structure is detected. §.§.§ Signal dampening through PSF smearing effects and brightness of the inner disk It is worth noting that when corrected for the fall-off of the stellar illumination by multiplying the data with r^2, r being the projected distance from the stellar position (right side of Figure <ref>), the dust scattering close to the star seems to be significantly fainter than the outer disk. However, this does not take into account the dampening effect of PSF smearing in PDI. This can reduce the polarimetric flux close to the star when employing the PDI method <cit.>. The magnitude of this effect depends both on the distribution of scattered light itself and on the shape of the PSF. It is weaker for stable, high-Strehl PSFs and further away from the star. The ZIMPOL Very Broad Band filter (590-881nm) is more strongly affected by this problem compared to the near-IR IRDIS filters because of the significantly lower Strehl ratios at this wavelength. The inner dust structure is particularly affected because of its proximity to the star.In principle, the best way to understand these effects is a forward-modeling of scattered-light images produced with a radiative transfer code, which are then (Stokes Q and U vectors) convolved with the PSF retrieved from the observations. Because developing a radiative transfer model is beyond the scope of this paper, specifically for the complex asymmetric dust structure we observe, we instead use the derived Q_ϕ image in order to estimate the magnitude of signal suppression within our scattered-light data. The process works as follows:* Produce an azimuthally averaged image Q_ϕ, avg of the Q_ϕ image and smooth it with a small (∼25 mas) Gaussian kernel in order to avoid effects from small-scale structures and noise* Split this image into the Stokes Q and U vectors using the inverse of the formulas shown in Section <ref>* Convolve the obtained Stokes vectors with the PSF obtained from the unsaturated science frames* Calculate Q_ϕ, avg, damp from these convolved Stokes vectors* Calculate the local damping factor as F_ damp = Q_ϕ, avg/Q_ϕ, avg, damp. We then get an approximation of the real (undamped) polarimetric scattered-light signature by multiplying Q_ϕ with F_ damp. Both Q_ϕ and Q_ϕ· F_ damp are displayed alongside each other in Figure <ref>. As can be seen, the inner dust structure brightens up significantly with this processing (factor of ∼5). The outer disk brightens as well (showing that PSF smearing has an effect even at >1), but by a smaller factor. We then produce averaged radial surface brightness plots again, which show that the inner disk is indeed as bright as the outer disk when corrected for the drop-off in stellar illumination (see Figure <ref>, left side).The inner dust structure is in fact very bright. When compared to the outer disk, it scatters more light than the entire outer region of the disk as seen in our images. To show this, we divide our image into three regions: The inner dust structure (0.025-0.2), the gap region (0.2-0.55), and the outer disk (0.55-1.35). Using the corrected Q_ϕ image and conservative error estimates constructed from the corrected U_ϕ image, we calculate a polarized flux ratio of F_ inner/F_ outer = 1.43±0.36. Most of this flux is very close to the star, in the region between 25mas and 50mas. If we further divide the inner disk based on this into a region of 25mas-50mas and 50mas-200mas, we get F_ 25-50/F_ outer = 1.04±0.23 and F_ 50-200/F_ outer = 0.37±0.07. The gap is dark compared to the outer disk, with F_ gap/F_ outer = 0.062±0.007. Figure <ref> shows the chosen regions of the disk for reference.Comparing the scattered light within 1.35 to the total flux (star and scattered light, both polarized and unpolarized) in the same region, we get a ratio of F_ pol, scattered/F_ total≈ 0.83%. This ratio is affected by both the albedo of the grains and the polarization fraction. The scattered, unpolarized light is also seen in the pure intensity image. Lacking a suitable reference PSF for subtracting the stellar halo, we can not repeat the analysis performed in <cit.> and <cit.>. However, a rough estimate based on subtracting scaled versions of the Q_ϕ image from the intensity image points to the polarization fraction showing qualitatively the same behavior as noted in those works - the polarization fraction is higher on the eastern and lower on the western side. §.§.§ Dust within the gap Besides the structure within the innermost ∼200 mas, we also detect a signal in Q_ϕ at >8σ throughout the gap, consistent with previous findings in <cit.>, where the signal-to-noise ratio was however much weaker (<3σ). There are two possible sources of this signal: It can either arise from light from the outer or inner disk that has been smeared into the gap region by the PSF, or it is scattered light from within the gap region. In order to distinguish between these two possibilities , we artificially map out a gap between 0.2and 0.4in our corrected data, setting this region to zero, and fold it (Stokes Q and U individually) with the PSF. The result is displayed in Figure <ref> alongside the other surface brightness plots. While some seeping in occurs, this could account for a signal on the order of ∼0.5% of the peak of the outer disk (in r^2-scaling), but not for the observed signal of ∼2% of the outer disk peak. We thus conclude that most of the observed signal (∼75%) must come from dust scattering in this region, and only a minor fraction (∼25%) of it results from PSF smearing.This means that dust scattering within the gap region is detected at a significant level of ∼6σ after taking into account possible smearing of signal from the in- or outside of the gap. The gap is not completely devoid of dust, as already hinted at in <cit.>. The dust scattering is weak (the signal is weaker than the signal from the outer / inner disk by a factor of ∼50-70 when corrected for the r^2 drop-off in illumination), meaning that the dust is either shadowed or optically very thin.In an attempt to better understand the dust distribution within the gap, we create σ confidence maps by comparing the signal in the Qr image with the noise in the Ur image <cit.>. These images are generated as follows: 1) Smooth the image with a Gaussian kernel of 30 mas width (approx. the size of the beam) in order to get rid of high spatial frequency noise, 2) Calculate the variance in the (smoothed) Ur image over an area five times as wide as the smoothing kernel, 3) divide the signal in Qr by the square root of the obtained variance. The results are shown in Figure <ref>. This technique is not perfect, namely because the results depend on the used smoothing kernel for the variance calculation, but allows us to give an estimate of the signal strength within the gap. We notice that there is significant scattering close to the outer edge of the gap, potentially from gas streaming into the gap and carrying small dust grains with it. This can also be seen from the spiral-like features at the outer edge of the gap in the southwest, consistent with the fact that the inner disk would not be able to support the accretion rate of the star for extended periods of time and thus material has to accrete from the outer to the inner disk <cit.>. This feature could not be clearly seen in previous observations <cit.>. The signal of the outer disk is very strong (up to 150σ). On the other hand, locating the dust within the gap, while clearly detected in the azimuthal averages, remains fairly inconclusive given the low signal-to-noise ratio and the fact that butterfly patterns can be easily introduced into Q_ϕ and U_ϕ images if the correction for instrumental polarization is not perfect (these do not affect azimuthal averages, as the positive and negative butterfly wings cancel out). What can be seen, though, is that neither in the 2015 nor in the 2016 or combined data, any shadow can be seen in the direction of the outer disk shadows. §.§ Comparison of 2015 and 2016 epochsThe 2015 and 2016 epochs have similar SNR across the disk and thus can be compared well. The appearance of the disk is very similar, as is to be expected, and combining the data increases the SNR of both the inner dust structure and outer disk (see Figure <ref>). However, there are minor differences worth pointing out.First, the inner dust structure, while appearing elongated in the SE-NW direction in both epochs, appears broader in the 2016 images in the N-S direction. This can be seen in both the SE and NW extension (Figure <ref>) and is independent of reduction parameters. Second, the structure of the gap in the σ maps is different. While there seems to be a bridge-like structure extending from the SE to the NW in the 2015 dataset, the opposite is true for the 2016 epoch - the bridge extends from the NE to the SW. However, as pointed out in the previous section, butterfly patterns can easily accidentally be introduced into these images. The exact structure depends on the reduction parameters, specifically the inner and outer radius used for the instrumental polarization correction. While for any specific set of parameters used, the dust appears differently in the two epochs, we are not convinced that these differences are significant. § DISCUSSIONWe have presented the first unambiguous detection of scattered light off circumprimary dust within 30au of HD142527. That dust must exist at these separations was already known <cit.>, but it had so far escaped direct detection in scattered light because of its proximity to the star, the required contrast performance, and the fact that the PSF smearing effect strongly suppresses the polarization signal at these separations. Our study is also an attempt at localizing scattered light within the large gap.§.§ Inner disk or dust structure and relation to shadowing <cit.> linked the two prominent local minima in the north and south of the outer disk to a shadowing from an highly inclined inner disk. Within the context of our observations, this presents two challenges: 1) the inner dust structure we observe does not resemble a disk that could conceivably be responsible for that shadowing; 2) no shadow is observed within the gap. <cit.> find that to reproduce their data, they need to invoke not only an inner disk (which is not inclined in their model due to the knowledge at the time), but also a dust halo close to the star. <cit.> show that in order to fit the SED using a two-component (inclined inner and outer disk) model without a halo, they need to extend the scale height of the inner disk beyond the expected hydrostatic equilibrium in order to fit the near-IR flux. They do not discuss, but also not exclude the possibility of a halo which would enhance the near-IR flux. We furthermore know from <cit.> that the secondary M-dwarf HD142527B is likely to be in an orbit co-planar with the suggested, highly inclined inner disk. This means that what we observe could in fact be both the inner disk or an extended, not necessarily spherical halo. We note at this point that <cit.> pointed out that a halo could be replenished by a highly excited debris disk. The stellar companion of HD142527 was not known at the time, but would excite any debris material in the inner disk. A dust halo producing grey extinction also helps to explain the unusually high infrared flux ratio of F_ IR/F_ star = 0.92.In this interpretation, the inner disk would then still be within or very close to our IWA, and the east-west extended structure seen would not be part of the inner disk, but part of the (dynamically excited) halo.This does not explain the "missing" shadow in the gap, but we have to remember that the scattering signal we observe, while a clear detection when azimuthally averaged, is locally still of very low SNR.While the dust scattering seems to be faint at first sight, our ad-hoc correction for PSF smearing effects reveals that the polarimetric flux from the inner disk region is in fact greater than that of the outer disk. Inclination effects could play a role here, and we do not know the line-of-sight position of the inner dust, but we do know that the scattering angles in the outer disk are close to 90^∘, the ideal scattering angle for strong polarization, due to the low inclination of the disk. Dust grain properties (e.g. a higher albedo) could also play a role, but to first order we have to assume that as much or more light is re-processed in the inner dust structure than is in the outer disk. Thus, the inner dust structure must be close to optically thick - at least very close to the star, where most of the polarized flux stems from.Combining these findings with our results, we deem the following scenario conceivable: 1) A highly inclined inner disk close to the star, potentially at least partly responsible for the high polarimetric flux very close to the star (25-50mas); 2) A dust halo which is responsible for the weaker polarimetric signal, mostly in the ∼50-200mas regime and where the unexpected east-west extension is explained by the fact that it is strongly distorted by interactions with the secondary M-dwarf and material accreting from the outer disk; 3) The well-known outer disk which re-processes large portions of light beyond 100au. The proposed size of the inner disk is consistent with recent sub-mm observations, in which an inclined disk with a radius of only 2-3 au is sufficient to explain the sub-mm measurements <cit.>.This scenario, however, does not explain the scattered light seen within the gap nor does it explain the lack of shadows therein. We thus could imagine as component 4) a larger-scale dust halo with dust not only located in the plane of the outer disk, but also high above the mid-plane which would dilute the shadows from the inner disk and thus be compatible with polarized flux from the entire gap region. The scenario would be similar to the scenario described by <cit.>, but with a smaller inner disk and an additional large, diluted halo of unexplained origin (it could, for example, stem from radiation pressure blow-out of small dust grains from the inner halo).We caution though that our data within the gap is of low signal-to-noise ratio and potentially affected by systematic effects from the unstable PSF of SPHERE in the optical (relatively, compared to the near-IR). Thus, the faint regions in the disk and specifically the gap are difficult to interpret conclusively, and the explanation for the "missing" shadows in the gap might simply be because of too low SNR. Ignoring the "missing" shadows, another explanation for faint, scattered light within the disk gap would be secondary scattering, i.e. light scattered in the inner region of the disk towards the mid-plane and then re-scattered by dust along the mid-plane within the gap region. However, <cit.> find a contrast ratio in the sub-mm between the outer disk peak and the gap region in excess of 1200, meaning that if there is any dust at the mid-plane in the gap region, it must be very thin. We show sketches of the two proposed possible explanations in Figure <ref>.While this explanation is compatible with our data, as far as we can see without producing a detailed radiative-transfer model, there could be other explanations for this complex circumstellar environment. The exact geometrical shape of the dust we detect remains unknown, also because of possible projection and complicated shadowing effects. It has to tie in, however, with the known kinematic signatures of the gas as studied by ALMA <cit.>. To fully understand this, a coupling of a hydrodynamical model which takes into account both the gas and dust dynamics as well as the orbit of HD142527B with a radiative-transfer code (beyond current state-of-the-art capabilities), or at least either adequate hydrodynamical simulations which can handle both gas and dust or radiative-transfer modeling would be necessary.§.§ Dust around HD142527B Contrary to the results of <cit.>, we do not see an enhancement of dust close to the location of the M-dwarf companion detected by <cit.>. Thus, we can not confirm their findings. We note that the contrast limits we reach are better (<cit.> did not detect the faint dust scattering we report here) and that quick tests with inserting their detection into our images show that we should have detected such a bright point-like source. A disappearance of a circumbinary disk on such short timescales is unlikely. We therefore can not confirm circumsecondary dust scattering, even though a circumsecondary disk is also needed to fit the SED of the secondary <cit.>. As a side note, we would like to mention that a detectable signal would stem from light from the primary scattered off the disk of the secondary. The secondary itself is too faint compared to the primary and the butterfly pattern it would produce in Stokes Q and U would be so small that it would be washed out by convolution with the PSF.It is worth mentioning that <cit.> detect an additional point-source of sub-mm emission within the gap, towards the north of the star. A possible explanation for this is dust surrounding a third object within the system, however we do not detect any additional scattered light in this region. §.§ Dust entering the gap from the outer disk Besides being able to show unambiguously that scattered light exists within the gap of HD142527, which could also stem from a larger halo as described above, we are also able to tentatively localise the polarized flux. The strongest signal is found close to the outer edge of the gap, which could be caused by gas that is dragged into the gap to cross the gap and eventually accrete onto the star (given the accretion rate of HD142527), and dragging micron-sized dust along.However, <cit.> found that the M-dwarf companion is unlikely to be responsible for truncating the outer disk (due to inclination), and thus also unlikely to be responsible for dragging dust into the gap via companion-disk interactions, despite the line-of-sight proximity of its possible apocenter and the dust signature entering the gap. Unseen planets within the gap would offer a potential explanation. §.§ Differences between 2015 and 2016 epochs The difference in the appearance of the inner dust and the potential difference in the appearance of dust in the gap between the 2015 and 2016 epoch points to a possible astrophysical difference between the two epochs. A time baseline of 11 months, while short, is still significant compared to the orbital timescale at the location of the inner working angle (∼4 au, ∼5 yr). It is very short compared to the orbital timescale at 0.1 (∼36 yr), though.It is possible that variations are caused by differential shadowing from dust inside our inner working angle, closer to the star than the dust structure we observe. Dust at that location would evolve on its own, shorter orbital timescale. However, a variable shadowing would not only influence the gap region, but also the outer disk. Variable shadowing on the outer disk is not observed. Thus, future observations will be required to confirm variations of the observed dust structure and understand their origins. § CONCLUSIONIn this paper, we present new observations of the HD142527 disk with ZIMPOL in the VBB filter. The disk is clearly detected, and for the first time, unambiguous detections of dust scattering within the gap and an inner dust structure are presented.However, our detection of the dust structure close to the star raises several questions. While it is in reasonable agreement with the model of <cit.>, it is not easy to reconcile with the findings of <cit.> and <cit.>, which predict an inclined inner disk, which we do not see with the right inclination. It is unclear whether the dust we see resembles a disk, and if it does, the inclination and position angle are such that it is unlikely to be responsible for shadowing the two prominent local minima in the outer disk. A model consisting of both an inner disk and a dust halo, which are overlaid in our images and cannot be properly separated due to limitations in signal-to-noise ratio, resolution and inner working angle, is conceivable and consistent with our data, but we have no proof for such an arrangement of the dust. The detection of the dust within the gap is not easily located, even though it is clearly detected in the azimuthally averaged maps. There is no obvious shadow towards the shadows in the outer disk. This could be explained by a larger-scale dust halo outside the disk mid-plane, but it is also possible that our signal-to-noise ratio is simply too weak to detect the shadows. There also seems to be some difference between our two epochs, but this detection at this point is tentative and will require further investigation in the future.HD142527 is a system in dynamical evolution displaying complex interplay of its misaligned components - the outer disk, the gap region, an inclined inner disk, a halo, the secondary M-dwarf, and potentially more that we have not discovered yet, for example planets responsible for carving out the large gap. The dust and gas components of the disk are clearly not azimuthally symmetric. Understanding the individual components and their interplay remains a challenging task, but due to the proximity and brightness of the disk, HD142527 remains an important transition disk test case. HA acknowledges support from the Millennium Science Initiative (Chilean Ministry of Economy) through grant RC130007 and further financial support by FONDECYT, grant 3150643. JH is supported by ANR grant ANR-14-CE33-0018 (GIPSE) and FM by grant ANR-16-CE31-0013 (Planet-Forming-Disks). SPHERE is an instrument designed and built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France),INAFOsservatorio di Padova (Italy), Observatoire de Gen`eve(Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France)and ASTRON (Netherlands), in collaboration with ESO.SPHERE was funded by ESO, with additional contributions from CNRS (France),MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands).SPHERE also received funding from the European Commission Sixth andSeventh Framework Programmes as part of the Optical InfraredCoordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (20042008), grant number 226604 for FP7 (20092012) and grant number 312430 for FP7 (20132016).Part of this work has been carried out within the framework of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation. H.A., S.P.Q., and H.M.S. acknowledge the financial support of the SNSF. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We thank the staff at VLT for their excellent support during the observations.This work has made use of data from the European Space Agency (ESA) mission Gaia (<http://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <http://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. Facilities: VLT:Melipal (SPHERE) apj.bst | http://arxiv.org/abs/1705.09680v2 | {
"authors": [
"H. Avenhaus",
"S. P. Quanz",
"H. M. Schmid",
"C. Dominik",
"T. Stolker",
"C. Ginski",
"J. de Boer",
"J. Szulágyi",
"A. Garufi",
"A. Zurlo",
"J. Hagelberg",
"M. Benisty",
"T. Henning",
"F. Ménard",
"M. R. Meyer",
"A. Baruffolo",
"A. Bazzon",
"J. L. Beuzit",
"A. Costille",
"K. Dohlen",
"J. H. Girard",
"D. Gisler",
"M. Kasper",
"D. Mouillet",
"J. Pragt",
"R. Roelfsema",
"B. Salasnich",
"J. -F. Sauvage"
],
"categories": [
"astro-ph.EP",
"astro-ph.SR"
],
"primary_category": "astro-ph.EP",
"published": "20170526185401",
"title": "Exploring dust around HD142527 down to 0.025\" / 4au using SPHERE/ZIMPOL"
} |
Inexact Krasnosel'skii-Mann iterations]Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spacesNúcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003M. Bravo]Mario Bravo [Mario Bravo]Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile,Alameda Libertador Bernardo O'higgins 3363, Santiago, Chile. E-mail: [email protected] Acknowledgements. This work was partially supported by Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003.Mario Bravo was partially funded by FONDECYT 11151003. Roberto Cominetti and Matías Pavez-Signé gratefully acknowledge the support provided by FONDECYT 1130564 and FONDECYT 1171501. R. Cominetti]Roberto Cominetti [Roberto Cominetti]Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez,Diagonal Las Torres 2640, Santiago, Chile.E-mail: [email protected] M. Pavez-Signé]Matías Pavez-Signé [Matías Pavez-Signé]Departamento de Ingeniería Matemática, Universidad de Chile,Beauchef 851, Santiago, Chile. E-mail: [email protected] [2010]Primary: 47H09, 47H10; Secondary: 65J08, 65K15, 60J10We study the convergence of an inexact version of the classicalKrasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-pointresiduals, from which we derive their rate of convergence as well as the convergenceof the iterates towards a fixed point. The results are appliedto three variants of the basic iteration: infeasible iterationswith approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in orderto study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators. [ [=====[ [===== § INTRODUCTIONLet T:C→ C be a nonexpansive map defined on a closed convex domain C in a Banach space(X,·).The Krasnosel'skii-Mann iteration approximates a fixed point of T by the sequential averaging processkm x_n+1= (1-α_n+1) x_n + α_n+1 Tx_n,wherex_0∈ C is an initial guess and α_n∈ [0,1] is a given sequence of scalars.This iteration, introduced by Krasnosel'skii <cit.> and Mann <cit.>,arises frequently in convex optimization as many algorithms can be castin this framework. This is the case of the gradientmethod forconvex functions with Lipschitz gradient <cit.>,the proximal point method <cit.>, as well as different decomposition methods such as the forward-backwardsplitting method<cit.>, the alternating direction method of multipliers ADMM <cit.>, the Douglas-Rachford splitting <cit.>, and the Peaceman–Rachford splitting <cit.>. For acomprehensive survey of these methods and their numerous applications we refer to <cit.>. Note that (<ref>) also arises when discretizing the evolution equationu'(t)+[I - T]u=0 (see e.g. <cit.>), so that many results for (<ref>) admit natural extensions to continuous time.A central issue when studying the convergence of the iterates x_n towards a fixed point of Tis to establish the strong convergence of the residuals x_n-Tx_n→ 0, a property known as asymptotic regularity <cit.>.For an historical account of results in this area we refer to<cit.>. In the case of a bounded domainC, an explicit estimate for the residual was conjectured in <cit.> andrecently confirmed in <cit.>, namelyx_n-Tx_n≤(C)√(π∑_k=1^nα_k(1-α_k)),where the constant 1/√(π) is known to be tight (see <cit.>). This inequality implies that asymptotic regularity holds as soon as ∑_k=1^∞α_k(1-α_k)=∞. The bound (<ref>) can also be used to estimate the number of iterations required to attain any prescribed accuracy, as well as to establish the rate of convergence of the residuals. For instance, if α_n remains away from 0 and 1 the bound yieldsx_n-Tx_n= O(1/√(n)), whereas for α_n=1/n one gets an order O(1/√(ln n)). When the operator values Tx can only be computed up to some precision, one is naturally led to consider the inexact iteration ikm x_n+1= (1-α_n+1) x_n + α_n+1 ( Tx_n + e_n+1),where e_n+1 can be interpreted as an error in the evaluation of Tx_n, or as a perturbation of the iteration. Note that (<ref>) requires x_n∈ C so that itassumes implicitly that the iterates remain in C. This inexact iteration was used by Liu <cit.> to study the equation Sx=f, restated as a fixed pointofTx=f+x-Sx, with S demicontinuous and strongly accretive on a uniformly smooth space. Liu proved the strong convergence of the iterates assuming that Tx_n remains bounded, ∑_k≥ 1e_k<∞, and α_n→ 0 with ∑_k≥ 1α_k=∞.Weak convergence of (<ref>)was also established for T nonexpansive, first on Hilbert spaces <cit.> and then on uniformly convex spaces <cit.>, provided that (T)≠∅ and ∑_k≥ 1α_k(1-α_k)=∞ with ∑_k≥ 1α_ke_n<∞. The rate of convergence of (<ref>) was recently studied in a Hilbert setting by Liang, Fadili and Peyré <cit.>, proving that x_n-Tx_n=O(1/√(n)) under the stronger summability condition ∑_k≥ 1ke_k<∞ and with α_nbounded away from 0 and 1. The proof exploits the Hilbert structure and does not seem to carry over to general Banach spaces. §.§ Our contributionThe main result in this paper is an extension of the bound (<ref>) which holds for the inexact iteration (<ref>) in general normed spaces. From this extended bound we draw a number of consequences on the convergence of the iterates and the rateof convergence of the fixed point residuals, and we derive continuous time analogs for the asymptotics of evolution equations governed by nonexpansiveoperators. In all what follows we denote ϵ_n≥e_n a bound for the errors, and we let τ_n= We also consider thefunctionσ:[0,∞)→ defined by σ(y)=min{1,1/√(π y)}. With these notations, our main result can be stated as follows. Let (x_n)_n∈ be a sequence generated by (<ref>) and assume that _0Then, for all n∈ we havex_n-Tx_n≤κ σ(τ_n) + ∑_i=1^n2 α_iϵ_i σ(τ_n-τ_i) + 2 ϵ_n+1.In particular, if τ_n→∞ and ϵ_n→ 0 with ∑_k≥ 1α_kϵ_k<∞,then x_n-Tx_n→ 0. Clearly, in the exact case with ϵ_n≡ 0 the bound (<ref>) yields (<ref>). The proof of Theorem <ref> is presented in Section <ref> and uses probabilistic argumentsby reducing the analysis to the study of an associated Markov reward process in ℤ^2.As a first consequence of this result, in <ref> we explain how the property x_n-Tx_n→ 0 can be used to show that (T)≠∅ as well as the convergence of the iterates x_n towards a fixed point.The assumption (_0) above imposes two conditions: the iterates must remain in C and the images Tx_n are bounded. Some situations in which these conditions hold are discussed in <ref>, including the case when T is defined on the whole space and either it has a bounded range, or∑_k≥ 1α_kϵ_k<∞ and (T)≠∅. Note also that when C is bounded one can take κ=(C) so that it suffices to ensure that the iterates remain in C. Alternatively, in <ref> we consider an iteration that uses approximateprojections to deal with the case when x_n falls outside C.Section <ref> exploits the bound (<ref>) in order to establish several resultson the rate of convergence of the residuals. Theorem <ref>shows that if α_n remains away from 0 and 1 and ∑_k≥ 1k^a e_k<∞ then x_n-Tx_n= O(1/n^b) withb=min{1/2,a}. This extends the main result in <cit.> that covers only the case a=1 and is restricted to Hilbert spaces. On the other hand, fromknown properties of the Gauss hypergeometric function _2F_1(a,b;c;z),we obtain as a Corollary of Theorem <ref> that when e_n= O(1/n^a) with α_n bounded away from 0 and 1, the residual norm satisfiesx_n - Tx_n={[O(1/n^a-1/2); O(log n/√(n)); O(1/√(n)) ].Note that for a≤ 1 the assumption e_n= O(1/n^a) is very mild and allows for nonsummable errors.Similar rates are obtained for vanishing stepsizes of the form α_n=1/n^c with c≤ 1.Section <ref> explores three variants of the basic iteration (<ref>). In <ref> we deal with the case in which the iterates might fall outside C byusing a suitable approximate projectionof x_n onto C. Then, in <ref> we analyze the Ishikawa iteration which can be seen as a special case of the inexact scheme (<ref>). In <ref> we consider a diagonal version of (<ref>) in which the operator T might change at each iteration. The final Section <ref> presents the extension of the results to continuous time,establishing the rate of convergence for the nonautonomous evolution equationE u'(t)+(I-T)u(t)=f(t), u(0)=x_0. § PROOF OF THEOREM <REF> We begin by noting thatx_n-Tx_n = (x_n-x_n+1)/α_n+1+e_n+1 so that x_n-Tx_n≤x_n-x_n+1/α_n+1+e_n+1. In order to bound the term x_n-x_n+1 we follow a similar approach as in <cit.> by establishing a recursive bound for the differencesx_m-x_n for all 0≤ m ≤ n. In what follows we let α_0=1 andfor 0≤ i≤ n we denoteπ_i^n=α_iso that ∑_i=0^nπ_i^n=1. For i=n we use the standard convention∏_k=n+1^n(1-α_k)=1. The following is a slight variant of <cit.>, which itself extends a similar result by Baillon and Bruck <cit.>.Let (x_n)_n∈ be defined inductively byx_n+1=(1-α_n+1)x_n+α_n+1y_n with x_0∈ X and y_n∈ X. Then, settingy_-1=x_0,we havex_n=∑_i=0^nπ_i^ny_i-1 for all n≥ 0. Moreover, for 0≤ m≤ n it holds x_m-x_n=∑_i=0^m∑_j=m+1^nπ_i^mπ_j^n(y_i-1-y_j-1). The equalityx_n=∑_i=0^nπ_i^ny_i-1 follows by a straightforward inductive argument, while (<ref>) follows from this equalityand the identities ∑_i=0^nπ_i^n=1 andπ_i^m-π^n_i=∑_j=m+1^nπ_i^mπ_j^n for 0≤ i≤ m≤ n.We note that the sequence generated by (<ref>) corresponds to y_n=Tx_n+e_n+1, from whichwe deduce the following recursive bound. Let (x_n)_n∈ be generated by (<ref>). Assume (_0) and let ϵ_n≥e_n.For each n∈ define inductively w_m,n for-1≤ m ≤ n by settingw_-1,n=κ andw_m,n=∑_i=0^m ∑_j=m+1^n π_i^mπ_j^n(w_i-1,j-1 + ϵ_i + ϵ_j)m=0,…,n. Then x_m-x_n≤ w_m,n for all 0≤ m≤ n. The proof is by induction on n. The base case n=0 being trivial, let us suppose that x_i-x_j≤ w_i,j holds for all i,j with 0≤ i ≤ j<n. Applying Lemma <ref> with y_n=Tx_n+e_n+1and setting by convention Tx_-1=y_-1=x_0 and e_0=0, we get theinequality x_m-x_n≤∑_i=0^m∑_j=m+1^nπ^m_iπ^n_j(Tx_i-1-Tx_j-1+e_i+e_j).The terms with i=0 can be bounded as Tx_-1-Tx_j-1=x_0-Tx_j-1≤κ = w_-1,j-1, while for the remaining terms the nonexpansivity of T and the induction hypothesis give Tx_i-1-Tx_j-1≤x_i-1-x_j-1≤ w_i-1,j-1. Hence x_m-x_n≤∑_i=0^m∑_j=m+1^nπ^m_iπ^n_j(w_i-1,j-1+ϵ_i+ϵ_j)=w_m,n which completes the induction step. §.§ Reduction to a Markov reward processThe main step in the proof of Theorem <ref>relies on a probabilistic reinterpretation of w_m,n in terms of a Markov reward process evolving in ℤ^2. The process is similar to the Markov chain used in <cit.>, except that we add rewards to account for the presence of errors, which requires a finer analysis. Namely, let m<n be positive integers and consider a race between a fox at position n that is trying to catch a hare located at position m. At each integer i∈ the fox jumps over a hurdle to reach the position i-1. The jump succeeds with probability (1-α_i) in which case the process repeats, otherwise the fox falls at i-1 where it gets a reward ϵ_i. The fox catches the hare if it jumps successfully down to m or below. Otherwise, the hare gets a chance to run towards the burrow located at -1 by following the same rules and with the same rewards. The process alternates until either the fox catches the hare, or the hare reaches the burrow. In the latter case the hare gets an additional reward κ.This yields a Markov chain with state space𝒮={(m,n): 0≤ m<n}∪{f,h} with f, h two absorbing states that represent respectively the cases in which the fox or the hare win the race (see Figure <ref>). Specifically, starting from a transient state (m,n) with 0≤ m<n, the process moves with probability π_i^mπ_j^n toa new state of the form (i-1,j-1) with 1≤ i ≤ m<j≤ n,and otherwiseit is absorbed in state f with probability ∑_j=0^mπ_j^n and in state h with probability π_0^m∑_j=m+1^nπ_j^n. When the process visits a transient state (i-1,j-1) the hare gets a reward ϵ_i and the fox gets a reward ϵ_j, which combined yield a total reward R_i-1,j-1=ϵ_i+ϵ_j. If the process reaches state h the hare gets areward κ and the fox gets nothing, whereas at the absorbing state f there is no reward.Then the total expected reward when the process starts atposition (m,n) with 0≤ m≤ n satisfies exactly the recursion (<ref>) with boundary condition w_-1,n=κ corresponding to the reward collected at the absorbing state h. Thisprovides an alternative way to compute w_n,n+1 and allows to establish the following bound.Let w_m,n be defined recursively by (<ref>) with w_-1,n=κ,where ϵ_n≥ 0 and ϵ_0=0.Then, for all n∈ we havew_n,n+1α_n+1≤κ σ(τ_n) + ∑_i=1^n2α_iϵ_i σ(τ_n-τ_i)+ϵ_n+1. Consider the process starting at state (n,n+1). Let R^H_i denote the eventin which the harecollects the reward at the site i-1, so that the total expected rewardof the hare can be expressed asT^H=κ (R_0^H)+∑_i=1^nϵ_i (R^H_i).The event R^H_i occurs iff the process visits a state (i-1,j-1) for some j>i, that is to say, if the hare is not captured before the i-th hurdle and it fails this i-th jump. Let F_i and H_i denote independent Bernoulli variables representing the failure of the jump over the i-th hurdle for the fox and hare respectively,with (F_i=1)=(H_i=1)=α_i.Denoting S_i+1the event that the hare is not captured before the (i+1)-th hurdle we have R_i^H={H_i=1}∩ S_i+1 so that (R^H_i)=α_i (S_i+1). Now, the event S_i+1 can be written asS_i+1={}which translates the fact that thehare is notcaptured provided that the fox falls more often than the hare. For j=n+1 the condition above amounts to F_n+1=1 so that denoting Z_k=F_k-H_k we can writeS_i+1={F_n+1=1}∩{}andwe mayuse<cit.> to get (S_i+1)=α_n+1()≤α_n+1 σ(τ_n-τ_i).From this we obtain (R^H_i)≤α_iα_n+1σ(τ_n-τ_i) and, noting that α_0=1, we getT^H≤α_n+1[κ σ(τ_n)+∑_i=1^nα_iϵ_i σ(τ_n-τ_i)]. A similar argument can be used to bound the total reward collected by the fox. Indeed, denoting R^F_j the event in which the harecollects the reward at the site j-1,the total expected reward of the fox isT^F=∑_j=1^n+1ϵ_j (R^F_j).In this case the event R^F_j corresponds to the fact that the process visits a state (i-1,j-1) for some i∈{1,…,j-1}. This requires that the fox fails the jump of the j-th hurdle, that the hare has not been captured,and that after the fox rests at j-1 the hare falls before reaching the burrow. Ignoring the latter conditionwe getthe inclusionR^F_j⊆{F_j=1}∩{}.For j=n+1 this gives R^F_n+1⊆{F_n+1=1} so that (R^F_n+1)≤α_n+1. Now,for j=1,…,n the condition ∑_k=i^n+1F_k>∑_k=i^nH_kis superflous when i=j as it followsfromthe same condition for i=j+1 and the fact that F_j=1. Hence we have R^F_j⊆{F_j=1}∩ S_j+1 which yields as before the upper bound (R^F_j)≤α_jα_n+1 σ(τ_n-τ_j).From these bounds we getT^F≤α_n+1[∑_j=1^nα_jϵ_j σ(τ_n-τ_j)+ϵ_n+1].Since w_n,n+1=T^H+T^F, combining (<ref>) and (<ref>) we readily get (<ref>). §.§ Proof of Theorem <ref>Using (<ref>) and Corollary <ref> we getx_n-Tx_n≤w_n,n+1/α_n+1+ϵ_n+1which combined with (<ref>) yields(<ref>).It remains to show that x_n-Tx_n→ 0 when τ_n→∞, ϵ_n→ 0, and ∑_k≥ 1α_kϵ_k<∞. Using (<ref>)and considering a fixed m∈, we can use the bound σ(τ_n-τ_i)≤ 1 for the terms i=m+1,…,n to getx_n-Tx_n ≤ κ σ(τ_n) +∑_i=1^m2α_iϵ_i σ(τ_n-τ_i) +∑_i=m+1^n2α_iϵ_i +2ϵ_n+1.Since σ(τ_n-τ_i)→ 0 as n→∞ we obtain lim sup_n→∞x_n-Tx_n≤∑_i=m+1^∞2α_iϵ_i so that the conclusion follows by letting m→∞. §.§ Convergence of the iteratesUnder some additional conditions, the fact that x_n-Tx_n→ 0 impliesthe existence of fixed points and the convergence of the (<ref>) iteration. The arguments are quite standard (see e.g.<cit.>)but for completeness we sketch the proof.We recall that X is said to have Opial's property if for every weakly convergent sequencex_n⇀ x we havelim inf_n→∞x_n-x<lim inf_n→∞x_n-y∀ y≠ x.Let (x_n)_n∈ be generated by (<ref>) and suppose thatx_n-Tx_n→ 0 and ∑_k≥ 1α_ke_k<∞. a) If T(C) is relatively compact then x_n converges strongly to a fixed point of T.b) If X is uniformly convex and x_n remains bounded then (T)≠∅.Moreover, if X satisfies Opial's property then x_n converges weakly to a fixed point of T. From (<ref>) we see that forx∈(T) the sequence x_n-x+∑_k>nα_ke_kdecreases with n and hence it converges. Since the tail ∑_k> nα_ke_k tends to 0 it follows thatthe limit ℓ(x)=lim_n→∞x_n-x is well defined. Now, in case a) we may extract a strongly convergent subsequence T(x_n_k)→x.Since x_n-Tx_n→ 0 we also have x_n_k→x and therefore x∈(T).It follows that lim_n→∞x_n-x=lim_k→∞x_n_k-x=0 so that x_n→ x in the strong sense.In case b) we can extract a weakly convergent subsequence x_n_k⇀ x andsince I-T is demiclosed (see <cit.>) the assumption x_n-Tx_n→ 0implies that x is a fixed point, hence (T)≠∅. Moreover, Opial's property implies that x_n has only one weak cluster point: if x_n'_k⇀ y is anotherweakly convergent subsequence with y≠ x thenℓ(x)=lim inf_k→∞x_n_k-x<lim inf_k→∞x_n_k-y=ℓ(y) ℓ(y)=lim inf_k→∞x_n'_k-y<lim inf_k→∞x_n'_k-x=ℓ(x)which yields a contradiction and therefore x_n⇀ x. 1) The existence of fixed points when T(C) is relatively compact goes back tothe original work of Krasnosel'skii in 1955, whereas for C bounded in a uniformly convex spacethis was proved in 1965 independently by Browder, Göhde, and Kirk.2) Without the summability condition ∑_k≥ 1α_ke_k<∞ the iterates might failto converge as illustrated by the trivial example Tx=x where x_n=∑_k=1^nα_k e_k. §.§ The assumption (_0) Theorem <ref> is based on assumption (_0) which requires simultaneously that i) the iterates remain in C, and ii) Tx_n-x_0≤κ for some constant κ.For a bounded domain C property ii) holds with κ=(C) so that one only needs to check i).This holds automatically for the exact iteration (<ref>) and more generally when the errors e_n+1are such that Tx_n+e_n+1∈ C. Note also that if X is a Hilbert space one could replace T by T̃=T∘ P_C where P_C is the projection onto the closed convex set C so that T̃:X→ C and (_0)holds with κ=(C). When X is not a Hilbert space the projection P_C might not exist and, even if it does, it might fail to be nonexpansive. Moreover, even in a Hilbert setting the exact projection P_C might be difficult to compute exactly. To deal with these cases, in <ref>we will show how (<ref>) can be adapted using approximate projections. When T is defined on the whole space condition i) is trivial and one only has to check ii). The following result describes two simple situations where this holds.Let T:X→ X be a nonexpansive map.a) If T has a bounded range then (<ref>) holds with κ=sup_x∈ XTx-x_0.b) If (T)≠∅ and the sum S=∑_k=1^∞α_ke_k is finite then (<ref>) holds with κ=2 (x_0,(T))+S. Property a) is self evident. In order to prove b) we note that for any givenx∈(T) we havex_n-x = (1-α_n)x_n-1+α_n(Tx_n-1+e_n)-x≤ (1-α_n)x_n-1-x+α_nTx_n-1-Tx+α_ne_n≤ x_n-1-x+α_ne_nfrom which we get inductively x_n-x≤x_0-x+e_k≤x_0-x+Sand then Tx_n-x_0≤Tx_n-x+x-x_0≤x_n-x+x-x_0≤ 2x_0-x+S.The conclusion follows by taking the infimum over x∈(T).From (<ref>) we get x_n∈ B(x_0,κ) soin part b) of the previous Proposition it suffices Tto be defined on a domain C that contains this ball. § RATES OF CONVERGENCE In this section we use the bound (<ref>) to estimate the rate of convergence of the fixed point residuals.Assume (_0). Supposethat ∑_k≥ 1e_k<∞ andthatα_n is bounded away from 0 and 1. Then there exists a constant ν≥ 0 such that x_n-Tx_n≤ν/√(n)+. Moreover, ifφ:[0,∞)→[0,∞) is nondecreasing and μ=∑_k≥ 1φ(k)e_k<∞, thenx_n-Tx_n≤ν/√(n)+2μ/φ(⌊n/2⌋).In particular, if ∑_k≥ 1k^a e_k<∞ for some a≥ 0 then x_n-Tx_n= O(1/n^b) with b=min{1/2,a}. Take β>0 such that α_n(1-α_n)≥β for all n≥ 1 and defineν=(κ+2√(2) ∑_k≥ 1e_k)/√(πβ). Since σ(τ_n-τ_i)≤ 1/√(πβ(n-i)) and α_i≤ 1, the inequality (<ref>) follows directly from(<ref>) by taking m=⌊n/2⌋, while (<ref>) follows from this and the inequality φ(m)∑_k≥ me_k≤∑_k≥ mφ(k)e_k≤μ.The last claim x_n-Tx_n= O(1/n^b) follows from (<ref>)by taking φ(k)=k^a. Note that in (<ref>) the tail μ_m=∑_k≥ mφ(k)e_k tends to 0 so that∑_k≥ me_k=o(1/φ(m)). In particular, for a<1/2 the last claim in the previous resultcan be strengthened to x_n-Tx_n= o(1/n^a).The previous result derives a rate of convergence from a control on the sum ∑_k≥ 1φ(k)e_k<∞.The next Lemma deals with the case where we control the errors e_n≤ϵ_n rather than their sum.Letη=√(1+4/π). If ϵ_n ≤ (1-α_n)f(τ_n) with f : [0,∞) → [0,∞) nonincreasing, then∑_i=1^n2α_iϵ_i σ(τ_n-τ_i)≤η∫_0^τ_nf(s)/√(τ_n-s) dsFor s∈[τ_i-1,τ_i] we have τ_n - s≤τ_n - τ_i-1=τ_n -τ_i+α_i(1-α_i).Since τ_n -τ_i≤1/π σ(τ_n-τ_i)^-2 andα_i(1-α_i)≤1/4≤1/4 σ(τ_n-τ_i)^-2 it follows that τ_n - s≤ () σ(τ_n -τ_i)^-2. This, combined with the monotonicity of f(·), yields 2f(τ_i)σ(τ_n-τ_i)≤ηf(s)/√(τ_n- s) so that (<ref>) follows by integrating over the interval [τ_i-1,τ_i] and then summing for i=1,…,n. Assume (_0). Suppose that τ_n→∞ and e_n= O((1-α_n)/τ_n^a). a) If 1/2≤ a<1 then x_n - Tx_n= O(1/τ_n^a-1/2).b) If a=1 then x_n - Tx_n= O(logτ_n/√(τ_n)).c) If a>1 then x_n - Tx_n= O(1/√(τ_n)). Let us consider the three terms in the bound (<ref>). The first term is of order κ σ(τ_n)= O(1/√(τ_n)) while the third term is 2e_n+1= O(1/τ_n+1^a) so that for a≥ 1/2 it is also O(1/√(τ_n)).To estimate the sum in the middle term we note that e_n≤ (1-α_n) f(τ_n) with f(s)=K/(s+1)^a for some constantK≥ 0. Hence, denoting I_a(t)=∫_0^t1/(s+1)^a√(t-s) ds and using Lemma <ref> we get∑_i=1^n2α_i e_i σ(τ_n-τ_i)≤ηKI_a(τ_n).For a=1 we have I_1(t)=2arcsinh√(t)/√(t+1)= O(log t/√(t)) which yields b). For a≠ 1 we may use the change of variables s=t (1-x) to express I_a(t) using theGauss hypergeometric function _2F_1(a,b;c;z), namelyI_a(t)=∫_0^11/(1-t/t+1 x)^a√(x) dx=.Now, for z∼ 0 we have _2F_1(a,b;c;z)∼ 1, while setting d=a+b-c we have the following identity (see <cit.>) which is valid whend is not an integer and |(z)|<π_2F_1(a,b;c;1-z)=z^-d +Taking z=1/t+1in this identity with b=1/2 and c=3/2, it follows that for t large I_a(t)∼+ From this we deduce both a) and c), except when a≥ 2 is an integer since in this case Γ(1-a) has a pole. However for a≥ 2 the rate e_n= O((1-α_n)/τ_n^a) is stronger than the same condition with a∈ (1,2) so that we still get the conclusionx_n - Tx_n= O(1/√(τ_n)).Assume (<ref>). Suppose thatα_n is bounded away from 0 and 1, and e_n= O(1/n^a). a) If 1/2≤ a<1 then x_n - Tx_n= O(1/n^a-1/2).b) If a=1 then x_n - Tx_n= O(log n/√(n)).c) If a>1 then x_n - Tx_n= O(1/√(n)). Since α_n is far from 0 and 1 we have τ_n= O(n) and the result follows directly from Theorem <ref>. For 1/2≤ a≤ 1 the conditione_n= O(1/n^a)isvery mild and allows for nonsummable errors.However, this only implies a rate for x_n-Tx_n andnotthe convergence of the iterates whichin generalrequires the errors to be summable (see Theorem <ref> and Remark <ref>).Theorem <ref>also gives rates of convergence for vanishing stepsizes of the form α_n=1/n^c with c≤ 1.We record the case α_n=1/n which is often used. Assume (<ref>) and α_n=1/n, and suppose that e_n= O(1/log^a n). a) If 1/2≤ a<1 then x_n - Tx_n= O(1/log^a-1/2n).b) If a=1 then x_n - Tx_n= O(loglog n/√(log n)).c) If a>1 then x_n - Tx_n= O(1/√(log n)). § VARIANTS OF THE (<REF>) ITERATION§.§ Inexact projectionsUp to now we assumed (_0) which requires that the iterates x_n remain in C. This is a nontrivial assumption that has to be checked independently. Alternatively one might use the metric projection P_C:X→ C, namely P_C(x)=min_z∈ Cx-z,and consider theiteration IKM_px_n+1=(1-α_n+1)x_n+α_n+1(T∘ P_Cx_n+e_n+1).As noted in <ref>, if P_C is well defined and nonexpansive, which is the case when X is a Hilbert space, the results in the previous sections apply directly by considering the map T∘ P_C instead of T. However, in more general spaces the projection might not exist and even if it exists it might fail to be nonexpansive.On the other hand, even in a Hilbert setting the projection might be hard to compute. To overcome these difficultiesone may consider to perform an inexact projection by choosing a sequence γ_n≥ 0 andstarting from x_0∈ Citerate as followsIKM_z{x_n+1=(1-α_n+1)x_n+α_n+1(Tz_n+e_n+1)..In general, finding z_n∈ C as above requires a specific algorithm. Simple cases where this can be done are when C is a ball or the positive orthant in an L^p space with 1≤ p≤∞. In these casesthe projection might fail to be nonexpansive and might even be nonunique.Let the sequence (x_n,z_n) be given by (<ref>) with e_n→ 0 and ∑_k≥ 1(α_ke_k+γ_k)<∞. Suppose that∑_k≥ 1α_k(1-α_k)=∞ and x_0-Tz_n≤κ for some κ≥ 0. Thenx_n-z_n→ 0 and z_n-Tz_n→ 0. Let us denote δ_n=d(x_n,C)+ γ_n. Lemma <ref> in Appendix <ref> shows that δ_n tends to 0 so that x_n-z_n≤δ_n→ 0. On the other handz_n-Tz_n≤z_n-x_n+x_n-Tz_n≤δ_n + x_n+1-x_n/α_n+1+e_n+1 so that it remains to show that x_n+1-x_n/α_n+1 tends to 0. We proceed as before by establishing a recursive bound x_m-x_n≤ w_m,n. Taking y_n=Tz_n+e_n+1 with y_-1=x_0 and using Lemma <ref>we getx_m-x_n≤∑_i=0^m∑_j=m+1^nπ_i^mπ_j^my_i-1-y_j-1.Set w_-1,n=κ for all n∈ and denote ϵ_n=e_n+δ_n-1 with ϵ_0=0. The terms with i=0 in the previous sum can be bounded asy_-1-y_j-1≤x_0-Tz_j-1+e_j≤ w_-1,j-1+ϵ_0+ϵ_j.On the other hand, since z_i-z_j≤x_i-x_j +δ_i+δ_j, the nonexpansivity of T implies that for j>i≥ 1y_i-1-y_j-1≤Tz_i-1-Tz_j-1 +e_i+e_j≤x_i-1-x_j-1 +ϵ_i+ϵ_j.Proceeding as in Corollary <ref> we get x_m-x_n≤ w_m,n with w_m,n defined recursively by (<ref>), and then Proposition <ref> yieldsx_n+1-x_n/α_n+1≤w_n,n+1/α_n+1≤κ σ(τ_n)+∑_i=1^n2α_iϵ_i σ(τ_n-τ_i)+ϵ_n+1.Since Lemma <ref> shows that ϵ_n→ 0 and ∑_k≥ 1α_kϵ_k<∞, by arguing as in the proof of Theorem <ref> we deduce that x_n+1-x_n/α_n+1 converges to 0as claimed.Under the same conditions of Theorem <ref> the following holds. a) If T(C) is relatively compact then x_n converges strongly to a fixed point of T.b) If x_n remains boundedand X is uniformly convex with Opial's property, then x_n converges weakly to a fixed point of T.Let ϵ_n=e_n+δ_n-1 as in the previous proof so that ∑_k≥ 1α_kϵ_k<∞.For each x∈(T) a simple computation yields x_n-x≤x_n-1-x+α_nϵ_n so that the sequence x_n-x+∑_k>nα_kϵ_k decreases with n, and thenx_n-x converges. Since z_n-x_n→ 0 it follows that z_n-x converges as well. Then, since z_n-Tz_n→ 0, we may argue as in the proof of Theorem <ref> to get the strong/weak convergence of z_n, and hence the corresponding convergence of x_n.The bound for δ_n in Lemma <ref>, together with (<ref>) and (<ref>), provide an explicit estimate for z_n-Tz_n from which one can study its rate of convergence using similartechniques as in Section <ref>.§.§ Ishikawa iterationIn <cit.> Ishikawa proposed an alternative method to approximate a fixed point ofa nonexpansive T:C→ C. Namely, given two sequencesα_n,β_n∈ (0,1) and starting from x_0∈ C, the Ishikawa process generates a sequenceby the following two-stage iterationI{y_n =(1-β_n+1)x_n+β_n+1Tx_nx_n+1 =(1-α_n+1)x_n+α_n+1Ty_n.In this subsection we assume that C is bounded and we denote κ=(C). Let the sequence (x_n) be given by the iteration (<ref>) with β_n→ 0 and ∑_k≥ 1α_kβ_k<∞, and assumethat ∑_k≥ 1α_k(1-α_k)=∞. Then x_n-Tx_n→ 0 and the following estimate holds x_n-Tx_n≤κ[σ(τ_n)+∑_i=1^n α_iβ_i σ(τ_n-τ_i)+2 β_n+1]. We observe that (<ref>) can be written as an (<ref>) iteration with errors given by e_n+1= Ty_n-Tx_n.Since Ty_n∈ C the iterates x_n remain in C while by nonexpansivity we have e_n+1≤y_n-x_n=β_n+1x_n-Tx_n≤κ β_n+1so the result follows directly from Theorem <ref>.Ishikawa proved in <cit.> that if C is a convex compact subset of a Hilbert space X, the iteration (<ref>) converges strongly to a fixed point as soon as 0≤α_n≤β_n≤ 1with β_n→ 0 and ∑_k≥ 1α_kβ_k=∞.Interestingly,Corollary <ref> together with Theorem <ref> impliestheconvergence when ∑_k≥ 1α_kβ_k<∞ which is complementary to Ishikawa's condition. Note also that we do not require α_n≤β_n. On the other hand, Ishikawa's theorem holds for the larger class of Lipschitzian pseudo-contractivemaps, whereas our result is restricted to nonexpansive maps butis valid in more general spaces and it yields the rate of convergence of the fixed-point residual as in<ref>. §.§ Diagonal KM iterationLet T_n:C→ C be a sequence of nonexpansive maps converging uniformly to T so that ρ_n=sup{T_nx-Tx:x∈ C} tends to 0. Starting fromx_0∈ C consider the diagonal iteration dkmx_n+1=(1-α_n+1)x_n+α_n+1T_n+1x_n.Let x_n be a sequence generated by (<ref>) with ρ_n→ 0 and∑_k≥ 1α_kρ_k<∞. Suppose that∑_k≥ 1α_k(1-α_k)=∞ and x_0-Tx_n≤κ for some κ≥ 0. Then x_n-Tx_n→ 0 and the following estimate holds x_n - Tx_n≤κ σ(τ_n)+ ∑_i=1^n2α_i ρ_i σ(τ_n-τ_i) + 2ρ_n+1. Note that (<ref>) corresponds to an (<ref>) iteration with errors given by e_n+1=T_n+1x_n-Tx_n.Since T_nx_n∈ C the iterates remain in C, and e_n≤ρ_n. Hence the result follows again from Theorem <ref>.The diagonal iteration (<ref>) was introduced in <cit.> in order to compute a solution for the split feasibility problem in Hilbert spaces. Weak convergence of (<ref>) was established in <cit.>for uniformly convex spaces with a differentiable norm, under the same assumptions of Corollary <ref>. Our result shows that this also holds for uniformly convex spaces with Opial's property, and moreover it yields rates of convergence for the residualsin the same way as in <ref>. § APPLICATION TO NONAUTONOMOUS EVOLUTION EQUATIONS Let T:X→ X be a nonexpansive map and f:[0,∞)→ X acontinuous function.Let u:[0,∞)→ X be the unique solution of the evolution equationE u'(t)+(I-T)u(t)=f(t), u(0)=x_0.In the autonomous case with f(t)≡ 0, Baillon and Bruck <cit.>used the Krasnosel'skii-Mann iteration to prove that u'(t)=O(1/√(t)),assuming that T:C→ C with C a bounded closed convex domain.In the nonautonomous case u(t) could leave the domain C so we assume that T is defined on the whole space.In order to deal with the unboundedness of the domain, and inspired from Proposition <ref>,we consider a continuous scalar function ϵ(t)≥f(t) and we assume one of the following alternativeconditions [ (_2'); (”_2) ]Under either one of these conditions we have the following analog of Theorem <ref>. Let u(t) be the solution of (<ref>) and assume (_2') or (_2”).Then u'(t)≤κ σ(t)+∫_0^t2 ϵ(s)σ(t-s) ds+ϵ(t).Moreover, if ϵ(t)→ 0 and ∫_0^∞ϵ(s) ds<∞ then u'(t)→ 0 as t→∞. Fix t>0 and set λ^n=t/n. Let us consider the sequence (x_k^n)_k≥ 0 defined by x_0^n=x_0 andx_k+1^n-x_k^n/λ^n=-(I-T)x_k^n+f((k+1)λ^n).It is well known that x_n^n→ u(t) and (x_n+1^n-x_n^n)/λ^n→ u'(t) as n→∞.On the other hand, x_k^n corresponds to the k-th term of an (<ref>) iteration with errors e_k^n=f(kλ^n) and constant stepsizes α_k≡λ^n. We claim that (<ref>) holds with κ defined as in (_2') or (_2”). Indeed, in the case (_2') this follows directly from Proposition <ref> a), whereas in the case (_2”) it follows from Proposition <ref> b) and the estimate∑_k=1^∞λ^ne_k^n≤∑_k=1^∞λ^nϵ(kλ^n)≤∫_0^∞ϵ(s) ds=S.Hence, letting τ^n_k=∑_i=1^kα_i(1-α_i) and invoking Proposition <ref> we getx_n+1^n-x_n^nλ^n≤κ σ(τ^n_n)+2∑_i=1^n ϵ(i) σ(τ^n_n-τ^n_i)+ϵ((n+1)).Since τ_n^n=t(1-t/n)→ t as n→∞ the first term κ σ(τ^n_n) converges to κ σ(t), while for the third term we have ϵ((n+1))→ϵ(t). Also τ_n^n-τ_i^n=(1-t/n)(t-it/n) so that the middle term is a Riemann sum for the function h_n(s)=2ϵ(s)σ((1-t/n)(t-s)). Since h_n(s) converges uniformly for s∈ [0,t] towards h(s)=2ϵ(s)σ(t-s), this Riemann sum converges as n→∞ to the integral ∫_0^t 2ϵ(s)σ(t-s)ds. Thereforeby letting n→∞ in (<ref>) we obtain(<ref>).To prove the last claim u'(t)→ 0 we note that σ(t)→ 0 for t→∞ while ϵ(t)→ 0 by assumption,so that it suffices to prove that ∫_0^t 2ϵ(s)σ(t-s)ds tends to 0 as t→∞. Denoting h_t(s)=2ϵ(s)σ(t-s)1_[0,t](s) this integral is exactly ∫_h_t(s)ds. Now, the definition of σ(·) implies h_t(s)→ 0 pointwise as t→∞, and since h_t(s)≤ 2ϵ(s), the conclusion follows from Lebesgue's dominatedconvergence theorem. Clearly, from (<ref>) we also getu(t)-Tu(t)≤κ σ(t)+∫_0^t2 ϵ(s)σ(t-s) ds+2ϵ(t)so thatu(t)-Tu(t)→ 0 as soon as ϵ(t)→ 0 and ∫_0^∞ϵ(s) ds<∞.As in the discrete setting, from this one can deduce that (T)≠∅ as well as the convergence of u(t) to a fixed point of T.Let u(t) be the solution of(<ref>). Suppose that ∫_0^∞ϵ(s)ds<∞and u(t)-Tu(t)→ 0. a) If T(C) is relatively compact then u(t) converges strongly to a fixed point of T.b) If X is uniformly convex and u(t) remains bounded then (T)≠∅.Moreover, if X satisfies Opial's property then u(t) converges weakly to a fixed point of T.We claim that for all x∈(T) the limit ℓ(x)=lim_t→∞u(t)-x exists.To prove this let θ(t)=1/2u(t)-x^2 and g(t)=√(2θ(t)+1)+∫_t^∞ϵ(s) ds.In order to establish the existence of the limit ℓ(x) it suffices to show that g(t) is decreasing. Let us prove that g'(t)≤ 0, that is to say, d/dt√(2θ(t)+1)≤ϵ(t).We recall that the duality mapping onXisthe subdifferential J(x)=∂ψ (x) of the convex function ψ(x)=1/2·^2.Choosing u^*(t)∈ J(u(t)-x), the subdifferential inequality gives(∀ v∈ X)so that taking v=u(t-h) with h>0 we get⟨ u^*(t),u(t-h)-u(t)⟩≤θ(t-h)-θ(t).Dividing by h and letting h↓ 0 it follows that θ'(t)≤⟨ u^*(t),u'(t)⟩. Then, using the equation (<ref>) and the fact that x∈(T), the nonexpansivity of T givesθ'(t) ≤ ⟨ u^*(t),(I-T)x-(I-T)u(t)+f(t)⟩= ⟨ u^*(t),Tu(t)-Tx⟩-⟨ u^*(t),u(t)-x⟩+⟨ u^*(t),f(t)⟩≤ u^*(t)u(t)-x -⟨ u^*(t),u(t)-x⟩+u^*(t)f(t).Now, from well known properties of the duality mapping we have ⟨ u^*(t),u(t)-x⟩=u(t)-x^2 and u^*(t)=u(t)-x so that θ'(t)≤u^*(t)f(t) =u(t)-xf(t)≤√(2θ(t)+1) ϵ(t)which proves our claim d/dt√(2θ(t)+1)≤ϵ(t). This implies the existence of ℓ(x)=lim_t→∞u(t)-x, from which the rest of the proof followsthe same pattern asthe proof of Theorem <ref>.The estimate (<ref>) can also be used to derive the following continuous time analogs of the rates of convergence in Theorem <ref> and Theorem <ref>. Let u(t) be the unique solution of (<ref>). Assume (_2”) and let ν=(κ+2√(2) S)/√(π). Then, for all t≥ 1 we have u'(t)≤ν√(t)+∫_t/2^∞4 ϵ(s) ds.Moreover, ifφ:[0,∞)→[0,∞) is nondecreasing and μ=∫_0^∞φ(s)ϵ(s) ds<∞, thenu'(t)≤ν√(t)+4μ/φ(t/2).In particular, if ∫_0^∞ s^a ϵ(s) ds<∞ for some a≥ 0, then u'(t)= O(1/t^b) with b=min{1/2,a}. Let us fix t≥ 1 and consider the bound (<ref>). Splitting the integral ∫_0^t2ϵ(s)σ(t-s) ds into [0,t/2] and [t/2,t], and noting that σ(t-s)≤√(2/π t) on the first interval and σ(t-s)≤ 1 on the second, we getu'(t)≤ν/√(π t)+∫_t/2^t2 ϵ(s) ds+ϵ(t).Since ϵ(·) is nonincreasing and t≥ 1, we have ϵ(t)≤∫_t/2^t2 ϵ(s) ds which plugged into (<ref>) yields (<ref>). Now, since φ(·) is nondecreasing, (<ref>) follows directly from (<ref>) using the inequality φ()∫_t/2^∞ϵ(s) ds ≤∫_t/2^∞φ(s)ϵ(s) ds≤μ,while the last claim u'(t)= O(1/t^b) follows from (<ref>)by taking φ(s)=s^a.Let u(t) be the solution of (<ref>) and assume (_2') or (_2”), and ϵ(t)= O(1/t^a) with a≥1/2.a) If 1/2≤ a<1 then u'(t)= O(1/t^a-1/2).b) If a=1 then u'(t)= O(log t/√(t)).c) If a>1 then u'(t)= O(1/√(t)).This follows from (<ref>) and from the analysis of the asymptotics of theintegral I_a(t)=∫_0^t1/(s+1)^a√(t-s) ds established in the proof of Theorem <ref>.§ BOUND FOR APPROXIMATE PROJECTIONSThe goal of this Appendix is to establish the next technical Lemma used in the proof of Theorem <ref>.Let (x_n,z_n) be given by (<ref>), and denote δ_n=d(x_n,C)+ γ_nand ξ_n=e_n+γ_n/α_n with δ_0=ξ_0=0. If ∑_k≥ 1α_k=∞ and ∑_k≥ 1α_kξ_k<∞,then δ_n→ 0 and ∑_k≥ 1α_kδ_k-1<∞. Starting from the identityx_n=(1-α_n)z_n-1+α_nTz_n-1+(1-α_n)(x_n-1-z_n-1)+α_ne_nand since (1-α_n)z_n-1+α_nTz_n-1∈ C, we get(x_n,C)≤(1-α_n)(x_n-1-z_n-1)+α_ne_n≤ (1-α_n)δ_n-1+α_ne_n.It follows that δ_n≤ (1-α_n)δ_n-1+α_nξ_n so that letting ρ_n=∏_j=1^n(1-α_j) with ρ_0=1 we getδ_n/ρ_n≤δ_n-1/ρ_n-1+α_n/ρ_nξ_n.Iterating this inequality we get δ_n/ρ_n≤∑_i=0^nα_i/ρ_iξ_iwhich yields δ_n≤∑_i=0^nα_iξ_i ρ_n/ρ_i.This inequality can be written as δ_n≤∫_f_n dμ with μ the finite measure ondefined by μ({i})=α_iξ_i, andf_n:→ given byf_n(i)=ρ_n/ρ_i for i≤ n and f_n(i)=0 for i>n. Since f_n(i)→ 0 as n→∞ and f_n(i)≤ 1, Lebesgue's dominated convergence theorem implies that ∫_f_ndμ tends to zero so that δ_n→ 0.It remains to show that the sum S=∑_k≥ 1α_kδ_k-1 is finite.Using the previous bound for δ_k-1 and exchanging the order ofsummation we getS≤∑_k=1^∞α_k∑_i=0^k-1α_iξ_i ρ_k-1/ρ_i =∑_i=0^∞α_iξ_i∑_k=i+1^∞.The term q_i+1^k=α_k∏_j=i+1^k-1(1-α_j) in this last sum can be interpreted as a probability. Namely, suppose that at every integer j we toss a coin that falls head with probability α_j. Then, q_i+1^k is the probability that starting at position i+1 the first head occurs exactly at position k. Hence ∑_k=i+1^∞ q_i+1^k=1 and therefore S≤∑_i=0^∞α_iξ_i<∞. ims | http://arxiv.org/abs/1705.09340v2 | {
"authors": [
"Mario Bravo",
"Roberto Cominetti",
"Matías Pavez-Signé"
],
"categories": [
"math.OC",
"47H09, 47H10 (Primary), 65J08, 65K15, 60J10 (Secondary)"
],
"primary_category": "math.OC",
"published": "20170525195617",
"title": "Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces"
} |
G↑↓ [email protected] Key Laboratory of Surface Physics, Institute of Nano-electronics and Quantum Computing, and Department of Physics, Fudan University, Shanghai 200433, ChinaCollaborative Innovation Center of Advanced Microstructures, Nanjing 210093, ChinaI study cross dimensionality of p-orbital atomic fermions loaded in an optical square lattice with repulsive interactions. The cross-dimensionalityemerges when the transverse tunneling of p-orbital fermions is negligible.With renormalization group analysis, the system is found to support two dimensionalcharge, orbital, and spin density wave states with incommensurate wavevectors. Thetransition temperatures of these states are controlled by perturbations near a one dimensional Luttinger liquid fixed point.Considering transverse tunneling, the cross-dimensionality breaks down and the density wave (DW) orders become unstable, and I findan unconventional superconducting state mediated by fluctuation effects. The superconducting gaphas an emergent nodal structure determined by the Fermi momentum, which is tunable by controlling atomic density. Taking an effective description of the superconducting state,it is shown that the nodal structure of Cooper pairing can be extracted from momentum-space radio-frequency spectroscopy in atomic experiments. These results imply that p-orbital fermions could enrich the possibilities of studying correlated physics in optical lattice quantum emulators beyond the single-band Fermi Hubbard model.Cross Dimensionality and Emergent Nodal Superconductivity withp-orbital Atomic Fermions Xiaopeng Li December 30, 2023 =========================================================================================Introduction.—Investigation of strongly correlated physics with ultracold atoms has attracted considerable efforts in the last decade. Bose- and Fermi-Hubbard models have been achieved in experiments by loading ultracold atoms in the lowest band of optical lattices <cit.>. For bosons, the Mott-superfluid quantum phase transitionhas been observed <cit.>. For fermions although cooling to the ground state is more experimentally challenging <cit.>,recent developments <cit.> have mounted to a new milestone with the long-sought anti-ferromagnetic ordered Mott insulator <cit.> finally accomplished through the technique of fermionic quantum microscope <cit.>. Upon doping the system is expected to show d-wave superconductivity as in cuperates. This opens up a new window to explore correlated physics of fermionic atoms previously thought unpractical.Apart from the single-band Fermi Hubbard model, orbital degrees of freedom also play important roles in solid state materials <cit.>. Unconventional superconductivity in iron-based superconductors <cit.>,competing ordersin complex oxides and heterostructures <cit.>,and the chiral p-wave topological state proposed in strontium ruthenates <cit.>all have multi-orbital origins.In optical lattices, ultracold atoms have successfully been put in higher orbitals in search of exotic quantum phases <cit.>. For example, a bosonic analogue of the chiral p-wave state <cit.> and an anomalous phase-twisting condensate have been observed <cit.>.In theory, various aspects of p-orbital fermions, such as orbital orders <cit.>, magnetism <cit.>, and topological phases <cit.> have been largely investigated. It has been found that spontaneous translational symmetry breaking with incommensurate wavevectors is generally favorable in the system of p-orbital fermions. However how the incommensurate DWsinterplay with superconductivity in repulsive p-orbital fermions, as analogous to the emergence of d-wave superconductivity in the doped anti-ferromagnetic s-band Mott insulator, remainsto be understood.In this work, I study two component (referring to atomic hyperfine states) fermionic atoms, e.g., ^40K loaded in p-orbital bands of an optical lattice.The orbital cross dimensionality of this system leads to incommensurate spin, orbital and charge density wave phases. Finite transverse tunneling of p-orbital fermions will destroy the cross dimensionality and weaken the DW orders, and a nodal superconducting (NSC) state is found to emerge when the incommensurate spin density wave (ISDW) is suppressed, which is analogous to the emergence of d-wave superconductivity (SC) in the dopeds-bandMott insulator <cit.>.A crucial difference worth emphasis is that the NSC state in the p-orbital setting appears at generic filling instead ofrestricted to near half filling. The nodal structure of the SC pairing is related to Fermi momentum and isthus tunable via controlling the atom number density.The spectroscopic properties of the NSC state are predicted, and can be tested with momentum-resolved radio-frequency spectroscopy in atomic experiments. I also point out that the theory in this work may also shed light on the superconductivity in certain oxide heterostrucutres, e.g.,LaAlO_3-SrTiO_3, where d_xz and d_yz orbitals on the interface can be modeled as p-orbitals effectively. Model.—A minimal tight binding model on a square latticedescribing p_x and p_y orbital fermions is considered here with theHamiltonian H = H_0 + H_ int, where the tunneling term isH_0 = ∑_r, μ, ν = x,y t_μν [c_μα,r ^† c_μα,r + ê_ν+ h.c.]. Here the annihilation operators c_x, α and c_y, αdescribe the p_x and p_y orbitals with spinpolarization α = ,, ê_x and ê_y are two reciprocal lattice vectors for the square lattice, andthe tunneling matrix is t_μν = t_∥δ_μν - t_⊥ (1-δ_μν ) <cit.>.The lattice spacing is set to be 1 throughout.Assuming spin SU(2) symmetry, the local interaction takes the form H_ int = U ∑_r, ν n_ν ( r ) n_ν( r ) + U' ∑_ r n_x ( r ) n_y ( r) + J ∑_ rS_x (r) · S _y ( r) + J' ∑_ r[ c_x ,r ^† c_x ,r ^† c_y ,r c_y ,r + h.c. ], wheren_να ( r) = c_να,r ^† c_να,r,n_ν ( r) = ∑_α n_να ( r),and S_ν ( r) = ∑_αβ c_να,r ^†σ _αβ c_νβ,r . The local interactionrespects conservation of the fermion parity of each orbital, ∏_ r (-1) ^n_x ( r) and ∏_ r (-)^ n_y ( r).This work focuses on the repulsive interaction with U>0. The low temperature phase diagram is analyzedwith renormalization group (RG) techniques.Due to the fermion parityandSU(2) symmetries <cit.>, the renormalized interaction takesa general form,H_ int = 1/2∫∏_i= 1,2,3d k _i /(2π)^2 [] V( [ k_1 k_2 k_3; ν_1 ν_2 ν_3 ]) C_ν_1 α ^† ( k_1) C_ν_2 β ^† ( k_2) C_ν_3 β ( k_3) C_κ(ν_1, ν_2, ν_3) , α( k_1 +k_2 -k_3) , whereC_να ( k) is the Fourier transform of c_να,r,and κ is a permutation symmetric function defined byκ(ν, ν, ν) = ν and κ(ν, -ν, ν ) =- ν(the `-' sign in front of the orbital index ν means switching between p_x and p_y).The frequency dependence in the effective interactions is neglected for such dependence is expected tobeunimportant <cit.>.The renormalization of effective interactions is given in Supplementary Information.Cross-dimensional limit.—First consider the limit t_⊥→ 0.In the renormalization to the low-energy limit, the coupling J' is strongly suppressed due tothe particle-particle channel, provided that |J'| <U, which holds for cold atoms with contact interactions.I will thus take J'→ 0.In this limit, the system is dynamically cross-dimensional, meaning thateach orbital tunnels only in the direction along itsown elongationwhereas they are coupled via inter-orbital interactions.Mathematically, the cross-dimensionality is characterized by atransverse-sliding-phase (TSP) symmetry,c_x,r→ e^i θ(r_y)c_x,r, c_y,r→e^i θ(r_x)c_y,r .In the absence of inter-orbital interactions, this cross-dimensional system reduces toan ensemble of decoupled one-dimensional systems, each being described by a Luttinger liquid theory. Resulting from the one-dimensionality of Fermi surfaces,perfect nesting wavevectors Q = (± Q, ± Q), with Q = 2 k_f(Fig. <ref>(a)).Thisgivesrise to leading instabilities in DWchannels []V _1 ( k,k') ≡V( [ k k' +Qk'; ν-ν-ν ]), V_2( k,k') ≡V( [k k' + Qk + Q;ν -νν ]) .Divergence of V_2 in the RG flow leads to formation ofSDWorders at low temperature,while divergence of V_2 -2 V_1 leads to charge densitywave (CDW) or orbital density wave (ODW) orders, depending on the sign of this coupling. The low temperature phases are determined by the effective interactions near the Fermi surface.The momentum is rewritten in terms of components parallel (e_∥) and perpendicular(e _⊥) to the Fermi surface as k = ke_∥ + χ (k_f + l)e_⊥, with the chirality χ = ± index thetwo Fermi surfaces for each orbital.Note thate_∥ and e_⊥ should be defined in an opposite way fortwo orbitals—e_∥ = ê_y, e_⊥ = ê_x for p_x ande_∥ = ê_x, e_⊥ = ê_y for p_y.Each single-particle mode is now labeled by three indices—parallel momentum k, chirality χ, and orbitalν.The dispersion near the Fermi surface readsϵ_ν ( k) = v_f l +O (l^2),with v_f the Fermi velocity. The interactions V_1( k, k^') and V_2( k, k^') projected onto the Fermi surfacetake a more explicit formg_⊥, 1(ν; χ, k , χ',k' )=[]Γ( [ k k' -χ Qk'k-χ' Q; χ- χ'χ'-χ; ν-ν-ν ν ]),g_⊥, 2(ν; χ, k, χ', k') =[]Γ( [k k' - χ Qk-χ'Q k';χ-χ' -χ χ';ν -νν -ν ]),where Γ (m_1,m _2,m_3,m_4), withm_j a collective column of indices( k_j,χ_j,ν_j )^T,describes scattering from modes m_3 and m_4 to m_1 and m_2.Pictorial illustration of these couplings is given in Supplementary Information.Due to the two dimensional nature of these inter-orbital couplings,the parallel momentum is not conserved.The TSPsymmetry implies g_⊥, j=1,2 (ν; χ, k, χ', k') = g_⊥, j (ν; χ, k + p, χ', k' + p').The inter-orbital couplings g_⊥, j thus have no k (or k') dependence. Further considering point group D_4 symmetry, g _⊥, j does not depend on ν, χ or χ'.Under renormalization, the inter-orbital couplings g_⊥, jare intertwined with intra-orbital ones g_∥, 1 ( ν, χ, k, k', q)= []Γ( [k k'+q k'k+q;χ -χχ -χ;νννν ]) , g_∥, 2 (ν, χ, k, k' , q)= []Γ( [k k'+qk+q k';χ -χ -χχ;νννν ]).Here, the parallel momentum is conserved, which is different from inter-orbital couplings.The other difference comes from the consequence of the TSP symmetry. For the intra-orbitalcouplings, TSP symmetry implies g_∥, j (ν, χ, k, k', q) = g_∥, j (ν, χ, k+p, k'+p, q).These couplings could therefore have more complicated momentum dependence.On the other hand, starting with on-site interactions, the momentum dependence ing_∥, j generated in the renormalization is mainly from intertwinedscattering with inter-orbital couplings, which causes the leading momentum dependence of g_∥, j (ν, χ, k, k', q)on the exchangeterm q, i.e.,g_ ∥, j (ν, χ, k, k', q) ≈ g_∥, j (q). This is justified by a complete treatment of momentum dependence.To explicitly extract the CDW or ODW channel, I introduce g̃_⊥, 1 = 2 g_⊥, 1 - g_⊥, 2,g̃_∥, 1 (Q) = 2 g_∥, 1 (Q) - g_∥, 2 (Q).At tree level, the g-ology couplings are related to the lattice model as g_⊥, 1 = U'-J, g_⊥, 2= -2J, g_∥, j = U.From the renormalization at one-loop level (see Supplementary Information),the renormalization group (RG) equations of the g-ology couplingsare obtained to bed g̃_⊥,1/ds = 1/π v_f{ - g̃_∥, 1 (Q)g̃_⊥, 1} d g_⊥, 2/ds = 1/π v_f{ g_⊥,2 g_∥, 2 (Q) } d g̃_∥, 1 (Q) /ds = 1/2 π v_f{-g̃ _⊥,1 ^ 2-g̃_∥, 1^2 (Q).. + g_∥,1 ^2 (q)+ g_∥, 2 ^2 (q) -4 g_∥, 1 (q)g_∥,2(q)} d g_∥,2 (Q)/ds = 1/2π v_f{ g_⊥,2 ^2+ g_∥,2 ^ 2(Q) -g_∥,2 ^2 ( q )- g_∥,1 ^2 (q)} .A shorthand notation is adopted f(q)= ∫d q/2π f(q). The RG low of g_∥, j (q) for the momentum q away from Q is obtained asdg_∥, 1(q )/ds= 1/π v_f{ -g_∥,1 ^2(q)+ g_∥,1 (q) g_∥,2 (q). .-g_∥, 1 (p) g_∥, 2(p)},d g_∥, 2 (q)/ds =1/2π v_f { g_∥,2 ^2(q)- g_∥,2 ^2 (p) - g_∥,1^2 (p) }. The RG flow ofintra-orbital couplings with q ≠ Q isnot affected by inter-orbital couplings.(More rigorously, q≠ Q means |q-Q| > Λ, with Λ≪ 2π for weak interactions.) Since the strong momentum dependence in g_∥, j near q ≈ Q is restricted to very limited phase space, I approximate g_∥, j (q)g_∥, j' (q) by[ g_∥, j (q) ] [g_∥, j' (q) ], and the deviation is quantified byε_jj' = Δ g_∥, j (q) Δ g_∥, j' (q), withΔ g_∥, j (q) = g_∥, j (q) -g_∥, j. The solution for g_∥, j is obtained to be[g_∥, 1(s ) ]^-1= [ g _∥, 1 (0)]^-1+ s/π v_f +O (ε), g_∥, 2 (s)=g _∥, 2(0) + 1/2g_∥, 1 (s)- 1/2g_∥, 1 (0) . With repulsive interaction, the intra-orbital coupling g_∥, 1 will renormalize to 0,and g_∥, 2 will renormalizeto a constant becauseg_∥,c≡1/2g_∥, 1 (0) - g_∥, 2 (0)is invariant in the RG flow.This leads to a fixed line at (g_⊥, 1 = g_⊥,2 =0, g_∥, 1 = 0,g_∥,2= -g_∥,c), whichcorresponds to a critical Luttinger liquid phase.Around this fixed line, the RG flow is determined byd/dsg̃_⊥,1∝ -g_∥,cg̃_⊥,1,d/ds g_⊥, 2∝ -g_∥,cg_⊥, 2,andd/dsΔ g_∥, j (q) ∝ - g_∥,cΔ g_∥, j (q).The two cases with g_∥, c <0 and g_∥, c >0 are different.On-site repulsion leads a negativeg_∥, c, for which the Luttinger liquid phase is unstable towards DW orders. Which DW is favorable at low temperature is determined by the comparison amongthese couplings, g̃_⊥,1, g_⊥, 2, and Δ g_∥, j (q). For p-orbital fermions, the inter-orbital interactions g̃_⊥,1 and g_⊥,2 are more likely to be dominant, sincethey are related to local interactions.A dominant coupling g_⊥,2 would support (2k_f, 2k_f)-SDW orders, in whichspin polarizations in both p_x and p_y fermions exhibit finite ground-state expectation values, ⟨ S_x ⟩ and ⟨ S_y ⟩,andoscillate with a wavevector (2 k_f , 2k_f). For g_⊥,2<0(g_⊥,2 >0), the spin polarizations ⟨ S_x ⟩ and ⟨ S_y ⟩areantiparallel (parallel).Analyzing the RG flow (see Supplementary Information) thescaling of transition temperatures, T_ c,1 for ODW/CDW andT_ c,2 for SDW is obtained to be, T_ c,1∝| g̃_⊥,1|^π v_f/g_∥,c,T_ c,2∝| g_⊥, 2|^ π v_f/g_∥,c. To complete the theory, I also discuss the possibilities with momentum dependent intra-orbital couplings to be dominant.If Δ g_∥, 2(q) is dominantand peaked at some momentum q_0, a (q_0, 2k_f)-SDW order is favorable(⟨ S_x⟩ and⟨ S_y⟩would oscillate with wavevectors (2k_f, q_0)and (q_0, 2k_f), respectively). For example, with nearest neighbor spin exchange interactions,J_nn∑_ r[ S⃗_x ( r) ·S⃗_x ( r+ê_y) + x↔ y],Δ g_∥, 2 (q) could bepeaked at q_0 = π and the ground state then has a (π, 2k_f)-SDW order, which is consistent withprevious Bosonization analysis of coupled 1D chains <cit.>.Otherwise if Δ g_∥, 1 (q_0) is dominant, a (q_0, 2k_f)-CDW/ODW order is favorable.With g_∥,c>0, the Luttinger liquid fixed line is stable against infinitesimal perturbations in DW channels discussed above, and a renormalized one dimensional Luttinger liquid state emerges in the bare two-dimensional system.But when the magnitude of the inter-orbital coupling g_⊥, 2 is larger than some critical value,|g_⊥,2| >g_ ^c,RG flow would escape the attractionof the stable fixed point.Finite g_⊥,2 thus lead to phase transitions towards two dimensional SDW states (Fig. <ref>(b)).One remark is that the renormalized one dimensional state is only stable in the strict cross-dimensional limit.Including J' would generate an effective coupling in the SC channel. The system then develops superconductivity at very low temperature. *Transverse tunneling and a nodal superconducting state.With finite transverse tunneling t_⊥≠ 0, the perfect Fermi surface nesting no longer holds and the g-ology RG description breaks down, nonetheless the SDW states could be stabilized by finite repulsive interactions. A functional RG approach is adopted to approximately solve the full renormalization equation (see Supplementary Information) by a patching scheme <cit.>, in which the interaction (Eq. (<ref>))is approximated by its projection to the Fermi surfaces. To determine the transition temperature more precisely, a temperature renormalizationscheme is implemented, where the renormalized interactionsexplicitly represent temperature dependence ofeffective scatterings of low-energy modes <cit.>.In the calculation, the SDW and SC channels—V_2 ( k,k') and[]V_ SC (ν,k; ν',k')=V( [k- k - k';νν ν' ]).The strengths of instabilities are characterized by their eigenvalues λ_ SDW andλ_ SC of largest magnitude. The low temperature phaseis determined by which channel is the most divergent.The transition temperature is the point where the most dominant channel diverges (in numerics,it is determined by which eigenvalue |λ| reaches 20 t_∥ first.)The phase diagram is shown in Fig. <ref>. The momentum dependence of the SC pairing structure function,⟨ C_ν↑ (k) C_ν↓ (- k)⟩,is related tothe eigenvector ψ_ν ( k ) of V_ SC.Rewriting the momentum on the Fermi surface in terms of parallel momentum (k) and chirality (χ), the paring function follows ψ_ν (χ, k) ∝⟨ C_ν↑ (χ, k ) C_ν↓ (-χ, -k) ⟩ .The relative sign between ψ_x and ψ_y is determined by the sign of J'.The pairing structure function is found to exhibit nontrivial momentum dependence as shown inFig. <ref>(c).At each Fermi surface,there are four nodal points in ψ_ν (χ, k), approximately located atk_ node= ± k_f, π± k_f,with k_f the Fermi momentum defined at the limit of t_⊥→ 0. The emergence of this nodal structure could be understood by considering the overlap channels betweenV_ SC and V_2,V_ SC (ν,k; ν,k) = V_2 ( k,k ),for those special momenta k as described in Eq. (<ref>).In the SC state, the system still has strong tendency towards formingSDWs with the wavevector Q≈ (2 k_f, 2k_f). This means there are large and positive,but otherwisenot divergent, couplings in V_2.Due to the overlap between the SDW and SC channels, these special points form “hot spots” forSC pairing, which give rise to nodal pointsin the ground state eigenvector of V_ SC (ν, k; ν'k').In the aspect of spin symmetry, this SC state is a singlet. *Experimental signatures of the nodal superconducting state.To observe the NSC state in atomic experiments, spectroscopic properties are studied through an effective description. Deep in the NSC phase, the low energy physics is expected to be described by an effective Bogoliubov de-Gennes (BdG) Hamiltonian, H_ BdG= ∑_ν k{ϵ_ν( k)C_να ^† ( k) C_να ( k) . + . [Δ ^* ( k_-ν ) C_ν↑ (k) C_ν↓ (- k)+ H.c.] }. Here the dependence of Cooper pairing on the momentum perpendicular to the Fermi surface is neglected for simplicity. Owing to the form of the Cooper pair observed in Fig. <ref>, I use an ansatz Δ (k) = Δ_1 cos (k) + Δ _2 cos (2k). Keeping higher harmonics is more precise but does not change the physics to be presented below. Taking the BdG Hamiltonian, the momentum resolved spectra function A( k, ω) is calculated through Green function methods <cit.>. The momentum dependence of A( k, ω) at zero energy is shown in Fig. <ref> (a),where the peaks of A( k, 0) reveal nodes of the SC pairing. The `V'-shape feature of the density of statesρ(ω) = ∑_ k A( k, ω) shown in Fig. <ref>(b) is a signature ofthe gapless Bogoliubov quasi-particles near the nodal points of SC gap. These properties can be tested in momentum-space resolved radio-frequency spectroscopy <cit.>.*Conclusion. I have derived cross-dimensional g-ology RG flow for p-orbital fermions. Charge, orbital, spin density waves andtheir transition temperatures are describedwithin our g-ology study at the limit of vanishing transverse tunneling.Atfinite transverse tunneling, DW orders are found to be suppressed, giving rise to an unconventional nodal superconductingstate.*Acknowledgement. The author thanks helpful discussions with W. Vincent Liu and Bo Liu. The work is supported by the Start-Up Fund of Fudan University.apsrev4-1§SUPPLEMENTARY INFORMATION § RENORMALIZATION EQUATION The renormalization of couplings upon integrating out high-energy modes is given in this section.is described by a flow equation <cit.> Λd/d Λ[]V( [ k_1 k_2 k_3; ν_1 ν_2 ν_3 ]) = ∑Π̇_ pp(ϵ_μ ( q) , ϵ_μ_ pp ( q_ pp) ) []V( [ k_1 k_2 q; ν_1 ν_2 μ ]) V( [ q_ pp q k_3; μ_ pp μ ν_3 ])- ∑ Π̇ _ ph( ϵ_μ ( q), ϵ_μ_ ph ( q_ ph ) ) { -2 []V( [ q k_2 k_3; μ ν_2 ν_3 ]) V( [ k_1 q_ ph q; ν_1 μ_ ph μ ]). +[]. V ( [ q k_2 k_3; μ ν_2 ν_3 ]) V( [ q_ ph k_1 q; μ_ ph ν_1 μ ]) + V( [ k_2 q k_3; ν_2 μ ν_3 ]) V( [ k_1 q_ ph q; ν_1 μ_ ph μ ]) }-∑ Π̇_ ph(ϵ_μ ( q), ϵ_μ_ ph '( q_ ph ' ))[]V( [ k_1 q k_3; ν_1 μ ν_3 ]) V( [ q_ ph ' k_2 q; μ_ ph ' ν_2 μ ]) . Here∑ means ∑_μ∫d^2q/(2π)^2,q_ pp =k_1 +k_2 -q,q_ ph =q +k_2 -k_3,q_ ph ' =q + k_1 -k_3,μ_ pp = κ ( ν_1, ν_2, μ),μ_ ph = κ (ν_2, ν_3, μ),μ_ ph' = κ (ν_1, ν_3, μ).The particle-particle/hole functions are Π̇_ ph (pp)(ϵ, ϵ')= ΛΘ (∓ϵϵ') /|ϵ| + |ϵ'|{Θ ( |ϵ|-Λ ) δ (|ϵ'|-Λ) + ϵ→ϵ'}, with Θ(x) the heavyside step function.§ TRANSITION TEMPERATURES OF DENSITY WAVE STATES The transition temperatures of DW states are estimated to beT_c ≈Λ_0 e^-s^*,with s^* the point where the corresponding coupling diverges. In this section, the bare inter-orbital interactions are assumed to be much weaker compared to intra-orbital ones. To calculate the transition temperature for SDW, the RG flow of g_⊥, 2 is derived assuming the intra-orbital couplings are at the Luttinger liquid fixed point, i.e.,g_∥,2(q) = -g_∥,c, g_∥,1 = 0. (if the bare couplings are not at the fixed point, the bare couplings can simply be replaced byrenormalized ones).Then the RG equation reads,d/ds[ g_∥,2 (Q) ± g_⊥,2]= 1/2π v_f{[ g_∥,2 (Q) ± g_⊥,2] - g_∥, c ^2 },from which it followsg_∥,2 (Q; s) + | g_⊥,2(s) |= g_∥,cα + e^ s g_∥, c /(π v_f) /α - e^ s g_∥, c /(π v_f) ,with α defined byα + 1/α-1 = [ g_∥,2 (Q; 0) + | g_⊥,2 (0) | ] /g_∥,c.For |g_⊥,2| /g_∥,c≪ 1,α^-1 = 1/2g_⊥,2/g_∥,c+O( ( g_⊥,2/g_∥,c)^2 ).The scale s^* is then obtained to bee^-s^*≈( 1/2|g_⊥,2| /g_∥,c)^ π v_f/g_∥,c ,from which the scaling of transition temperature for SDW follows,T_ c,2∝| g_⊥, 2|^ π v_f/g_∥,c.With similar analysis for RG flow of g̃_⊥,1,the transition temperature for CDW/ODW is obtained as, T_ c,1∝| g̃_⊥, 1|^ π v_f/g_∥,c. | http://arxiv.org/abs/1705.09686v1 | {
"authors": [
"Xiaopeng Li"
],
"categories": [
"cond-mat.quant-gas",
"cond-mat.str-el",
"quant-ph"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170526191728",
"title": "Cross Dimensionality and Emergent Nodal Superconductivity with $p$-orbital Atomic Fermions"
} |
[email protected] Department of Physics, University of Portland, Portland, Oregon 97203, [email protected] Departamento de Física, Universidade Federal FluminenseAv. Litorânea s/n,Niterói, 24210-340, RJ, [email protected] Departamento de Física, Universidade Federal FluminenseAv. Litorânea s/n,Niterói, 24210-340, RJ, Brazil In this work we analyze the ground-state properties of the s=1/2 one-dimensional ANNNImodel in a transverse fieldusing the quantum fidelity approach.We numerically determined the fidelity susceptibility as a function of the transverse field B_x and the strength of the next-nearest-neighbor interaction J_2, forsystems of up to 24 spins.We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic,floating,2,2 phases, and we predict an infinite number of modulated phases in the thermodynamic limit (L →∞). Paramagnetism only occurs for larger magnetic fields. The transition lines separating the modulated phases seem to be of second-order,whereas the line between the floatingand the 2,2 phases is possibly of first-order. 75.10.Pq,75.10.JmQuantum fidelity approach to the ground state properties ofthe 1D ANNNI model in a transverse field J. Florencio December 30, 2023 ===================================================================================================== §INTRODUCTION At very low temperatures, quantum fluctuations play an important rolein the characterization of the ground-state properties of quantumsystems <cit.>.These fluctuations are induced by varying the relative strengthof competing interactions among the constituents of the systemor by changing the strength of the applied fields. When large enough,quantum fluctuations dramaticallychange the nature of a given ground-state.A quantum phase transition may occur, thereby creating a boundarybetween distinct ground-states. The one-dimensional axial next-nearest neighbor Ising (1D ANNNI) model in a transverse field is one of the simplest models in which competing interactionslead to modulated magnetic orders, frustration, commensurate-incommensuratetransitions, etc. These features are known to appear in the ground-state of themodel in the one-dimensional case.Frustration in the 1D ANNNI model arises from the competition betweennearest-neighbor interactions which favor ferromagnetic alignment of neighboring spins, while an interaction with opposite sign betweenthe next-nearest-neighbors fosters antiferromagnetism. At T = 0, the presence of a transverse magnetic field gives riseto quantum fluctuations that play an analogous role as that of temperaturein thermal magnetic systems that are responsible for triggeringphase transitions. In one dimension, the ANNNI model in a transverse field is actually an extension of the transverse Ising model. The latter consists of Ising spins with nearest-neighborinteractions in the presence of a magnetic field in the transverse direction.The transverse Ising model was initially used to explain the order-disordertransitions observed in KDP ferroelectrics <cit.>.An experimental realization of that model in real magneticsystems was observed in LiHoF_4 in an external field <cit.>.An exact solution to the model in one dimension was subsequentlyfound by Pfeuty by mapping the set of the original spin operators onto anew set of noninteracting spinless Fermi operators <cit.>.Recently, a degenerate Bose gas of rubidium confined in a tilted optical lattice was used to simulate a chain of interacting Ising spins in the presence of both transverse and longitudinal fields <cit.>.It has also been proven that the ground-state properties of the d-dimensional Ising model with a transverse field, is equivalent to the (d + 1)-dimensional Ising model without a magnetic field at finite temperatures <cit.>.In the case of the 1D ANNNI model in a transverse field at T=0 andthe 2D ANNNI model (without transverse field) at finite T, such equivalencemay only exist in the limit of very strong transverse field and in the weak-couplinglimit of the NN- and NNN-interactions of the 1D model <cit.>. There is no guarantee that the ground-state phase diagramsof those models bear any resemblances to each other. Therefore we shall not compare the phase diagrams of these two models in this work. The transverse 1D ANNNI model has been the subject of great interest <cit.>, in part due to the number of quantum phases with unusual and intriguing features it displays.Several analytical and numerical methods have been employed to establish its phase diagram.Among those studies, there are analysis using quantum Monte Carlo <cit.>, exact diagonalization of small lattice systems <cit.>,interface approach <cit.>,scaling behavior of the energy gap <cit.>, bosonization and renormalization groupsmethods <cit.>,density matrix renormalizationgroup <cit.>, perturbation theory <cit.>,and matrix product states <cit.>.The phase diagrams from those works do not necessarily agree with each other.In the following we discuss the common features as well as some of the differences between them. In most of the studies, there is ferromagnetism for J_2 < 0.5 and 2,2 antiphase for J_2 > 0.5. The transition lines usually end at the multicritical point (J_2,B_x)=(0.5,0.0). The phase diagram of Dutta and Sen shows antiferromagnetism instead of the2,2 antiphase for J_2 > 0.5 <cit.>.That is a rather surprising result not to show the antiphase, since even in the classical case, B_x = 0, that antiphase is energetically favorable. Some authors obtain diagrams with 5 phases, namely, ferromagnetic, paramagnetic, modulated paramagnetic,floating, and antiphase.Such are the diagrams of Arizmendi et al. <cit.>, Sen et al. <cit.>,and Beccaria et al. <cit.>.On the other hand, Rieger and Uimin <cit.>, Chandra and Dasgupta <cit.>, andNagy <cit.>present diagrams with 4 phases, ferromagnetic, paramagnetic, floating, and antiphase. In Refs.<cit.> and <cit.> the boundary lines meet at the multicritical point,whereas in Ref. <cit.> the paramagnetic phase is restricted to suficiently high B_x, thus its boundary lines do not reach the multicritical point. In the studies by Sen <cit.> and Guimarães et al. <cit.>, one findsdiagrams with 3 phases only, ferromagnetic, paramagnetic, and antiphase, where their transition lines end at the multicritical point. The phase diagram of Dutta and Sen <cit.> displays ferromagnetism, a spin-flop phase, a floating phase, and an antiferromagnetic phase.In that work, the floating phase lies between the antiferromagnetic and the spin-flop phases. Such spin-flop and antiferromagneticphases do not appear in any of the other phase diagramsin the literature. In addition, their transition lines do not end at the multicritical point. As one can see, there is not a consensus on the ground-state phase diagram of the model.The number,nature, or locationof the phases usually vary from one work to another. In any case, all the studies in the literature report on a finite number of phases.As we shall see below, our phase diagram agrees with some of the worksin the literaturewith regard to the existence of ferromagnetic, floating, and theantiphase.However, our numerical results suggest that there are an infinite number of modulated phases between the ferromagnetic and the floating phase. Such scenario is similarto the one found in the the work of Fisher and Selke <cit.>on the low-temperature phase diagram of an Ising model with competing interactions. In that study the phasediagram shows an infinite number of commensurate phases.While the identification of the usual thermal phasetransitions relies mostly on the behavior of an orderparameter or on an appropriate correlationfunction, quantum phase transitions can also be characterized solely by the properties of the ground-stateeigenvectors of the system on each side of the boundarybetween two competing quantum mechanical states. We usefidelity susceptibility to determine the phase boundary lines, as well as a direct inspection of the eigenvectors to understand the nature of the phases. In our work, paragnetism only occurs at high fields B_x, hence it does not appear in our phase diagram, which covers the low field region only. In addition, our numerical analysispoints to the the existenceof a region of finite width for the floating phase. §THE MODELThe one-dimensional ANNNI model in the presence of atransverse magnetic field is defined asH = -J_1∑_iσ^z_iσ^z_i+1 + J_2∑_iσ^z_iσ^z_i+2 - B_x ∑_iσ^x_i.The system consists of L spins, with s=1/2,where σ^α_i (α = x,y,z) is the α-component of a Pauli operator located at site i in a chain where periodic boundary conditions are imposed. We considered ferromagnetic nearest-neighbor Ising coupling J_1 > 0 and antiferromagnetic next-nearest-neighbor interaction J_2 > 0.B_x isthe strength of a transverse applied magnetic field along the x-direction. We set J_1 = 1 as the unit of energy.At T = 0 and in the absence of an externalmagnetic field (B_x=0), the model is trivially solvable andpresents several ordered phases.For J_2 < 0.5, the ground-state ordering is ferromagnetic, and for J_2 > 0.5, the ordering changes to a periodic configuration with two up-spins followed by two down-spins which is termed the 2,2-phase, or antiphase.In this work we have used the notation p,q to represent a periodic phase, with p up-spins followed by q down-spins.AtJ_2 = 0.5, the model has a multiphase point where theground-state is infinitely degenerateand a large numberof p,q-phases are present, as well as otherspin configurations. The number of phases increasesexponentially with the size of the system <cit.>.On the other hand, for a non-zero external magnetic field andJ_2 = 0, the model reduces to the Ising modelin a transverse field, which was solved exactly by Pfeuty <cit.>.The transverse magnetic field induces quantum fluctuations that eventually drivethe system through a quantum phase transition.Its ground-state undergoes asecond-order quantum phasetransition at B_x = 1, separating ferromagnetic fromparamagnetic phases. In the 1D transverse ANNNI model, next-nearest-neighborinteractions introduces frustration to the magnetic order.A much richer variety of phases becomespossible when one varies the strength of the interactions among the spins or their couplings to the magnetic field. Given that insofar there is not a definite answer to the problem of the ground-state properties of the transverse ANNNI model, where different approaches yield distinct phase diagrams, we use quantum fidelity method together with direct inspection of the ground-state eigenvector to shed some light into the problem. We believe our approach is suitablebecause both the fidelity susceptibilityand ground-state eigenvector provide detailed direct information aboutboundary and nature of the ground-state phases.We investigate how the phase diagram evolves as we consider larger and larger lattices.Our results are consistent with some known results, such as the classical multicritical point, the Pfeuty quantum transitionpoint, and the exact Peschel-Emery line which runs between those two points in the phase diagram <cit.>. From our results for finite sized systems we can inferwhich phases will be present in the thermodynamic limit.§THE FIDELITY METHOD Suppose the Hamiltonian of the system depends ona parameter λ, which drives the system througha quantum phase transition at a critical valueλ= λ_c.Quantum fidelity is defined as the absolute valueof the overlap between neighboringground-sates of the system <cit.>,F(λ,δ) = |ψ(λ - δ)|ψ(λ + δ)|.Here ψ is the quantum non-degenerateground-state eigenvector that is evaluated at somevalue of λ, shifted by an arbitrary smallquantity δ around it.In addition to the dependence on λ and δ,the quantum fidelity is also a function of the size of the system.The basic idea behind the fidelity approach is that theoverlap of the ground-state for values of theparameter λ between the two sidesof a quantum transition, exhibits a considerabledrop due to the distinct nature of the ground stateson each side of the phase boundary.Quantum fidelity has been used in quantum informationtheory <cit.> as well as in condensed matterphysics, in particular in the study of topologicalphases <cit.>. For a fixed value L and in the limit of very small δ,the quantum fidelity may be written asa Taylor expansion, F(λ,δ) = 1 - χ(λ)δ^2 +O(δ^4),where the ground-state eigenvector is normalized to unity.The quantity χ(λ) is called the fidelity susceptibilityand will reach a maximum at the boundary between adjacentquantum phases.We used the fidelity susceptibility to find thephase boundary lines the (J_2,B_x)-plane andcompare them with the results obtained by othermethods.To determine the ground-state energy and eigenvectoras a function of λ,we employed both Lanczos and the conjugate-gradient methods. The latter is known to be a fast and reliable computational algorithm.It has been used in statistical physics, especiallyin the context of Hamiltonian models and of transfer-matrixtechniques <cit.>. Both methods give the sameground-state eigenvalues and eigenstates within a given precision.Depending on the size of the system,the ground-state energy is calculatedwith precision between 10^-10 and 10^-12.We have used δ = 0.001 in all calculationsinvolving thefidelity susceptibility. For the location of each point at the criticalboundary, we calculated the maximum valueof the fidelity susceptibility as defined by Eq. <ref>.In order to identify the nature of the quantum phase,we examined howthe ground-stateeigenvectors are written in terms of a completeset of appropriate basis vectors. To find the eigenstates and correspondingeigenvalues of the system we needed to choosea complete set of orthogonal basis vectors andwrite the Hamiltonian in matrix form using this basis set. The eigenvalues and eigenstates are found by exact numerical diagonalization.A convenient basis consists of the tensor productof L eigenstates of the z-component of thelocal spin-operator acting on each site.Denoting the eigenstates by |s>_i,wheres=1,is the eigenstate label of theoperator σ^z_i for an up-spinand s = 0 for the a down-spin at site i.A generic basis eigenstate for the full systemwith L spins can be written as |n> = ∏_i^L|s>_i, where n labels the basis state and hasthe values n = 0, 1,..., N-1, and where N = 2^Lrepresents the dimensionof the Hilbert space.The basis index n, if written in binary notation,can also be used to specify the configurationof the spins forming that basis.That is, when n is written in binary notation,the position and value of a bit will indicatewhether the spin at that position (site) is up (1)or down (0).For instance, for a chain of 12 spinsthe state |1755> in binary notation is written as|011011011011>, which representsa periodic configuration with one down-spin (0) followed by twoup-spins (11).In this notation, an arbitrary eigenstate of the Hamiltonianmay be cast as |ϕ_α> = ∑_n=0^N-1a_α(n)|n>,whereα = 0, ...,N-1, labels the quantum states,withα = 0 assigned to the ground-state. Since the matrix Hamiltonianisreal and symmetric,the coefficients a_α(n) are real.As a result, the quantum state |ϕ_α> can bevisualized in a single graph by plottinga_α(n) as a function of the quantum state index n.The graph will completely identify the spatial distributionof spins in the quantum state <cit.>.§RESULTS In the following we present our results for system sizes L= 8, 12, 16, 20, and 24.We chose those sizes in order to avoid the effects of frustration and preservethe symmetry of the 2,2 antiphase, which has periodicity of 4 lattice spacings.Still we are able to draw reliable conclusions as well as predictions about the quantum model in the thermodynamic limit. Let us consider first the case L=8.Figure <ref> shows the fidelity susceptibility plotted against thenext-nearest-neighbor interaction J_2for a fixed transverse field B_x = 0.2.The two peaks in the graph givethe locations of the critical points where quantum phase transitions occur.By calculating the susceptibility for severalvalues of B_xand J_2, we obtain the phase diagram shown in Fig. <ref>.There, we readily identify three distinct phases for low magnetic fields. Theregion farthest to the left (F) is ferromagnetic, while the middle (P_1) has a modulated phase, andthe region farthest to the right has the antiphase(2,2). The transition line bordering the ferromagneticphase is close to the exact Peschel-Emeryline <cit.>.As we shall see, for larger system sizes we obtain results which closer to that line. Notice that all the phase boundary lines meet at (J_2,B_x)=(0.5,0.0), the known multicritical point. Finally, for large enough magnetic fields, the modulated phase becomes paramagnetic. Such a feature does not appear in the phase diagram shown, which covers relatively low magnetic fields, where lies the interesting physics. That is also true for all the following phase diagrams below, which are valid at the low field region, where weare concerned with the onset and further evolutionof modulated phases as the system sizes increases.Consider now L=12.Figure <ref> shows the fidelity susceptibility versus J_2, forB_x = 0.2.The three peaks on the graph givethe locationswhere thephase transitions occur.Proceeding in a similar way for various values of B_x we determine the phase diagram, which is shown in Fig. <ref>. Alternately, by keeping J_2 fixed and sweeping with B_x we obtain the same phase diagram. As an example of this we present Figs. <ref>, <ref>, and <ref>, which shows the susceptibilities along B_x. The peaks are at the same locations as those obtained earlier with J_2 sweeps. As can be seen, there appears an additional phase boundary line, as compared to the case L=8. There is a modulated phase in the region P_2, and a floating phase P_1.These phases are separated by the boundary line thatmeets at the multicritical point.For very large fields B_x we expect the system to be paramagnetic.The ferromagneticand antiphase regions remain basically the same, apart from a slight shift in theirborders, due to finite size effects. The boundary line between the ferromagnetic and its neighboring modulated phase is now closer tothe Peschel-Emery line than that of the case L=8. The spin configurations in each of the phases can be inferred from a plot of the amplitudes a_0(n) of the ground-state eigenvector versus the basis index n for a point deep within a given phase.For instance, consider the point in the phase diagram(J_2,B_x)=(0.345,0.200), which is in the F-phase. Figure <ref> shows a_0(n) vs n for that point.The two largest contributions to the ground-state correspond to the ferromagnetic spin configurations, n=0 and n=4095, which havebinary representations 000000000000 and 111111111111, respectively.The other basis states with smaller amplitudes are induced by the transverse magnetic field. Those amplitudes increase withB_x. Consider now (J_2,B_x)=(0.438,0.200), which lies in the region P_2 of Fig. <ref>.The amplitudes of the ground-state basis vectors are depicted in Fig. <ref>.The largest contributions come from ferromagnetic orderings, whilethe second largest amplitudes are from the basis state 000000111111 and its cyclic permutations of the spins. The third largest amplitudes are very close to the second. They come from the states 000000011111,111111100000, and all the others were obtained by their cyclic relatives. The boundary line separating the F-phase from the neighboring modulated phase starts out at the multiphase point (J_2,B_x)=(0.5,0.0) and ends close to the Pfeuty transition point (J_2,B_x)=(0.0,1.0).We find that as the transverse field becomes sufficiently large the system enters a paramagnetic phase, where the spins tend to point in the same direction as the field.That is a general feature of the model. No matter which phase the system is in when B_x is small, eventually it will become paramagnetic as the field increases.We do not find any evidence of a sharp transition to paramagnetism. It seems that paramagnetism is achieved through a crossover mechanism, so that no transition line is observed. Fig. <ref> shows the ground-state eigenvector amplitudes for 3 cases: B_x=0.200, 2.000, and 20.00. The figures were obtained for L=12 and J_2=0.565, but similar behavior is expected for any other set of parameters L and J_2. The top figure (B_x=0.2) shows 6 largest amplitudes that correspond to that basis vectors containing periodic sequences of 3 up- followed by 3 down-spins.The next largest amplitudes stem from spin arrangements not periodic. As the field becomes sufficiently large, the amplitudes for the ordered phase disappear, while all the other amplitudes becomes larger, as can be seen in the middle figure of Fig. <ref>. There, most of the spins are equally likely to align themselves with the transverse field. Finally, for very large fields (e. g., B_x=20.00),nearly all the spins align themselves with the field, resulting in a more evenly distributed amplitudes of the basis vectors.Clearly the system is in an induced paramagnetic phase. As we shall see later, when we consider larger lattices, nonperiodic configurations will dominate the low-B_x phase. That amounts to the so-called floating phase. In that phase there is not any periodic spin order commensurate with the underlying lattice. Finally, the ground-state of the rightmost phasein Fig. <ref> is dominated by four amplitudes correspondingto the 2,2-phase. The dependence of the amplitudes with the state indexfor (J_2,B_x)=(0.675,0.200) in that phase,is depicted in Fig. <ref>. Again, small amplitudes are due to the transverse magnetic field and, as in the other cases,and they get larger as B_x increases.Both the F-phase and the 2,2-phase are presentin all the cases we considered (B_x ≤ 1.2), for all lattice sizes L. They are expected to be presentin the thermodynamic limit. This is in agreement with the resultsfound by other methods <cit.>. However, as we consider larger lattices,other modulated phasesappear in between the ferromagneticand the floating phase.It should be noted that all the transitionlines start out at the multiphase point and then spreadoutwards as B_x increases.For sufficiently large B_x the phase is expected to be paramagnetic. Let us consider now the model with size L=16. Figure <ref> shows the fidelity susceptibility as a functionofJ_2, for B_x=0.2. The susceptibility exhibits four peaks, thus indicatingfive distinct phases.Again, by numerically varyingB_x and J_2,we obtained the phase diagram for the system, depicted in Fig. <ref>. At the two far sides of the diagram we obtained the F- and 2,2-phases, as in the previous case.The positions of the boundaries of the F- and 2,2-phases with their neighboring phasesare weakly dependent on the system size,especially the boundary of the F-phase. The slope of the boundary line of the2,2-phase for L=16 diminishes a little as compared with the previous case L=12. We find an additional modulated phase, which is dominated by states with theordered pattern4,4. There appears to be other contributions to the ground-state of much smaller weights which are not ordered, but which will increase with the applied field B_x.Again, all the transition lines start at the multicritical point.For L=20 and B_x= 0.2 the fidelity susceptibilityshows 5 peaks, as seen in Fig. <ref>.The plot indicates the existence of five phase transitionsfor this lattice size. The phase diagram J_2–B_xis shown in Fig. <ref>. We observe that anothermodulated phase has appeared.Now, in addition to the ferromagnetic, floating,and2,2-antiphase, the system now has three modulated phases. The floating phase P_1 for this lattice size isdominated by the orderings 3,2 and2,3. Again, the modulated phases eventuallybecomeparamagnetic for large enough transverse fields.For larger system sizes, we observe a pattern thatallows us to make inferences about the phases of the system in thethermodynamic limit. Due to computer limitations, the largest system studied is L=24. Figure <ref> shows the fidelity susceptibility as a function ofJ_2, forB_x=0.2. There are 6 peaks, indicating an equal number of phase transitions.The phase diagram is shown in Fig. <ref>. We now identify 4modulated phases in the figure, P_2, P_3, P_4, and P_5, in addition to the floating P_1, ferromagnetic F, and the 2,2 phases. The paramagnetic phase only occurs for high B_x, where the phases lose their characteristics as the spins tend to align with the transverse field. The modulated phases are characterized by several periodicities, among them 4,4 for P_3, and 3,3 for P_2. The floating phase P_1 is now dominated by configurations which do not exhibit any periodicity within the system size. No particular ordering seems to take place as L increases, hence no commensurateorder emerges in the floating phase. As the system size increases, more modulated phases appear. For sufficiently large transverse magnetic fields one expects the system to become paramagnetic.The origin of the modulated phases follows from the degeneracy of the ground-state at J_2= 0.5 and B_x=0.0. There, the ground-state is highly degenerate, with the number of configurations exponentially increasing with the size of the system, as mentioned before. The transverse magnetic field lifts the degeneracies, thus separating the phases.At finite sizes, some of the phases become visible.As one consider larger systems, more of thosephases appear.The ferromagnetic as well as the 2,2 phases should be obviouslypresent for any system size in the cases J_2 <0.5 and J_2>0.5, respectively, since they are energetically favorable in thosesituations. Our numerical analysis was done with amaximum of 24 spins due to computer limitations.Yet, we can expect that as the numberof spins increases there will appear more andmore distinct modulated phases.We predict that at the thermodynamic limitthere will be a (denumerable) infinite numberof modulated phases. At criticality the fidelity susceptibilty shows power-law behaviorwith the lattice size, indicating that the transition is of second-order;otherwise it is of first-order <cit.>.For instance, for thetransition line closest to the feromagnetic phase we observe a power-lawbehavior, which is shown in Fig. <ref>.The solid line is the numerical fit χ = 59.2L^2.It seems that all the transition lines between modulated phases are of second-order.In particular, the transition between the modulated phase P_2 and the floating phase (P_1)seems to be of second-order, contrary to the claims that it is of BKT type. Finally, the transition line separating the floating and2,2 antiphase is of first-order, since the behavior of the susceptibilitydeviates frompower-law, as can be seen in Fig. <ref>.The scaling behavior of the fidelity susceptibility in the vicinityof a quantum critical point has been found to be <cit.>:χ(λ_c) ∼ L^2/ν.whereν is the critical exponent describing the divergenceof the correlation function.For the case of the transition lineclosest to the ferromagnetic phase (see Fig. <ref>),the behavior of the fidelity susceptibiliy at criticality is quadraticimplying that ν =1. Hence, in this region the model is inthe same universality class as the transverse Ising model. §SUMMARY AND CONCLUSIONS We have studied the ground-stateproperties of the one-dimensional ANNNI modelin a transverse magnetic field.The phase diagrams in the (J_2, B_x) plane wereobtained using the quantum fidelity methodfor several lattice sizes. A new picture emerged that is distinct from previouslyreported results. In addition to the known phases, namely, ferromagnetic,floating, and the 2,2 phase,it seems that there will be an infinite number of modulated phases of spin sequences commensurate with the underlyinglattice in the thermodynamic limit.We do not find paramagnetism for small values of the applied field.Paramagnetism is expectedto occur at sufficiently high fields, not shown in our phase diagrams. The transitions between the modulated phases seem to be of second-order. On the other hand,the transition between the floating and 2,2 phase appears to be of first-order.1cm ACKNOWLEDGEMENTS 0.5cm We thank C. Warner for critical reading of the manuscript. We also thank FAPERJ, CNPq and PROPPI (UFF)for financial support. O.F.A.B. acknowledges support from the Murdoch College of Science Research Program and a grant from theResearch Corporation through the Cottrell College ScienceAward No. CC5737. Sac99 S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, Cambridge, England, 1999).Gen63 P. G. de Gennes, Solid State Comm. 1, 132 (1963).Bit96 D. Bitko, T.F. Rosenbaum, and G. Aeppli, Phys. Rev. Lett.77, 940 (1996).Pfe70 P. Pfeuty, Ann. Phys. (N.Y.) 57, 79 (1970).Sim11 J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss,and M. Greiner, Nature (London) 472, 307 (2011).You75 A. P. Young, J. Phys. C8, L309 (1975).Her76 J. A. Hertz, Phys. Rev. B 14, 1165 (1976).Suz76 M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976).Ruj81 P. Rujan, Phys. Rev. B 24, 6620 (1981).Bar81 M.N. Barber and P.M. Duxbury, J. Phys. A 14, L251 (1981).Bar82 M.N. Barber and P.M. Duxbury, J. Stat. Phys. 29, 427 (1982).Suz13 S. Suzuki, J. Inoue, and D.K. Chakrabarty, Quantum Ising Phases and Transitions in Transverse Ising Models, Lecture Notes in Physics, 2nd ed., Heidelberg, Springer, 2013.Dut15 A. Dutta, G. Aeppli, B.K. Chakrabarti, U. Divakaran, T.F. Rosenbaum, and D. Sen,in Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information, Cambridge University Press, Delhi, (2015).Ari91 C. M. Arizmendi, A. H. Rizzo, L. N. Epele, and C. A. Garcia,Z. Phys. B: Condens. Matter 83, 273 (1991).Sen92 P. Sen, S. Chakraborty, S. Dasgupta, and B.K. Chakrabarti,Z. Phys. B: Condens. Matter 88, 333 (1992).Rie96H. Rieger and G. Uimin, Z. Phys. B: Condens. Matter101, 597 (1996).Sen97 P. Sen, Phys. Rev. B, 55, 11367 (1997).Gui02 Paulo R. Colares Guimarães, J. A. Plascak, F. C. Sá Barreto,and J. Florencio, Phys. Rev. B, 66, 064413 (2002).Dut03A. Dutta and D. Sen, Phys. Rev. B, 67, 094435 (2003).Bec06 M. Beccaria, M. Campostrini, and A. Feo, Phys. Rev. B73, 052402 (2006).Bec07 M. Beccaria, M. Campostrini, and A. Feo, Phys. Rev. B 76, 094410 (2007).Cha07 A. K. Chandra and S. Dasgupta, Phys. Rev. E 75, 021105 (2007).Nag11 A. Nagy, New J. Phys. 13 023015 (2011).Fis80 M. E. Fisher andW. Selke, Phys. Rev. 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Saguia, B. Boechat,J. Florencio, Phys. Rev. E 90, 032101 (2014).You11 W.L. You and Y. L. Dong, Phys. Rev. B 84, 174426 (2011). Sch09 D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. 103, 170501 (2009).Alb10 A.F. Albuquerque, F. Alet, C. Sire, and S. Capponi,Phys. Rev. B 81,064418 (2010). | http://arxiv.org/abs/1705.09679v2 | {
"authors": [
"O. F. de Alcantara Bonfim",
"B. Boechat",
"J. Florencio"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170526185026",
"title": "Quantum fidelity approach to the ground state properties of the 1D ANNNI model in a transverse field"
} |
Experimental status of the nuclear spin scissors modeP. Schuck Received/ Accepted====================================================== We consider four definitions of solution to the initial–boundary value problem for a scalar balance laws in several space dimensions. These definitions are generalised to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.2010 Mathematics Subject Classification: 35L65, 35L04, 35L60Keywords: Initial–boundary value problem for balance laws; Entropy–entropy flux pairs; Boundary conditions§ INTRODUCTIONThis paper is concerned with the relations among different definitions of solution to the Initial–Boundary Value Problem (IBVP) for a general scalar balance law in several space dimensions:{[ ∂_t u (t, x) + ∇· f(t,x,u (t,x)) = F (t,x,u (t,x))(t,x)∈_+ ×Ω; u (0, x) = u_o (x)x∈Ω;u (t,ξ) = u_b (t,ξ) (t, ξ)∈_+ ×∂Ω. ].Above and hereinafter, Ω is an open bounded subset of ^N, with smooth boundary ∂Ω, and _+ = [0, +∞[.The way the boundary condition is satisfied is going to be precised further on and constitutes a key issue addressed in this paper.The pioneering work by Bardos, le Roux and Nédélec <cit.> introduces a definition of solution to (<ref>) following the spirit of the one given by Kružkov in <cit.> in the case without boundary. The idea of the authors is to include in a unique integral inequality both Kružkov definition and the boundary condition. However, the BLN–definition considers only functions admitting a trace at the boundary, for instancefunctions. In <cit.>, the authors explain the way the boundary condition has to be understood and introduce a key inequality on the boundary, which we call the BLN condition, relating the boundary datum to the trace of the solution, see (<ref>).It is also possible to consider a definition of solution to (<ref>) analogous to the BLN–one, though involving classical (regular) entropy–entropy flux pairs, see <cit.> and Definition <ref>. In this way there is a sort of symmetry between, on one side, the BLN–definition and this one for IBVPs as (<ref>) and, on the other side, Kružkov definition and the definition of weak entropy solution for initial value problems on all ^N.On all ^N, solutions to the initial value problem for a general scalar balance law are usually found in Ł∞, therefore a question naturally arises: is it possible to find a concept of solution to the IBVP (<ref>) in this function space?The key difficulty is that, in general, a function in Ł∞ does not necessarily admit a trace at the boundary. A first proposal to overcome this issue is given by Otto in his PhD thesis (a summary is presented in <cit.>, while more details and proofs can be found in <cit.>). In the case of autonomous scalar conservation laws, Otto replaces the Kružkov entropy–entropy flux pairs as exploited in <cit.> with the so-called boundary entropy–entropy flux pairs and bases on them his definition of solution. In this paper we provide a generalisation of this definition to deal with non autonomous fluxes and arbitrary source terms, see Definition <ref>.Still looking for solutions to (<ref>) in Ł∞, in the case of scalar conservation laws with divergence free flux, Vovelle introduces in <cit.> a definition of solution using the so-called Kružkov semi-entropy–entropy flux pairs. This definition is then extended by Martin in <cit.> to deal with general scalar balance laws. The resulting MV–definition, exploiting Kružkov semi entropies, resembles the BLN–definition, although not requiring the existence of the trace of the solution at the boundary.To sum up, there are mainly four definitions of solution to (<ref>), which can be classified as follows: involving the trace at the boundary (BLN–solutions and classical entropy–solutions, i.e. Definition <ref>) or not (Otto-type solutions, i.e. Definition <ref>, and MV–solutions); dealing with regular entropies (entropy–solutions and Otto–type ones) or with Lipschitz ones (BLN and MV–solutions). Definition <ref> and MV–definition share the interesting feature of being stable under Ł1-convergence, see Remark <ref> and also <cit.>.In this paper, we prove first the equivalence of Definition <ref> and MV–definition, and then focus on the way the boundary conditions are fulfilled according to those definitions. We proceed similarly for the definitions of solution requiring the existence of the trace, that is entropy–solutions and BLN–solutions.The main result of this paper is the proof of the equivalence among all the presented definitions of solution to (<ref>). Of course, this can be done only when the existence of the trace of the solution at the boundary is assumed. Further information on the existence of the trace at the boundary can be found in <cit.>, see Section <ref> for more details. As an intermediate step, we also prove the equivalence among the way the boundary conditions are understood according to the various definitions. The paper is organised as follows. Section <ref> collects the notation used throughout the paper. Sections <ref> and <ref> are devoted to the Otto-type definition of solution and Martin–Vovelle–one: the first section contains the definitions themselves and the theorem stating the equivalence between them, while results on the way the boundary conditions are fulfilled constitute the latter one. Section <ref> deals with the definitions of solution with traces and provides also the main equivalence result. In Section <ref> we give the definition of strong solution to (<ref>) and a related results. Section <ref> provides further details on the one dimensional case, while Section <ref> summarises the existence results that can be found in the literature. We collect the detailed proofs of our results in Section <ref>. § NOTATIONThe space dimension N, with N ≥ 1, is fixed throughout. We set _+=[0,+∞[. We denote by ν (ξ) the exterior normal to ξ∈∂Ω. For w,k ∈ setℐ[w,k] = { z ∈ (w-z) (z-k) ≥ 0 } = {θw + (1 - θ) k θ∈ [0,1] }.In other words, ℐ[w,k] denotes the closed interval with end points w and k.For the divergence of a vector field, possibly composed with another function, we use the following notation:∇·f(t,x,u (t,x)) = ÷f(t,x,u (t,x)) + ∂_u f(t,x,u (t,x)) ·∇u (t,x).We use below the following standard assumptions: (IC) u_o ∈Ł∞ (Ω;);(BC) u_b ∈Ł∞ ([0,T] ×∂Ω;);(f) f ∈2 ([0,T] ×Ω×; ^N), ∂_u f ∈∞ ([0,T] ×Ω×; ^N) and ∇·∂_u f ∈Ł∞ ([0,T] ×Ω×; ^N× N);(F) F ∈2 ([0,T] ×Ω×; ) and ∂_u F ∈Ł∞ ([0,T] ×Ω×; ).Following <cit.>, we set(s) =1 s>0, 0 s ≤ 0,(s) =0 s ≥ 0, -1 s < 0, [s^+=max{s, 0} ,;s^-= max{-s, 0} . ]We often use below the equalities (-s)^- = s^+ and (-s)^+ = s^-. Introduce moreover the following notation: if g: ^2 →, for all z, w ∈, set∂_1 g (z,w) =lim_h → 0g (z+h,w)-g (z,w)/h,∂_2 g (z,w) =lim_h → 0g (z,w+h)-g (z,w)/h,and similarly for functions of more arguments.§ Ł1-STABLE DEFINITIONSBefore introducing the first definition of solution to (<ref>), we need to recall the notion of (classical) entropy–entropy flux pair, see <cit.>. The pair (η,q) ∈2 (;) ×2 ([0,T]×Ω×; ^N) is called an entropy–entropy flux pair with respect to f if * η is convex, i.e. η” (z)≥ 0 for all z∈;* for all t∈ [0,T], for all x ∈Ω, for all x∈, ∂_3 q (t,x,z) = η' (z)∂_3 f (t,x,z). The notion of boundary entropy–entropy flux pair is first introduced by Otto in <cit.>, see also <cit.>, for autonomous scalar conservation laws on bounded domains, and then extended to a more general case in <cit.>. We recall it here for completeness. The pair (H,Q) ∈2 (^2;) ×2 ([0,T]×Ω×^2; ^N) is called a boundary entropy–entropy flux pair with respect to f if * for all w ∈ the function z ↦ H (z,w) is convex;* for all t ∈ [0,T], x∈Ω and z,w ∈, ∂_3 Q (t,x,z,w) = ∂_1 H (z,w)∂_3 f(t,x,z);* for all t ∈ [0,T], x∈Ω and w ∈, H (w,w)= 0, Q (t,x,w,w) = 0 and ∂_1 H (w,w) = 0.Note that if H is as above, then H ≥ 0.We now extend the definition given by Otto (see <cit.> and also <cit.>) to account for non autonomous fluxes and arbitrary source terms. The concept of boundary entropy–entropy flux pairs introduced above characterises the definition. A regular entropy solution (RE–solution) to the initial–boundary value problem (<ref>) on the interval [0,T] is a map u ∈Ł∞ ([0,T] ×Ω; ) such that for any boundary entropy–entropy flux pair (H, Q), for any k ∈ and for any test function ϕ∈1 (]-∞,T[ ×^N; _+)∫_0^T ∫_Ω[ H (u (t,x), k ) ∂_t ϕ (t,x) + Q (t,x,u (t,x), k ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)x̣ṭ+ ∫_Ω H(u_o (x), k )ϕ (0,x) x̣ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_0^T ∫_∂Ω H (u_b (t,ξ), k )ϕ (t,ξ) ξ̣ṭ≥ 0,where 𝒰 is the interval 𝒰=[-U,U], with U=u_Ł∞ ([0,T] ×Ω;). A comment on the constant appearing in the last line of the integral inequality above is at the end of Section <ref>. Observe that an equivalent definition of solution can be obtained considering test functions ϕ∈1 (×^N; _+) and the following integral inequality: ∫_0^T ∫_Ω[ H (u (t,x), k ) ∂_t ϕ (t,x) + Q (t,x,u (t,x), k ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)x̣ṭ + ∫_Ω H(u_o (x), k )ϕ (0,x) x̣ - ∫_ΩH(u (T,x), k )ϕ (T,x) x̣ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_0^T ∫_∂Ω H (u_b (t,ξ), k )ϕ (t,ξ) ξ̣ṭ≥ 0.A similar definition of solution is given by Vovelle in <cit.>, see also <cit.>, using the so called Kružkov semi-entropy–entropy flux pairs, which are Lipschitz continuous functions, thus less regular than the boundary entropies considered in the definition of RE–solution.A semi-entropy solution (MV–solution) to the initial–boundary value problem (<ref>) on the interval [0,T] is a map u ∈Ł∞ ([0,T] ×Ω; ) such that for any k ∈ and for any test function ϕ∈1 (]-∞,T[ ×^N; _+)∫_0^T ∫_Ω(u (t,x) - k )^± ∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω^± (u (t,x) -k)(f(t,x,u (t,x) ) -f(t,x,k )) ·∇ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω^± (u (t,x) -k) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_Ω(u_o (x) - k )^± ϕ (0,x) x̣ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_0^T ∫_∂Ω(u_b(t,ξ) - k)^± ϕ (t,ξ) ξ̣ṭ≥ 0,where 𝒰 is the interval 𝒰=[-U,U], with U=u_Ł∞ ([0,T] ×Ω;). Before entering into the details of the link between these two definitions of solution to (<ref>), we emphasise a feature they share. Both Definitions <ref> of RE–solution and <ref> of MV–solution are stable under Ł1-convergence. This remarkable feature is underlined in <cit.> for the particular definition given by Otto, but it is immediate to see that it extends to both Definitions <ref> and <ref>. More precisely, let u_o^n and u_b^n be sequences of initial and boundary data converging in Ł1 to u_o and u_b respectively. Let u^n be a solution to (<ref>), according to either of the two definitions, with initial datum u_o^n and boundary datum u_b^n. Assume that u_n converges to u in Ł1. Then, this limit function u is a solution to (<ref>), according to the same definition, with initial datum u_o and boundary datum u_b.Our first aim is to establish a connection between Definition <ref> of RE–solution and Definition <ref> of MV–solution. An intermediate step is constituted by the following Lemma, which gives a link between the boundary entropy–entropy flux pairs exploited in Definition <ref> and the Kružkov semi-entropy–entropy flux pairs used in Definition <ref>.Let η∈2 (;) be a convex function such that there exists w ∈ [A,B] with η(w) = 0 and η' (w)=0. Then η can be uniformly approximated on [A,B] by applications of the kinds ↦∑_i=1^p α_i (s-κ_i)^- + ∑_j=1^q β_j(s-κ̃_j)^+,where α_i ≥ 0, β_j≥ 0, κ_i,κ̃_j ∈ [A,B].Conversely, there exists a sequence of boundary entropy–entropy flux pairs which converges to the Kružkov semi-entropy–entropy flux pairs.Thanks to Lemma <ref>, the equivalence between RE–solution and MV–solution follows immediately. For the detailed proof we refer to Section <ref>. Let u ∈Ł∞ ([0,T] ×Ω; ). Then u is a RE–solution to (<ref>), in the sense of Definition <ref>, if and only if u is a MV–solution to (<ref>), in the sense of Definition <ref>.§ BEHAVIOUR AT THE BOUNDARYWe now focus our attention on the way the boundary conditions are fulfilled according to the definitions of solution to (<ref>) introduced in Section <ref>.All the proofs are deferred to Section <ref>. The following Lemma is a generalisation to problem (<ref>) of <cit.>. It states the way the boundary conditions are satisfied in the case of a RE–solution to (<ref>).Let u ∈Ł∞ ([0,T] ×Ω; ) be a RE–solution to (<ref>), according to Definition <ref>. Then, for all boundary entropy–entropy flux pairs (H,Q) and for all β∈Ł1 (]0,T[ ×∂Ω; _+)_ρ→ 0^+∫_0^T ∫_∂Ω Q (t, ξ, u(t, ξ - ρ ν (ξ)), u_b (t,ξ)) ·ν (ξ)β (t, ξ) ξ̣ṭ≥ 0,ν (ξ) being the exterior normal to ξ∈∂Ω. Due to the equivalence between RE–solution and MV–solution proved in Theorem <ref>, the boundary conditions are satisfied in the sense of (<ref>) also in the case of a MV–solution to (<ref>). See <cit.>, and also <cit.>, for a different proof of (<ref>), starting from Kružkov semi-entropy–entropy flux pairs. An alternative formulation of the boundary conditions is also possible, both in the case of RE–solution and MV–solution, see <cit.> and <cit.>. Let u ∈Ł∞ ([0,T] ×Ω; ) be a RE–solution (or MV–solution) to (<ref>) in the sense of Definition <ref> (Definition <ref>).Define the function ℱ∈0 ([0,T] ×Ω×^3; ^N):ℱ (t, x, z, w, k) =f (t,x,w) - f (t,x,z) z ≤ w ≤ k, 0 w ≤ z ≤ k, f (t,x,z) - f (t,x,k) w ≤ k ≤ z, f (t,x,k) - f (t,x,z) z ≤ k ≤ w, 0 k ≤ z ≤ w, f (t,x,z) - f (t,x,w) k ≤ w ≤ z.Then, for all β∈Ł1 (]0,T[ ×∂Ω; _+) and for all k ∈_ρ→ 0^+∫_0^T ∫_∂Ωℱ(t, ξ, u(t, ξ - ρ ν (ξ)), u_b (t,ξ), k ) ·ν (ξ)β (t, ξ) ξ̣ṭ≥ 0. Observe that the function ℱ defined in (<ref>) can be written also as followsℱ (t, x, z, w, k) =12 [(z-w)(f (t,x,z) - f (t,x,w) ) .-(k-w)(f (t,x,k) - f (t,x,w) ) . +(z-k)(f (t,x,z) - f (t,x,k) ) ],and alsoℱ (t, x, z, w, k) =(z-max{w,k}) (f (t,x,z) - f(t,x,max{w,k}))+ (z-min{w,k}) (f (t,x,z) - f(t,x,min{w,k})). § SOLUTIONS WITH TRACES So far we have considered two definitions of solution to (<ref>), sought in Ł∞, and proved their equivalence. The RE–definition involves regular entropies, while the MV–definition deals with Lipschitz continuous ones. In this Section we present two additional definitions of solution to (<ref>), in which the trace of the solution at the boundary appears explicitly. The idea is to draw a parallel with RE and MV–solutions: indeed, the two definitions we are going to introduce are characterised by regular and Lipschitz continuous entropies respectively, and we prove that they are equivalent.Since the existence of the trace of the solution is required, more regularity is needed on the solution with respect to Definitions <ref> of RE–solutions and <ref> of MV–solutions. To this aim, introduce the following space: A function u belongs to the space 𝒯ℛ^∞ ([0,T] ×Ω;) if there exists a function u ∈Ł∞ ([0,T] ×∂Ω; ) such that_r → 0^+∫_0^T ∫_∂Ωu (t,ξ -r ν (ξ)) -u (t, ξ)ξ̣ṭ =0.We remark the following. Bardos, le Roux and Nédélec in <cit.> consider solutions in ([0,T] ×Ω;) ⊂𝒯ℛ^∞ ([0,T] ×Ω;): indeed,functions admit a trace at the boundary reached by Ł1 convergence, see <cit.>, <cit.>, <cit.> and <cit.>. Consider now the case of Ł∞ solutions, which are functions u satisfying in the sense of distribution on ]0,T[ ×Ω the inequality∂_t η(u) + ÷q (t,x,u) ≤0,for any (η,q) (classical) entropy–entropy flux pair (with respect to f, see Definition <ref>). Panov proves in <cit.> the existence of the trace at the boundary for Ł∞ solutions to (<ref>), under the following non-degeneracy condition on the flux: the function f is continuous and such that for a.e. (t,ξ) ∈_+ ×∂Ω and all (s,y) ∈_+ ×^N ∖ {(0,0)} the function u → s u + y f (t,ξ,u) is not constant on non-degenerate intervals, i.e. f satisfies the following genuine non linearity conditionℒ ({ u | s + y ∂_u f (t,ξ,u) = 0 }) = 0(s,y) ∈_+ ×^N ∖ {(0,0)},where ℒ is the Lebesgue measure and (t,ξ) ∈_+ ×∂Ω. As pointed out also in <cit.>, the above assumption allows to avoid flux functions whose restriction to an open subset is linear. The following definition uses the (classical) entropy–entropy flux pairs, see Definition <ref>. It extends the particular case of scalar conservation laws with autonomous fluxes considered in <cit.> and in <cit.>, see also <cit.>. An entropy solution (E–solution) to the initial–boundary value problem (<ref>) on the interval [0,T] is a map u ∈ (Ł∞∩𝒯ℛ^∞ )([0,T] ×Ω; ) such that for any entropy–entropy flux pair (η, q) and for any test function ϕ∈1 (]-∞,T[ ×^N; _+)∫_0^T ∫_Ω[ η(u (t,x))∂_t ϕ (t,x) + q(t,x,u (t,x) ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ωη'(u (t,x)) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ q (t,x,u (t,x) )ϕ (t,x) x̣ṭ + ∫_Ωη(u_o (x)) ϕ (0,x) x̣ - ∫_0^T∫_∂Ω q (t,ξ,u_b (t,ξ) ) ·ν (ξ)ϕ (t,ξ) ξ̣ṭ + ∫_0^T ∫_∂Ωη'(u_b(t,ξ)) ( f(t,ξ, u_b(t,ξ)) - f (t,ξ, u (t,ξ))) ·ν (ξ)ϕ (t,ξ) ξ̣ṭ≥ 0.We now recall the definition of solution to (<ref>) due to Bardos, le Roux and Nédélec <cit.>, which exploits the classical Kružkov entropy–entropy flux pairs. A Kružkov-entropy solution (BLN–solution) to the initial–boundary value problem (<ref>) on the interval [0,T] is a map u ∈ (Ł∞∩𝒯ℛ^∞ )([0,T] ×Ω; ) such that for any k ∈ and for any test function ϕ∈1 (]-∞,T[ ×^N; _+)∫_0^T ∫_Ωu (t,x) - k ∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k)(f(t,x,u (t,x) ) -f(t,x,k )) ·∇ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_Ωu_o (x) - kϕ (0,x) x̣ - ∫_0^T ∫_∂Ω(u_b (t,ξ) -k) (f(t,ξ, u(t,ξ)) - f (t,ξ,k)) ·ν (ξ)ϕ (t,ξ) ξ̣ṭ≥ 0.E–solutions, as in Definition <ref>, and BLN–solutions, as in Definition <ref>, are actually equivalent, see <cit.>. The proof of the equivalence between these two Definitions of solution is based on an analogous of Lemma <ref> and is briefly sketched in Section <ref>. The map u ∈ (Ł∞∩𝒯ℛ^∞ )([0,T] ×Ω; ) is an E–solution to (<ref>), in the sense of Definition <ref>, if and only if u is a BLN–solution to (<ref>), in the sense of Definition <ref>. Before studying the relation among all the considered definitions, we provide the analogous to Lemma <ref> and Lemma <ref>, explaining the way E–solutions and BLN–solutions to (<ref>) fulfil the boundary conditions. Concerning E–solutions, the following Lemma holds. Let u ∈ (Ł∞∩𝒯ℛ^∞)([0,T] ×Ω; ) be an E–solution to (<ref>) in the sense of Definition <ref>. Then, for all (classical) entropy–entropy flux pairs (η,q) and for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω,[ q(t,ξ, u (t,ξ)) - q(t,ξ, u_b (t,ξ)) - η'(u_b (t,ξ))( f(t,ξ, u (t,ξ)) - f (t,ξ, u_b (t,ξ)))] ·ν (ξ) ≥ 0. The proof is deferred to Section <ref>. Observe that condition (<ref>) is the generalisation to the multidimensional case of the boundary entropy inequality due to Dubois and LeFloch <cit.>.In the following Lemma we recall the well-known BLN condition, linking the boundary datum and the trace of the solution. The proof is in Section <ref>. Let u ∈ (Ł∞∩𝒯ℛ^∞)([0,T] ×Ω; ) be a BLN–solution to (<ref>) in the sense of Definition <ref>. Then, for all k ∈ and for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω,(( u (t,ξ) - k) - (u_b (t,ξ) - k) ) ( f(t,ξ, u (t,ξ)) - f (t,ξ,k) ) ·ν (ξ) ≥ 0.Moreover, condition (<ref>) is equivalent to the following: for all k∈ℐ[ u (t,ξ), u_b (t,ξ)] and a.e. (t,ξ) ∈ ]0,T[ ×∂Ω( u (t,ξ) - u_b (t,ξ)) ( f(t,ξ, u (t,ξ)) - f (t,ξ,k) ) ·ν (ξ) ≥ 0. The following Proposition constitutes the basis for the proof of the equivalence of all the definitions of solution to (<ref>) presented so far. It is a generalisation of <cit.> to problem (<ref>): it takes into account non autonomous fluxes and arbitrary source terms. In particular, this Proposition provides a connection among the ways the boundary conditions are understood according to the various definitions of solution introduced so far. However, we need to require the existence of the trace of the solution at the boundary. For further details about the trace, see the references at the beginning of Section <ref>. The proof is deferred to Section <ref>.Let u_b ∈Ł∞ ([0,T] ×∂Ω;) and u ∈ (Ł∞∩𝒯ℛ^∞) ([0,T] ×Ω;). Then the following statements are equivalent: * (<ref>) holds, for any boundary entropy–entropy flux pair (H,Q) and for any β∈Ł1 (]0,T[ ×∂Ω;_+); * (<ref>) holds, for any β∈Ł1 (]0,T[ ×∂Ω;_+) and for all k∈; *for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω and for all k ∈ it holdsℱ(t, ξ,u (t, ξ), u_b (t,ξ), k) ·ν (ξ)≥ 0,with ℱ as in (<ref>); * (<ref>) holds for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω and for all k ∈ℐ[ u (t,ξ), u_b (t,ξ)]; * (<ref>) holds for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω and for any entropy–entropy flux pair (η,q); *for a.e. (t,ξ) ∈ ]0,T[ ×∂Ω and all entropy–entropy flux pair (η, q), such that η' (u_b (t,ξ)) = 0 and q(t, ξ, u_b (t,ξ))=0, it holdsq(t,ξ,u (t,ξ)) ·ν (ξ) ≥ 0. We can now state our main result: given that u admits a trace in the sense of (<ref>), we prove that the Definitions of solution presented in this Section, that is E–solution and BLN–solution, are equivalent to the Definitions of solution introduced in Section <ref>, i.e. RE–solution and MV–solution. Let u ∈ (Ł∞∩𝒯ℛ^∞)([0,T] ×Ω; ). Then u is a RE–solution to (<ref>) according to Definition <ref>, or equivalently a MV–solution to (<ref>) in the sense of Definition <ref>, if and only if u is an E–solution to (<ref>) according to Definition <ref>, or equivalently a BLN–solution to (<ref>) in the sense of Definition <ref>.The proof is deferred to Section <ref>. Remark that, according to the results by Panov <cit.> recalled at the beginning of this section, Ł∞ solutions admit a trace at the boundary in the case of non-degenerate fluxes, thus in those cases there is no need to consider the intersection with the space 𝒯ℛ^∞([0,T] ×Ω; ).§ STRONG SOLUTIONSFor completeness, we recall below the definition of strong (smooth) solution to (<ref>).A strong solution to the initial–boundary value problem (<ref>) on the interval [0,T] is a map u ∈1 ([0,T]×Ω;) ∩0([0,T] ×Ω;) which satisfies pointwise the equation and the initial condition, and it is such that, for all (t,ξ)∈]0,T[ ×∂Ω and for all k ∈ℐ[u (t,ξ), u_b (t,ξ)],(u (t,ξ) - u_b (t,ξ)) ( f(t,ξ,u (t,ξ)) - f (t,ξ,k) ) ·ν (ξ) ≥ 0.Note that condition (<ref>) reduces to (<ref>): the difference is that strong solutions are defined up to the boundary, and therefore the notion of trace is not needed. For further details on the boundary conditions for smooth solution, including an heuristic derivation, see <cit.>.The following result holds. Let u ∈1 ([0,T]×Ω;) ∩0([0,T] ×Ω;) be a strong solution to (<ref>) in the sense of Definition <ref>. Then u is also a RE–solution to (<ref>), in the sense of Definition <ref>.Obviously, due to the equivalence among the definitions of solution proven in Theorem <ref>, every strong solution is also a MV–solution, an E–solution and a BLN–solution.The proof follows the line of the second part of the proof of Theorem <ref> and it is hence omitted. The main difference is that the solution itself at the boundary is considered, instead of its trace. § THE 1 DIMENSIONAL CASEIn this section we focus on the case N=1, i.e. Ω is the interval ]a,b[, with a, b ∈. The boundary datum is assigned at the end points of the interval: for t ∈_+u(t,a) = u_b (t,a), u (t,b)= u_b (t,b).We write explicitly how the last line of the integral inequality in the definition of solution reads in the case of the RE–definition and of the E–definition, the other two cases being completely analogous. Observe that the exterior normal to ∂Ω in a is -1, while in b is +1. * RE–definition:+ ∂_u f_Ł∞([0,T] ×Ω×𝒰;) ∫_0^T[ H (u_b(t,a), k ) ϕ(t,a) + H (u_b(t,b), k ) ϕ(t,b) ]ṭ.* E–definition:+ ∫_0^T q (t,a,u_b (t,a))ϕ (t,a)ṭ - ∫_0^T q (t,b,u_b (t,b))ϕ (t,b)ṭ- ∫_0^T η'(u_b (t,a)) (f(t,a,u_b (t,a)) - f(t,a, u (t,a))) ϕ (t,a) ṭ+ ∫_0^T η'(u_b (t,b)) (f(t,b,u_b (t,b)) - f(t,b, u (t,b))) ϕ (t,b) ṭ. What is immediately evident is the presence of the minus sign in the last case, while the first contains only sums. The minus sign is due to the scalar product with the exterior normal to ∂Ω, which occurs in the E–definition, and in the BLN–definition as well. It can be seen that there is no need for a minus sign neither in the RE–definition nor in the MV–definition.We can exploit the one dimensional setting to analyse a feature of the RE–definition and the MV–definition. Indeed, the integral inequalities of these two definitions involve the constant ∂_u f_Ł∞ ([0,T] ×Ω×𝒰;^N), where 𝒰=[-U,U], with U= u_Ł∞ ([0,T] ×Ω;). This is nothing but the Lipschitz constant of f with respect to u, therefore one may wonder whether it is possible to consider a different constant, either smaller or larger, and to still get a solution. One can see, through the following one dimensional example, that the above constant is indeed the smallest possible.Consider the following problem{[ ∂_t u (t, x) + ∂_x f (u (t,x)) = 0(t,x)∈ _+ × ]a,b[; u (0, x) = 1x∈]a,b[;u (t,a) = u_b (t,a) = 1t∈ _+; u (t,b) = u_b (t,b) = -1t∈ _+ ].f (u) = u^2/2.The solution is constant and equal to 1. Fix T>0. Observe that u_Ł∞ ([0,T] × [a,b];)=1. Consider the integral inequality (<ref>) of the MV–definition, and use the positive constant c instead of ∂_u f_Ł∞ ([0,T] × [a,b] ×𝒰;) = 1. Since we know that u =1 is the solution, simple computations show that c should be grater or equal to 1.Obviously, given a solution u, choosing a greater value of the Lipschitz constant still ensures that u is a solution: indeed the constant is multiplied by a non negative term, both in the RE and in the MV–definition.§ NOTES ON THE EXISTENCE OF SOLUTIONSWe recap the results present in the literature concerning the existence of solutions to problem (<ref>), specifying in each case the considered definition of solution.As far as it concerns the case of an autonomous scalar conservation laws, i.e. f=f (u) and F=0, Otto proves the existence and uniqueness of a RE–solution to (<ref>), see <cit.> and also <cit.> for more detailed proofs.In <cit.>, Martin proves the existence and uniqueness of a MV–solution to the general problem (<ref>), though imposing on the flux and the source terms a condition which leads to a (simple) maximum principle, see <cit.>.Bardos, le Roux and Nédélec prove in <cit.> existence and uniqueness of a BLN–solution to (<ref>) in the case of homogeneous boundary conditions, i.e. u_b=0. The result is then generalised in <cit.> to allow for (regular) non zero boundary data. The case of an autonomous scalar conservation laws with BLN–definition is considered also by Serre in <cit.>. Dafermos in <cit.> focuses on the same autonomous problem, although under homogeneous boundary conditions, and exploits the RE–definition of solution.§ TECHNICAL DETAILSTheorem <ref> A RE–solution is a MV–solution. It is enough to consider the following sequences of boundary entropy–entropy flux pairsH_n (z,k) =(((z- k)^±)^2 + 1/n^2)^1/2 -1/nQ_n (t,x,z,k) =∫_k^z ∂_1 H_n (w, k)∂_u f (t,x,w)ẉ.Indeed, as n goes to +∞,H_n (z,k) ⟶ (z-k)^±Q_n (t,x,z,k) ⟶(z-k)^± (f (t,x,z) - f (t,x,k)),so that, in the limit, (<ref>) becomes∫_0^T ∫_Ω(u (t,x) - k)^±∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω( u (t,x)-k)^± (f (t,x,u (t,x)) - f (t,x,k)) ·∇ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω( u (t,x)-k)^± ( F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ) ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω( u (t,x)-k)^±(÷ f (t,x,u (t,x) ) - ÷ f (t,x, k ))ϕ (t,x)x̣ṭ + ∫_Ω(u_o (x)- k )^± ϕ (0,x) x̣ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_0^T ∫_∂Ω(u_b (t,ξ)-k )^± ϕ (t,ξ) ξ̣ṭ≥ 0,where 𝒰=[-U,U], with U=u_Ł∞ ([0,T]×Ω;).Combining the second and the third lines in the inequality above yields exactly (<ref>). A MV–solution is a RE–solution. Since u ∈Ł∞ ([0,T]×Ω;), there exist A, B ∈, with A < B, such that A ≤ u (t,x) ≤ B for a.e. (t,x) ∈ [0,T]×Ω, and hence u ∈Ł∞ ([0,T]×Ω;[A,B]). We can then apply Lemma <ref>: each boundary entropy–entropy flux pair is uniformly approximated by a linear combination with positive coefficients of Kružkov semi-entropy–entropy flux pairs. Thus the inequality in (<ref>) is preserved and (<ref>) holds.Lemma <ref> The proof extends <cit.> to consider non autonomous fluxes and general source terms.Let (H,Q) be a boundary entropy–entropy flux pair, k ∈. The analogous of <cit.> is the following:∫_0^T ∫_Ω[ H (u (t,x), k ) ∂_t ϕ (t,x) + Q (t,x,u (t,x), k ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)x̣ṭ≥ 0,which follows directly from the Definition <ref> of RE–solution when considering a test function ϕ∈1 (]0,T[ ×Ω; _+).Similarly, the analogous of <cit.> is the following:∫_0^T ∫_Ω[ H (u (t,x), k ) ∂_t ϕ (t,x) + Q (t,x,u (t,x), k ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)x̣ṭ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_0^T ∫_∂Ω H(u_b (t,ξ), k )ϕ (t,ξ) ξ̣ṭ≥ 0,where 𝒰=[-U,U] with U=u_Ł∞ ([0,T] ×Ω;), which follows directly from the Definition <ref> of RE–solution when considering a test function ϕ∈1 (]0,T[ ×Ω; _+).We proceed as in the proof of <cit.>. For the sake of simplicity, we restrict ourselves to the case of a half–space, i.e.Ω ={ x = (x', s) ∈^N-1×y<0 },ν = (0, …, 0, 1) ∈^N,Γ = ]0,T[ ×^N-1,r =(t,x') ∈Γ, Q_T ={ p = (r,s)r ∈Γ , s<0 }.The general case can then be obtained by a covering argument, i.e. by considering that the boundary ∂Ω can be locally replaced by the border of a half-space.For any boundary entropy–entropy flux pair (H,Q), denote, for (t,x) ∈ [0,T] ×Ω, w ∈ℚ,η(z) = H (z,w) , q (t,x,z) = Q (t,x,z,w) .Choosing ϕ (t,x) = ϕ (r,s) = β (r) α (s), with α∈1 (]-∞,0[;_+), in (<ref>) yields-∫_-∞^0 ∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ α' (s)ṣ≤ C ∫_-∞^0 α (s) ṣ,whereC =η (u)_Ł∞ (Q_T;)∫_Γ∂_t β (r)ṛ + q(·,·,u)_Ł∞ (Q_T;^N)∫_Γ∇_x'β(r)ṛ + [ η' (u)_Ł∞ (Q_T;)(F - ÷ f)(·, ·, u)_Ł∞ (Q_T;) + ÷ q (·, ·, u)_Ł∞ (Q_T;)] ∫_Γβ (r)ṛ.Thanks to integration by parts on the left hand side of (<ref>) and to the fact that α≥ 0, we obtain that the functions ↦∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ - C sis non increasing on ]-∞,0[. Moreover, we have_s → 0^-inf∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ≥ - _Q_Tq (·,·,u)∫_Γβ (r)ṛ.Monotonicity (<ref>) and boundedness from below (<ref>) imply that the following quantity is finite_s → 0^-∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ.From (<ref>), for all α∈1 (;_+) we get-∫_-∞^0 ∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ α' (s)ṣ ≤ C ∫_-∞^0 α (s) ṣ + ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_Γη(u_b (r))β (r) ṛα (0).Choose α_n (s) = (n s +1) ]-1/n,0[, mollify it properly and insert it in (<ref>): in the limit n →∞ we obtain_s → 0^-∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ≥ - ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_Γη(u_b (r))β (r) ṛ.Let J ⊆1 (Γ;_+) be a countable set of functions such that for all β∈Ł1 (Γ;_+) there is a sequence (β_n) in J such that lim_nβ_n = β in Ł1 (Γ;_+). Therefore,lim_n ∫_Γq ( r, s, u (r,s) ) ·νβ_n (r) ṛ = ∫_Γq ( r, s, u (r,s) ) ·νβ(r) ṛuniformly in s ∈ ]-∞,0[ andlim_n∫_Γη(u_b (r)) β(r) ṛ = ∫_Γη(u_b (r)) β(r) ṛ.Due to (<ref>), there exists a set E_w of measure zero such that for all β∈ J there exists lim_s → 0^-s ∉ E_w∫_Γ q ( r, s, u (r,s) ) ·ν β (r) ṛ and moreoverlim_s →0^-s ∉E_w ∫_Γq ( r, s, u (r,s) ) ·νβ(r) ṛ ≥- ∂_u f_Ł∞([0,T] ×Ω×𝒰; ^N) ∫_Γη(u_b (r)) β(r) ṛ.Note that the set E_w depends on w because η and q depend on w. The above result can be extended to functions β∈Ł1 (Γ;_+), so that for all w ∈ℚ the quantitylim_s →0^-s ∉E_w ∫_ΓQ ( r, s, u (r,s), w ) ·νβ(r) ṛexists andlim_s → 0^-s ∉ E_w∫_Γ Q ( r, s, u (r,s), w ) ·ν β (r) ṛ≥ - ∂_u f_Ł∞ ([0,T] ×Ω×𝒰; ^N)∫_Γ H (u_b (r), w ) β (r) ṛ.Let v ∈Ł∞ (Γ;) and β∈Ł1 (Γ;_+) be given, and let (v_n) be a sequence of simple functions with values in ℚ which converges to v almost everywhere in Γ. Obviously, (<ref>) holds for all w=v_n. Moreover,lim_n ∫_ΓQ ( r, s, u (r,s), v_n (r)) ·νβ(r) ṛ = ∫_ΓQ ( r, s, u (r,s), v (r)) ·νβ(r) ṛuniformly in s ∈ ]-∞,0[ andlim_n ∫_ΓH (u_b (r), v_n (r) ) β(r) ṛ = ∫_ΓH (u_b (r), v (r) ) β(r) ṛ.Hence, the following inequality holdslim_s →0^-s ∉E_w ∫_ΓQ ( r, s, u (r,s), v (r) ) ·νβ(r) ṛ ≥- ∂_u f_Ł∞([0,T] ×Ω×𝒰; ^N) ∫_ΓH (u_b (r), v (r) ) β(r) ṛ.Choosing v=u_b and recalling the properties of the boundary entropy H (see Definition <ref>) conclude the proof. Lemma <ref> The proof follows immediately from Lemma <ref>. Indeed, for k ∈ and n ∈∖{0}, using the notation introduced in (<ref>), define the mapsΔ^k (u,w) =min_z ∈ℐ[w,k]u-zH_n^k (u,w) =((Δ^k (u,w))^2 + 1/n^2)^1/2 - 1/nQ_n^k (t,x,u,w) =∫_w^u ∂_1 H_n^k (z,w)∂_u f (t,x,z) ẓ.It can be easily proved that, for all k ∈, the sequence of boundary entropy–entropy flux pairs (H_n^k(u,w), Q_n^k(t,x,u,w)) converges uniformly to (Δ^k(u,w), ℱ (t,x,u,w,k)) as n goes to +∞. Applying (<ref>), with Q replaced by Q_n^k, yields the thesis in the limit n → +∞, for all k ∈ and β∈Ł1 (]0,T[ ×∂Ω;_+). Theorem <ref> An E–solution is a BLN–solution. It is sufficient to consider the following sequence of (classical) entropies: for k ∈η_n (z) =√((z-k)^2+1n),the corresponding entropy fluxes q_n being defined as in point 2. of Definition <ref>. A standard limiting procedure allows to obtain, in the limit n → + ∞,η_n (z) →z-kq_n (t,x,z) → (z-k)(f (t,x,z) - f (t,x,k)),so that, in the limit n → +∞, (<ref>) becomes∫_0^T ∫_Ω[ u (t,x) - k ∂_t ϕ (t,x) + (u (t,x) - k ) ( f(t,x,u (t,x) ) - f(t,x, k ) ) ·∇ϕ (t,x)]x̣ṭ + ∫_0^T ∫_Ω(u (t,x) - k ) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω(u (t,x) - k ) ÷( f (t,x,u (t,x) ) - f (t,x, k ) )ϕ (t,x) x̣ṭ + ∫_Ωu_o (x)-kϕ (0,x) x̣- ∫_0^T∫_∂Ω(u_b (t,ξ) - k ) ( f (t,ξ,u_b (t,ξ) ) - f(t,ξ,k)) ·ν (ξ)ϕ (t,ξ) ξ̣ṭ + ∫_0^T ∫_∂Ω(u_b (t,ξ) - k ) ( f(t,ξ, u_b(t,ξ)) - f (t,ξ, u (t,ξ))) ·ν (ξ)ϕ (t,ξ) ξ̣ṭ≥ 0.Combining in the inequality above the second line with the third and the fifth one with the sixth yields exactly (<ref>). A BLN–solution is an E–solution. Assume that u_Ł∞ ([0,T] ×Ω;)≤ U.It is immediate to see that u satisfies (<ref>) with η(u) = αu - k + β, for any α > 0 and k, β∈.Moreover, if u satisfies (<ref>) for two distinct locally Lipschitz continuous pairs (η_1,q_1) and (η_2, q_2), then the same inequality (<ref>) holds for u with (η_1+η_2, q_1+q_2). It can be proved by induction that u satisfies (<ref>) for any pair (η,q) with η piecewise linear and continuous on [-U,U].Furthermore, if u satisfies (<ref>) for the continuous pairs (η_n,q_n) and the η_n converge uniformly to η on [-U,U], then u fulfils (<ref>) also for the pair (η,q), where q is defined as in point 2. of Definition <ref>. To conclude, since any convex entropy η is the uniform limit on [-U, U ] of piecewise linear and continuous functions, we obtain the proof. Lemma <ref> The proof follows the lines of that of <cit.>. Indeed, let Φ∈1 (]0,T[ ×^N; _+) and ψ_h ∈1 (Ω̅; [0,1]), with ψ_h (ξ) = 1 for all ξ∈∂Ω, ψ_h (x)=0 for all x ∈Ω with B (x,h) ⊆Ω, and ∇ψ_h_Ł∞ (Ω;^N)≤ 2/h. Write (<ref>) with ϕ (t,x) = Φ (t,x)ψ_h (x) and take the limit as h → 0. For any entropy–entropy flux pair (η,q), thanks to the Dominated Convergence Theorem and to <cit.>, we get∫_0^T∫_Ω q(t,ξ, u (t,ξ)) ·ν (ξ)Φ (t,ξ) ξ̣ṭ - ∫_0^T∫_Ω q(t,ξ, u_b (t,ξ)) ·ν (ξ)Φ (t,ξ) ξ̣ṭ + ∫_0^T∫_Ωη' (u_b (t,ξ)) ( f(t,ξ,u_b (t,ξ)) - f(t,ξ, u (t,ξ)) ) ·ν (ξ)Φ (t,ξ) ξ̣ṭ ≥ 0.Hence, for any entropy–entropy flux pair (η,q), (<ref>) holds almost everywhere on ]0,T[ ×∂Ω. Lemma <ref> Proving that (<ref>) holds is done in the same way as in the proof of <cit.>.It is immediate to prove that (<ref>) reduces to (<ref>) when k ∈ℐ[ u (t,ξ), u_b (t,ξ)].On the other hand, assume that (<ref>) holds and consider the various possibilities. * If k ≤min{ u (t,ξ), u_b (t,ξ) } or k ≥max{ u (t,ξ), u_b (t,ξ) }: the quantity(u (t,ξ) - k) - (u_b (t,ξ)- k)is equal to 0, so (<ref>) clearly holds. * if u (t,ξ) ≤ k ≤ u_b (t,ξ): (<ref>) reads - ( f(t,ξ, u (t,ξ)) - f (t,ξ,k)) ·ν (ξ) ≥ 0, while ( u (t,ξ) - k)=-1 and (u_b (t,ξ)- k)=+1, so that (<ref>) clearly holds. * if u_b (t,ξ) ≤ k ≤ u (t,ξ): (<ref>) reads ( f(t,ξ, u (t,ξ)) - f (t,ξ,k)) ·ν (ξ) ≥ 0, while ( u (t,ξ) - k)=+1 and (u_b (t,ξ)- k)=-1, so that (<ref>) clearly holds.The proof is completed.Proposition <ref> <ref> ⇒ <ref>. It is proved in Lemma <ref>. <ref> ⇒ <ref>. From (<ref>) it follows that, for any β∈Ł1 (]0,T[ ×∂Ω;_+) and for any k∈,∫_0^T ∫_∂Ωℱ(t, ξ,u (t, ξ), u_b (t,ξ), k) ·ν (ξ)β (t,ξ) ξ̣ṭ = _ρ→ 0^+∫_0^T ∫_∂Ωℱ(t, ξ, u (t, ξ -ρ ν (ξ) ), u_b (t,ξ), k ) ·ν (ξ)β (t,ξ) ξ̣ṭ.Therefore, there is a set E ⊆ ]0,T[ ×∂Ω of zero measure such that, for all k∈ and for all (t,ξ) ∈ (]0,T[ ×∂Ω) ∖ Eℱ(t, ξ, u (t, ξ), u_b (t,ξ), k) ·ν(ξ) ≥0. <ref> ⇒ <ref>. It follows immediately from the definition (<ref>) of ℱ. <ref> ⇒ <ref>. For any entropy–entropy flux pair (η,q) and for any (t,ξ) ∈ ]0,T[ ×∂Ω, it holdsq (t,ξ,z) = q (t,ξ,w) + ∫_w^z η' (λ)∂_u f (t,ξ,λ)λ̣= q (t,ξ,w) + η' (w) ( f (t,ξ,z) -f (t,ξ,w) ) + ∫_w^z η” (λ)(f (t,ξ,z) -f (t,ξ,λ) )λ̣.The above formula and (<ref>) imply (<ref>). <ref> ⇒ <ref>. It is sufficient to apply (<ref>) to any entropy–entropy flux pair (η,q) withη' (u_b (t,ξ)) = 0q (t, ξ, u_b (t,ξ))=0. <ref> ⇒ <ref>. For any boundary entropy–entropy flux pair (H,Q) and β∈Ł1 (]0,T[ ×∂Ω;_+) it holds∫_0^T ∫_∂Ω Q (t,ξ,u (t,ξ), u_b (t,ξ)) ·ν (ξ)β (t,ξ) ξ̣ṭ = _ρ→ 0^+∫_0^T ∫_∂Ω Q (t,ξ, u (t,ξ - ρ ν (ξ)), u_b (t,ξ)) ·ν (ξ)β (t,ξ) ξ̣ṭ.The right hand side above is clearly positive, due to Definition <ref> and (<ref>), proving (<ref>).Theorem <ref> Thanks to Theorem <ref> and Theorem <ref>, we know that the following relations hold⟺⟺Therefore, we now prove that a MV–solution is a BLN–solution and that an E–solution is a RE–solution. A MV–solution is a BLN–solution. Let k∈ and ϕ∈1 (]-∞,T[ ×Ω; _+). Adding (<ref>) with '+' and (<ref>) with '-' yields the following inequality:∫_0^T ∫_Ωu (t,x) - k ∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k)(f(t,x,u (t,x) ) -f(t,x,k )) ·∇ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_Ωu_o (x) - kϕ (0,x) x̣≥ 0.Fix h > 0 and consider as a test function Φ_h (t,x) = ϕ (t,x) (1 - ψ_h (x)), with ϕ∈1 (]-∞,T[ ×^N; _+) and ψ_h ∈1 (Ω̅; [0,1]), with ψ_h (ξ) = 1 for all ξ∈∂Ω, ψ_h (x) = 0 for all x ∈Ω with B (x, h) ⊆Ω, and ∇ψ_h_Ł∞ (Ω;^N)≤ 2/h. Note that lim_h → 0(1-ψ_h (x))= Ω (x) and Φ_h ∈1 (]-∞,T[ ×Ω; _+). Using Φ_h into (<ref>) yields∫_0^T ∫_Ωu (t,x) - k(1-ψ_h (x)) ∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) (f(t,x,u (t,x) ) -f(t,x,k )) ·∇ϕ (t,x) (1 - ψ_h (x) ) x̣ṭ - ∫_0^T ∫_Ω (u (t,x) -k) (f(t,x,u (t,x) ) -f(t,x,k )) ·ϕ (t,x)∇ψ_h (x)x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) (1-ψ_h (x)) x̣ṭ + ∫_Ωu_o (x)-k ϕ (0,x) (1-ψ_h (x)) x̣≥ 0.Let now h tend to 0. Thanks to <cit.> we obtain∫_0^T ∫_Ωu (t,x) - k∂_t ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) (f(t,x,u (t,x) ) -f(t,x,k )) ·∇ϕ (t,x)x̣ṭ - ∫_0^T ∫_∂Ω ( u (t,ξ) -k) (f(t,ξ, u (t,ξ) ) -f(t,ξ,k )) ·ν(ξ)ϕ (t,ξ)ξ̣ṭ + ∫_0^T ∫_Ω (u (t,x) -k) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x)x̣ṭ + ∫_Ωu_o (x)-k ϕ (0,x)x̣≥ 0.Consider in particular the third line above:- ∫_0^T ∫_∂Ω ( u (t,ξ) -k) (f(t,ξ, u (t,ξ) ) -f(t,ξ,k )) ·ν(ξ)ϕ (t,ξ)ξ̣ṭ.Since u is a MV–solution to (<ref>), we can apply Lemma <ref>, or equivalently <ref>. Moreover, u ∈ (Ł∞∩𝒯ℛ^∞)([0,T] ×Ω; ) and thus Proposition <ref> and Lemma <ref> hold. Therefore, thanks to (<ref>) and the positivity of the test function ϕ, we get[(<ref>)] ≤- ∫_0^T ∫_∂Ω (u_b (t,ξ) -k) (f(t,ξ,u (t,ξ) ) -f(t,ξ,k )) ·ν(ξ) ϕ(t,ξ)ξ̣ ṭ,which inserted into (<ref>) yields (<ref>), concluding the proof. An E–solution is a RE–solution. Let ϕ∈1 (]-∞,T[ ×Ω;_+) and k ∈. For any boundary entropy–entropy flux pair (H,Q), set for any t ∈[0,T], x ∈Ω and z ∈η (z) =H (z,k), q (t,x,z) =Q (t,x,z,k).By Definition <ref> of boundary entropy–entropy flux pair, (η,q)isanentropy–entropy fluxpairwithrespectto f. Notice moreover that η(k)=0. Since u is an E–solution to (<ref>), it satisfies (<ref>) withthe above choice of thetestfunction, which,thanksto (<ref>),now readsas follows:∫_0^T ∫_Ω[ H(u (t,x), k)∂_t ϕ (t,x) + Q(t,x,u (t,x),k ) ·∇ϕ (t,x)] x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k ) [ F(t,x,u (t,x) ) - ÷ f (t,x,k ) ]ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x) x̣ṭ + ∫_Ω H(u_o (x), k) ϕ (0,x) x̣≥ 0. Apply now Lemma <ref> and Proposition <ref>. In particular, (<ref>) holds for any boundary entropy–entropy flux pair (H,Q) and for any β∈Ł1 (]0,T[ ×∂Ω;_+). We now follow the lines of the second part of the proof of <cit.> in order to prove that u satisfies (<ref>). The idea is to show that every u which is a solution inside the domain Ω, that is (<ref>) holds, and which satisfies the boundary condition in a suitable way, is indeed a RE–solution.Define the following maps: for z,w ∈H̃ (z,w) =η(z) - η(w)z ≤w ≤k, 0w ≤z ≤k, η(z)w ≤k ≤z, η(z)z ≤k ≤w, 0k ≤z ≤w, η(z) - η(w)k ≤w ≤z, and, for t ∈ [0,T], x ∈Ω,Q̃ (t,x,z,w) =q (t,x,z) - q (t,x,w)z ≤w ≤k, 0w ≤z ≤k, q (t,x,z)w ≤k ≤z, q (t,x,z)z ≤k ≤w, 0k ≤z ≤w, q (t,x,z) - q (t,x,w)k ≤w ≤z. It is easy to see that (H̃, Q̃) ∈0 (^2;) ×0 ([0,T] ×Ω×^2; ^N). Define, for n ∈∖{0},H_n (z,w) = { [ . [η(z) - η(w-1/n)z ≤w -1/n;0 w -1/n ≤z ≤k + 1/n;η(z) - η(k+1/n) k+1/n ≤z ] } w ≤k,;. [ η(z) - η(k-1/n)z ≤k-1/n; 0 k-1/n ≤z ≤w+1/n; η(z) - η(w+1/n)w + 1/n ≤z ]} k ≤w. ] .Then, (H̃, Q̃) can be locally uniformly approximated by (H̃_n, Q̃_n), defined as followsH̃_n (z,w) =∫_ H_n (λ, w)ρ_1/n (z-λ) λ̣,Q̃_n (t,x, z,w) =∫_w^z ∂_1 H̃_n (λ, w)∂_u f (t,x,λ) λ̣,where ρ_1/n is a smooth mollifier. The pair (H̃_n, Q̃_n) is clearly a boundary entropy–entropy flux pair. Since (<ref>) holds, we have, for all β∈Ł1 (]0,T[ ×∂Ω; _+),_ρ→0^+ ∫_0^T ∫_∂Ω Q̃_n (t, ξ,u (t,ξ- ρν(ξ)), u_b (t,ξ)) ·ν(ξ) β(t,ξ) ξ̣ṭ ≥0,which becomes, as n → + ∞,∫_0^T ∫_∂ΩQ̃(t, ξ, u (t,ξ), u_b (t,ξ)) ·ν (ξ)β (t,ξ) ξ̣ṭ≥ 0,where we use the hypothesis that u admits a trace at the boundary. Going through all the cases in the definition of Q̃ and exploiting the properties of η yieldQ̃ (t,x,z,w) - q (t,x,z) ≤∂_u f_Ł∞([0,T]×Ω×𝒰;^N) η(w),where 𝒰 is the interval 𝒰=[-U,U] with U=u_Ł∞ ([0,T] ×Ω;).Therefore, by (<ref>), for all β∈1 (]0,T[ ×∂Ω; _+)∫_0^T ∫_∂Ω q (t, ξ,u (t,ξ)) ·ν (ξ)β (t,ξ) ξ̣ṭ ≥ - ∂_u f_Ł∞ ([0,T]×Ω×𝒰;^N)∫_0^T ∫_∂Ωη(u_b (t,ξ)) β (t,ξ) ξ̣ṭ.Fix h>0. Consider (<ref>) with the test function Φ_h (t,x) = ϕ (t,x) (1 - ψ_h (x)), with ψ_h as in the first part of the proof of this Theorem, so that we obtain∫_0^T ∫_Ω H (u (t,x), k )(1 - ψ_h (x))∂_t ϕ (t,x)x̣ṭ + ∫_0^T ∫_Ω Q (t,x,u (t,x), k ) ·∇ϕ (t,x)(1 - ψ_h (x)) x̣ṭ - ∫_0^T ∫_Ω Q (t,x,u (t,x), k ) ·ϕ (t,x)∇ψ_h (x) x̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x)(1 - ψ_h (x))x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)(1 - ψ_h (x)) x̣ṭ + ∫_Ω H(u_o (x), k )ϕ (0,x) (1 - ψ_h (x)) x̣≥ 0.Let now h tend to 0: by <cit.> we get∫_0^T ∫_Ω H (u (t,x), k )∂_t ϕ (t,x)x̣ṭ + ∫_0^T ∫_Ω Q (t,x,u (t,x), k ) ·∇ϕ (t,x) x̣ṭ - ∫_0^T ∫_Ω Q (t,ξ,u (t,ξ), k ) ·ϕ (t,ξ) ξ̣ṭ + ∫_0^T ∫_Ω∂_1 H (u (t,x), k )[ F (t,x,u (t,x)) - ÷ f(t,x,u (t,x)) ] ϕ (t,x) x̣ṭ + ∫_0^T ∫_Ω÷ Q (t,x,u (t,x), k )ϕ (t,x)x̣ṭ + ∫_Ω H(u_o (x), k )ϕ (0,x) x̣≥ 0.Thanks to the definition of q (t,x,z) = Q (t,x,z,k) and to (<ref>) we obtain (<ref>), concluding the proof.Acknowledgement: The present work was supported by the PRIN 2015 project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications and by the INdAM–GNAMPA 2017 project Conservation Laws: from Theory to Technology.abbrv | http://arxiv.org/abs/1705.09109v1 | {
"authors": [
"Elena Rossi"
],
"categories": [
"math.AP",
"35L65, 35L04, 35L60"
],
"primary_category": "math.AP",
"published": "20170525094901",
"title": "Definitions of solutions to the IBVP for multiD scalar balance laws"
} |
Integration of Satellites in 5Gthrough LEO Constellations Oltjon Kodheli, Alessandro Guidotti, Alessandro Vanelli-Coralli Dept. of Electrical, Electronic, and Information Engineering (DEI), Univ. of Bologna, Bologna, ItalyEmail: {a.guidotti, alessandro.vanelli}@[email protected] December 30, 2023 ===================================================================================================================================================================================================================================================The standardization of 5G systems is entering in its critical phase, with 3GPP that will publish the PHY standard by June 2017. In order to meet the demanding 5G requirements both in terms of large throughput and global connectivity, Satellite Communications provide a valuable resource to extend and complement terrestrial networks. In this context, we consider a heterogeneous architecture in which a LEO mega-constellation satellite system provides backhaul connectivity to terrestrial 5G Relay Nodes, which create an on-ground 5G network. Since large delays and Doppler shifts related to satellite channels pose severe challenges to terrestrial-based systems, in this paper we assess their impact on the future 5G PHY and MAC layer procedures. In addition, solutions are proposed for Random Access, waveform numerology, and HARQ procedures. § INTRODUCTIONDuring the last years, wireless communications have been experiencing an exploding demand for broadband high-speed, heterogeneous, ultra-reliable, secure, and low latency services, a phenomenon that has been motivating and leading the definition of new standards and technologies, known as 5G. The massive scientific and industrial interest in 5G communications is motivated by the key role that the future system will play in the worldwide economic and societal processes to support next generation vertical services, e.g., Internet of Things, automotive and transportation sectors, e-Health, factories of the future, etc., <cit.>.Due to the challenging requirements that 5G systems shall meet, e.g., large throughput increase, global and seamless connectivity, the integration of satellite and terrestrial networks can be a cornerstone to the realization of the foreseen heterogeneous global system. Thanks to their inherently large footprint, satellites can efficiently complement and extend dense terrestrial networks, both in densely populated areas and in rural zones, as well as provide reliable Mission Critical services. While in the past terrestrial and satellite networks have evolved almost independently from each other, leading to a difficult a posteriori integration, the definition of the 5G paradigm provides the unique chance for a fully-fledged architecture. This trend is substantiated by 3GPP Service and system Aspects (SA) activities, which identified satellite systems as a possible solution both for stand-alone infrastructures and for complementing terrestrial networks, <cit.>, as well as by EC H2020 projects as VITAL (VIrtualized hybrid satellite-TerrestriAl systems for resilient and fLexible future networks), in which the combination of terrestrial and satellite networks is addressed by bringing Network Functions Virtualization (NFV) into the satellite domain and by enabling Software-Defined-Networking (SDN)-based, federated resources management in hybrid SatCom-terrestrial networks, <cit.>.In this context, the integration of terrestrial systems with Geostationary Earth Orbit (GEO) satellites would be beneficial for global large-capacity coverage, but the large delays and Doppler shifts in geostationary orbits pose significant challenges. In <cit.>, resource allocation algorithms for multicast transmissions and TCP protocol performance were analyzed in a Long Term Evolution (LTE)-based GEO system, providing valuable solutions. However, to avoid the above issues, significant attention is being gained by Low Earth Orbit (LEO) mega-constellations, i.e., systems in which hundreds of satellites are deployed to provide global coverage, as also demonstrated by recent commercial endeavors.In <cit.>, we considered a mega-constellation of LEO satellites deployed in Ku-band to provide LTE broadband services and analyzed the impact of large delays and Doppler shifts on the PHY and MAC layer procedures.Since 3GPP Radio Access Network (RAN) studies and activities are now providing significant results and critical decisions have been made during the last meetings on the PHY and MAC layers for the New Radio (NR) air interface, which will be finalized by June 2017, <cit.>, it is of outmost importance to assess the impact that these new requirements will have on future 5G Satellite Communications (SatCom). To this aim, in this paper, we move from the analysis performed in <cit.> and assess the impact of large delays and Doppler shifts in a LEO mega-constellation operating in Ku-band. In particular, we will focus on waveform design, Random Access, and HARQ procedures.§ SYSTEM ARCHITECTUREWe consider a system architecture similar to that addressed in <cit.>, with a mega-constellation of Ku-band LEO satellites, deployed at h=1200 km from Earth, beam size set to 320 km, and a minimum elevation angle of 45 degrees. In the proposed heterogeneous architecture, we introduce on-ground Relay Nodes (RN) that extend the terrestrial network coverage with reduced costs. RNs were introduced and standardized in LTE, <cit.>, and within the 3GPP standardization framework it is agreed that they will be also implemented in 5G, <cit.>.The following assumptions are made for the proposed system architecture, which is shown in Fig. <ref>: i) the terrestrial terminals are 5G User Equipments (UE) connected to the 5G RNs that provide the terrestrial access link; ii) we assume the deployment of Type-1 RNs, which have up to layer-3 capabilities, i.e., they can receive, demodulate, and decode data, apply another FEC, and then re-transmit a new signal. They also have their own cell ID, synchronization, broadcast, and control channels; iii) the satellites are assumed to be transparent and to provide backhaul connectivity to the on-ground RNs; and iv) the satellite gateway is connected to the satellites through ideal feeder links, providing access to the Donor gNB (DgNB) that connect the RN with the 5G Core Network. It is worthwhile to highlight that the RNs act as gNB (5G NodeB, as per 3GPP nomenclature) from the User Equipment (UE) perspective and as UE when communicating with a donor gNB (DgNB).The focus of this paper is on the backhaul link between the RN and the DgNB, and in particular on the impact of large delays and Doppler shifts on the PHY/MAC procedures. We assume the link to be a modified version of UE-RN interface, which will have the same protocols, but with different RF characteristic. For instance, it is already agreed that the 5G air interface will use Ciclic Prefix-Orthogonal Frequency Division Multiplexing (CP-OFDM) waveforms, coupled with scalable numerology, and UE-RN interface can use different numerologies with respect to RN-DgNB interface. We assume a Frequency Division Duplexing (FDD) frame structure, since Time Division Duplexing (TDD) is not suitable to scenarios affected by large delays. Finally, outband RNs are assumed, i.e., backhaul and the access links use different frequencies. This isolation is important, because the generic UE will covered by both the related RN and the satellite, which might cause interference without a proper frequency separation. It is worth noting that one DgNB can be connected with more than one RN and M≤ N, where M and N represent the number of DgNBs and RNs respectively. § SATELLITE CHANNEL In this section, we focus on the main satellite channel impairments, i.e. large Round Trip Time (RTT) and Doppler shifts, in order to assess their impact on the 5G MAC and PHY procedures.§.§ Delay In order to compute the RTT, we have to take into account the distance from RN to the satellite, and the distance from the satellite to the DgNB. There should be also the path between UEs and RNs, but it is negligible compared to the other terms. The minimum distance between the satellite and the RN is reached with an elevation angle of 90 degrees and is equal to h=1200 km. By approximating also the distance between the satellite and the DgNB d_S-D with the altitude h, we have:RTT = 2*h+d_S-D/c≈ 4*h/c≈ 16mswhere c is the speed of light. This value of is quite large if compared with the maximum RTT foreseen in a terrestrial network. This is a critical challenge for integrating satellites with 5G systems and will be addressed in the following sections. §.§ Doppler In order to compute the Doppler shift, two separate scenarios have to be treated. First, we consider the Doppler shift that the UE experiences due to its relative motion to RN. If v denotes the speed of the UE relative to the RN, f_c the carrier frequency, and θ the angle between velocity vectors, the maximum Doppler frequency is given by:f_d=v*f_c/c*cos(θ)The targeted mobility in 5G is 500 km/h for carrier frequencies below 6 GHz, <cit.>. This is the maximum UE speed at which a minimum predefined quality of service (QoS) can be achieved. Therefore, at 4 GHz carrier frequency, supporting a 500 km/h mobility for the UE would lead to a Doppler spread up to 1.9 kHz. Furthermore, transmissions in above 6 GHz bands will be more sensitive to frequency offset. It means that NR will be able to correctly estimate and decode the signal, even though a Doppler shift of 1.9 kHz is experienced.Second, we compute the Doppler shift experienced on the satellite channel. This Doppler shift will be caused only by the movement of satellite, because both the RN and the DgNB are fixed on Earth. A closed-form expression for the Doppler in LEO satellites is provided in <cit.> and is as follows:f_d(t)=f_c*w_SAT*R_E*cos(θ(t))/cwhere w_SAT is the angular velocity of the satellite w_SAT=√(G*M_E/(R_E+h)^3), R_E is the Earth radius, θ(t) is the elevation angle at a fixed time t, M_E is the mass of Earth, and G the gravitational constant. In the considered scenario, we can derive the maximum Doppler shift to be 11 GHz < f_d < 14 GHz in Ku-band. Clearly, this value is significantly larger than that experienced in a terrestrial link.§ CHALLANGES OF 5G AIR INTERFACE (NR) IN SATELLITE CONTEXT In this section, we assess the impact of the large RTT and Doppler shifts the NR MAC and PHY layers. There is an ongoing work towards 5G standardization, including MAC and PHY layer aspects. Some significant high-level agreements have been reached in the latest 3GPP meetings. On the basis, we highlight the challenges that the satellite link imposes to NR, in order to be “satellite friendly.” §.§ Waveform Three scenarios have been defined by ITU-R for IMT-2020 (5g) and beyond <cit.>: i) eMBB (enhanced Mobile BroadBand); ii) mMTC (massive Machine Type Communications); and iii) uRLLC (Ultra-Reliable and Low Latnecy Communications). It is agreed that CP-OFDM waveforms are supported for eMBB and uRLLC use cases, while Discrete Fourier Transform-spread-OFDM (DFT-s-OFDM) waveforms are also supported at least for eMBB uplink for up to 40 GHz <cit.>. With respect to the LTE waveforms, the NR waveforms are more flexible. This flexibility is given by different sub-carrier spacing (SCS) and filtering/windowing techniques. Depending on the use case, NR will use different numerologies across different UEs. Scalable numerology should allow at least from 15 kHz to 480 kHz subcarrier spacing, <cit.>. Basically, the values of SCS can be defined as: SCS = 15 kHz * 2^n, where n is a non negative integer.It is straightforward to say that larger SCS lead to increased robustness to Doppler shifts. Depending on the UE speed and the carrier frequency used, SCS should be large enough to tolerate the Doppler. For example, in LTE, with a 2 GHz carrier frequency, the maximum Doppler that can be supported is 950 Hz because the SCS is fixed to 15 kHZ. Basically, it corresponds to 6.3 % of the SCS. NR can cope with larger Doppler shift values because it can adopt SCS larger than 15 kHz. As outlined in the previous section, NR should support up to 1.9 kHz of Doppler shift for a 4 GHz of carrier frequency. This is totally possible, by using SCS of 30 kHz (6.3%*30kHz≈ 1.9kHz). When going above 6 GHz, the sensitivity to frequency offset is larger and higher SCS can be implemented. The Doppler shift in the considered satellite system is still very large even in when relying on the highest SCS made available by NR (480 kHz), which corresponds to a value of 30.4 kHz of tolerated frequency offset.It is worth noticing that, in our scenario, the only interface affected by high Doppler through satellite is the RN-DgNB interface. The Doppler experienced due to UE movement (high speed trains), is estimated and compensated by RNs. Therefore, no modification is needed in the interface UE-RN. Solutions should be found for RN-DgNB interface, in order to cope with extremely high Doppler in the satellite link. §.§ Random Access The Random Access (RA) procedure is used by the UEs to synchronize with the gNB and to initiate a data transfer. During 3GPP RAN meetings, the following high level agreements were made about RA procedure in NR, <cit.>: i) both contention-based and contention-free RA procedure should be supported in NR; and ii) contention-based and contention-free RA procedure follow the steps of LTE. Basically, contention-based RA is performed for initial access of the UE to the gNB, or for re-establishment of synchronization in case it is lost. Whereas, contention-free RA is performed in handover situation, when the UE was previously connected to a gNB.There are four steps for contention-based RA procedure in NR. Steps 1 and 2 mainly aim at synchronizing uplink transmission from UE to gNB. The UE randomly chooses a preamble from a predefined set and sends it to the gNB. The preamble, consists of a cyclic prefix of length T_cp, a sequence part of length T_SEQ, and a guard time T_G in order to avoid collision of RA preambles. The gNB receives the preamble and sends a RA Response (RAR) to the UE with information about timing advance (TA), a temporary network identifier (T-RNTI), and the resources to be used in step 3. For the moment being, there is no agreement on the size of RAR window in NR.In steps 3 and 4, a final network identifier CRNTI is assigned to the UE. In LTE, the contention timer in which the UE receives the final network identifier can be as large as 64 ms. In NR there is no agreement so far related to this value. In our scenario we have taken into account the presence of RN, so we have to analyze the RA procedure in two stages, as follows. §.§.§ UE random access procedureEach UE should perform the RA procedure with the corresponding RN. It is worth noticing that contention-free RA involves only steps 1 and 2. In these first two steps, the RN does not need to contact the DgNB. It can terminate all protocols up to Layer 3. The delay in the satellite link is not involved, therefore contention-free RA procedure can be implemented without modifications. On the other hand, in contention-based RA, in order to provide a final network identifier to the UEs (step 3 and 4), the RN should contact the Core Network through DgNB. In this case, the RTT time on the satellite channel should be considered. The contention timer in NR should be larger than 16 ms, in order to support RA procedure through satellite link. In LTE, there is a contention timer up to 64 ms. Even though, the tendency is to have lower value in NR, no drastic reduction is expected. Hence, UE-RN RA procedure is expected to be realizable without any modification.§.§.§ RN attach procedureWhen a new RN is installed in the network, it should automatically begin its start-up attach procedure, which is performed in two phases,<cit.>. In the first phase, in order to have information for the initial configuration, RN attaches to the network as a UE performing the same steps of RA procedure. In the second phase, the RN receives from DgNB further specification (e.g., IP address of the S-GW/P-GW), so as to be able to operate as a relay. In contrast with the UE RA procedure, here the RTT of satellite channel is present in all steps. Therefore, not only the contention timer should be more than 16 ms, but also the RAR window size. In LTE, the RAR window size is between 3 and 15 ms and lower values are expected for NR <cit.>. Possible solutions should be found about the RAR window size.Another thing to be considered, is the preamble format of the RA procedure for RNs. The length of the preamble in the RA procedure defines the coverage that can be supported by the standard. In LTE there are 4 preamble formats, and the largest one is designed to support a 100 Km radius cell, corresponding to 0.67 ms Timing Advance (TA), <cit.>. In NR, at least for deployments below 6 GHz, longer RA preambles should be supported, <cit.>. It means that NR should support at least the same coverage as LTE. Therefore, we have to calculate the TA needed in our scenario for RN attach procedure through satellite link. To do so, we have to find the difference in RTT between 2 RNs, one with the shortest and the other one with the longest path to the satellite. The worst scenario, where the differences of RTT between two RNs is at its maximum occurs at minimum elevation angle, which in our case is 45 degrees, as shown in Fig. <ref>. By geometrical considerations, we can find that the maximum allowable distance between the RN and the satellite is d_1 = 1580 km, by solving the following equation:(R_E+h)^2 - R_E^2 - |d_1|^2 = 2R_E|d_1|sin(θ)where θ is the minimum elevation angle. As the beam has a diameter of 320 km, this is also the max distance between 2 RNs. It can be easily computed that d_2 = 1372 km, which results to |d_1 - d_2|= 208 km. This value is almost twice larger than the cell size that can be supported in NR. Valuable solutions should be found. Finally, it is important to mention that at start-up deployment of network, TA and RAR window size does not pose any challenge in RN attach procedure, because all the RNs can be introduced in the network one-by-one. However, under certain circumstances (e.g., satellite handover, heavy rain, etc.) RNs can loose their synchronization to DgNB, therefore a repetition of RN attach procedure is needed and all the RNs will compete for the channel at the same time. §.§ Satellite Handover LEO satellites move at high speeds above the ground. Even though no mobility is foreseen for RNs, due to the movement of the satellite, RNs will be seen at a certain amount of time. Therefore, all the system should pass through handover process. It means that each RN should perform RA procedure periodically, in order to obtain the physical link with the DgNB through another satellite. Performing a RA procedure periodically for RNs is very time-expensive. Consequently, there can be a significant reduction of throughput. Other solutions should be found. A possible solution can be found in<cit.>. §.§ Hybrid Automatic Repetition Quest (HARQ) One of the main requirements for 5G is to improve the link reliability. To this aim, among other solutions, HARQ protocols will be implemented as in LTE, <cit.>. If the transmission block (TB) is decoded correctly at the receiver, it responds with an ACK, otherwise it will send a NACK and the packet will be re-transmitted by adding more redundancy bits. In order to improve system efficiency, multiple parallel HARQ processes are used. That means that, in order to be able to achieve peak data rate, the following formula should hold:N_HARQ,min≥T_HARQ/TTIwhere N_HARQ,min is the minimum number of HARQ processes, TTI is the transmission duration of one TB, and T_HARQ is the time duration between the initial transmission of one TB and the corresponding ACK/NACK. This is illustrated in Fig. <ref>.T_1 and T_2 are the processing time at the UE and gNB respectively, T_ack is the TTI duration of ACK/NACK and T_p is the propagation time from gNB to UE or vice versa.Based on these considerations, in FDD LTE up to 8 parallel HARQ processes are used both in the downlink and the uplink. The numerology is fixed, hence there is just one HARQ process configuration. It is agreed that NR <cit.><cit.> should be able to supportmore than one HARQ processes for a given UE and one HARQ process for some UEs. It means that, depending on the use case, different HARQ configurations can be supported. This gives us a desired degree of freedom, when considering HARQ procedure over a sat-channel.In our scenario, the critical issue is the high RTT (16 ms) in the satellite link, which will have a significant impact in HARQ process, especially to the overall time T_HARQ. In a terrestrial link, the impact of RTT is negligible to HARQ, because the processing time at gNB/UE is much larger than the propagation delay. Whereas, in the considered LEO system, RTT now has a major role in the HARQ protocol. If we consider a total processing time of 8 ms (like in LTE) and a 1 ms TTI we can calculate:N_HARQ,min^SAT≥T_HARQ/TTI=16 + 8 /1 =24As we can see, NR should be able to support 24 parallel HARQ processes in order to be compatible and to offer the desired peak data rate over a LEO satellite link. The possible large number of HARQ processes would have an impact on the following: i) soft-buffer size of the UE (N_buffer∝ N_HARQ,max*TTI); ii) bit-width of DCI fields (3 in LTE because there are 8 processes). On one hand, increasing UE buffer size can be very costly and on the other hand larger bit-widths of DCI field would lead to large DL control overhead. These issues should be treated carefully and solutions must be found.§ PROPOSED SOLUTIONS§.§ Waveform In order to deal with large Doppler shifts in the satellite channel, we have to find valuable solutions without changing 5G waveforms to assure forward compatibility. We propose to equip RNs with GNSS receivers, being able to estimate the position of the satellite. By doing so, we can compensate the Doppler shift significantly. For sure, whenever we deal with an estimator, we will have some error characterized by a variance of error. Due to the estimation error, a residual Doppler shift will be generated.Let us characterize the residual Doppler as a function of estimation error as follows: Given the position of the satellite X_SAT(t) at a certain time, we can calculate the distance |d(t)| and the elevation angle, θ(t) by simple geometry, as shown in Fig. <ref>. By substituting this value of elevation angle, we obtain the real Doppler shift. However, due to an error in estimating the position (R_B), we will have a new duplet (|d_e(t)|,θ_e(t)). These values can be related to the real duplet (|d(t)|,θ(t)) as follows:|d_e(t)|*cos(θ_e(t))=|d(t)|*cos(θ(t)) + R_B |d(t)|^2=|d_e(t)|^2+R_B^2 - 2*R_B*|d_e(t)|*cos(θ_e(t))By solving (7) and (8) we obtain:cos(θ_e(t)) = |d(t)|*cos(θ(t))+R_B/√(|d(t)|^2+R_B^2 + 2*R_B*|d(t)|*cos(θ(t)))By plugging this result to equation <ref>, we finally obtain the residual Doppler f̃_̃d̃(̃t̃)̃. The result is illustrated in Fig. <ref> for values of 0≤ R_B≤ 50 and 45≤θ≤ 90. The results are summarized in Table 1. It is worth mentioning that the highest Doppler is experienced at 90 degrees of elevation angle under a fixed estimation error. we canconclude that,increasing the SCS, larger estimation errors can be tolerated. Therefore, it is preferable a waveform with high SCS in RN-DgNB interface.§.§ RA Procedure For the RA procedure, the RAR window size and TA in case of RN-DgNB RA shall be considered.§.§.§ RAR window sizeTwo solutions can be identified.Option 1: NR should fix a value of the RAR window, taking into account the worst scenario, having the largest delay. Hence RAR window size should be larger than 16 ms for all use cases. This will allow to deal with large delays in a satellite channel, but will impose useless high delays in the network, where a terrestrial channel is used.Option 2: RNs, before starting the RA procedure, should be informed from their respective downlink control channel (R-PDCCH), about the presence of a satellite link. Depending on this, the RNs should be able to change the RAR window size. Clearly, adding more bits in the PDCCH will increase the overhead, but in this case it is needed just one bit more to indicate the presence of a sat-channel or not. §.§.§ Taming advance (TA)Two solutions are proposed.Option 1: the length of preamble format is also the indication of the maximum supported cell size. NR should use larger preamble formats in order to cope with larger TA due to satellite channel. A detailed analysis is needed about the preamble format.Option 2: by equipping RNs with GNSS receivers, they can estimate the satellite position and their distance. Therefore, it is possible to estimate the TA before sending the preamble to DgNB. By doing so, there is no need to change the preamble format. Due to the estimation error, RNs with receive the exact TA in the RAR message, which is going to be much lower. An estimation error up to 100 km (corresponding to the largest cell size) can be supported in order to keep the same preamble format.§.§ HARQ Retransmissions In order to deal with the large possible number of HARQ re-transmission, 4 solutions are identified. §.§.§ Increased number of HARQ processes and buffer sizeAs we highlighted in the previous section, for assuring continuous transmission and achieving peak data rate, we need at least 24 parallel HARQ processes. The increase on the buffer size and whether the UE will have this capability of memory, need further analysis. §.§.§ Increased number of HARQ processes with reduced buffer sizeIt is possible to increase the number of HARQ processes, by maintaining the buffer size under control. In LTE, we reserve 1 bit for ACK/NACK. By enhancing the feedback information using 2 bits<cit.>, we can inform the transmitter on how close the received TB is to the originally TB. Therefore, the number of re-transmission will be reduced, because the transmitter can add the redundant bits according to the feedback information. Reducing the number of re-transmission has a direct impact on the buffer memory size. §.§.§ Reduced number of HARQ processes with reduced buffer sizeIn case the first two solutions are not possible due to UE capability, another solution would be to keep the number of HARQ processes under a certain limit. It is worth noticing that the throughput of transmission will be reduced. §.§.§ No HARQ protocolIn case HARQ in not supported due to large delay in the satellite channel, a solution would be to replicate the TB a certain amount of times before sending to the receiver. This will reduce the impact of the delay, but the throughput will be reduced significantly.§ CONCLUSIONSIn this paper, we proposed a 5G-based LEO mega-constellation architecture to foster the integration of satellite and terrestrial networks in 5G. By referring to the latest 3GPP specifications, we addressed the impact of typical satellite channel impairments as large delays and Doppler shifts on PHY and MAC layer procedures, as well as on the waveform numerology. The impact of the Doppler shift on the waveform and the effect of large delays in RN-DgNB RA procedure, can be estimated and compensated by accurate GNSS receivers.Increasing SCS in the backhaul link, allows larger tolerated estimation error. We also propose that RNs should be informed by control channel (R-PDCCH) about the presence of the satellite link, in order to increase RAR window size. To achieve the peak data rate, at least 24 HARQ parallel processes are needed, posing a big challenge on UE buffer size and DCI field. Some solutions have been proposed for keeping the number of HARQ processes and buffer size under control. 15GPPP_2 5GPPP, “5G Empowering Vertical Industries,” February 2016. Aavailable at: https://5g-ppp.eu/roadmaps/5GPPP_2 5GPPP, “5G Innovations for new Business Opportunities,” March 2017. Available at: https://5g-ppp.eu/roadmaps/3GPPSA_1 3GPP TR 22.862 v14.1.0 (2016-09), “3rd Generation Partnership Project; Feasibility Study on New Services and Markets Technology Enablers for Critical Communications,” September 2016.3GPPSA_2 3GPP TR 22.891 v14.2.0 (2016-09), “3rd Generation Partnership Project; Feasibility Study on New Services and Markets Technology Enablers,” September 2016.VITAL H2020-ICT-2014-1 Project VITAL, Deliverable D2.3, “System Architecture: Final Report,” June 2016.Intro1 F. Bastia et al., “LTE Adaptation for Mobile Broadband Satellite Networks,” EURASIP Journal on Wireless Communications and Networking, November 2009.Intro2 G. Araniti, M. Condoluci, and A. Petrolino, “Efficient Resource Allocation for Multicast Transmissions in Satellite-LTE Network,” in IEEE Glob. Comm. Conf. (GLOBECOM), December 2013.Intro3 M. Amadeo, G. Araniti, A. 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"authors": [
"Oltjon Kodheli",
"Alessandro Guidotti",
"Alessandro Vanelli-Coralli"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170526071458",
"title": "Integration of Satellites in 5G through LEO Constellations"
} |
http://arxiv.org/abs/1705.09278v1 | {
"authors": [
"Weiqiang Yang",
"Narayan Banerjee",
"Supriya Pan"
],
"categories": [
"astro-ph.CO",
"gr-qc"
],
"primary_category": "astro-ph.CO",
"published": "20170525174955",
"title": "Constraining a dark matter and dark energy interaction scenario with a dynamical equation of state"
} |
|
𝐱 ŁL Amirsina Torfi, Jeremy Dawson, Nasser M. Nasrabadi West Virginia University [email protected],{jeremy.dawson,nasser.nasrabadi}@mail.wvu.eduText-Independent Speaker Verification Using 3D Convolutional Neural Networks [============================================================================ In this paper, a novel method using 3D Convolutional Neural Network (3D-CNN) architecture has been proposed for speaker verification in the text-independent setting. One of the main challenges is the creation of the speaker models. Most of the previously-reported approaches create speaker models based on averaging the extracted features from utterances of the speaker, which is known as the d-vector system. In our paper, we propose an adaptive feature learning by utilizing the 3D-CNNs for direct speaker model creation in which, for both development and enrollment phases, an identical number of spoken utterances per speaker is fed to the network for representing the speakers' utterances and creation of the speaker model. This leads to simultaneously capturing the speaker-related information and building a more robust system to cope with within-speaker variation. We demonstrate that the proposed method significantly outperforms the traditional d-vector verification system. Moreover, the proposed system can also be an alternative to the traditional d-vector system which is a one-shot speaker modeling system by utilizing 3D-CNNs. Speaker verification, 3D convolutional neural networks, text-independent, speaker model § INTRODUCTION Speaker Verification (SV), is verifying the claimed identity of a speaker by using their voice characteristics as captured by a recording device such as a microphone. The concept of SV belongs within the general area of Speaker Recognition (SR), and can be subdivided to text-dependent and text-independent types. In text-dependent mode, a predefined fixed text, such as a pass-phrase, is employed for all stages in speaker verification process. One the other hand, in text-independent SV, no prior constraints are considered for the spoken phrases by the speaker, which makes it much more challenging compared to text-dependent scenario. Generally, there are three steps in a SV process: development, enrollment, and evaluation. In the development step, the background model will be created for the speaker representation. In the enrollment step, the speaker models of new users are generated using the background model. Finally, in the evaluation phase, the claimed identity of the test utterances should be confirmed/rejected by comparing with available previously generated speaker models.Successful SV methods often employ unsupervised generative models such as the Gaussian Mixture Model-Universal Background Model (GMM-UBM) framework <cit.>. Some models, such as i-vector, based on GMM-UBM, have demonstrated effectiveness as well <cit.>. Although the aforementioned models proved to be effective for SV tasks, the main issue is the disadvantage of unsupervised methods in which the model training is not necessarily supervised by speaker discriminative features. Different approaches, such as the SVM model for GMM-UBMs <cit.> and PLDA i-vectors model <cit.>, have been developed as discriminative models to supervise the generative framework and demonstrated promising results. Recent research efforts on deep learning approaches have proposed data driven feature learning methods. Inspired by using Deep Neural Networks (DNNs) in Automatic Speech Recognition (ASR) <cit.>, other research efforts have been conducted on the application of DNNs in SR <cit.>, and have shown to be promising for learning task-oriented features. Convolutional neural networks (CNNs) have been applied for feature extraction, which has often been utilized for 2D inputs. However, 3D CNN architectures have recently been employed for action recognition <cit.> and audio-visual matching <cit.>. For the work presented here, we use 3D CNNs to capture within-speaker variations in addition to extracting the spatial and temporal information jointly.In this paper, we focus on the text-independent scenario where no prior information is available in the context of the speakers' utterances for all stages. The difficulty of the chosen setting is that the proposed system should be able to distinguish between the speaker and speech related information as different utterances (context-wise) from the same speaker that are fed to the system. In this paper, we extend the application of DNN-based feature extraction to a text-independent SV task, the objective of which is to build a speaker-related bridge between the development and enrollment stages to create more generalizable speaker models. Our source code is available online[https://github.com/astorfi/3D-convolutional-speaker-recognition] as an open source project <cit.>.§ RELATED WORKS We investigate the application of Convolutional Neural Networks <cit.> to speaker recognition which recently has been used in speech processing <cit.>. In previous studies regarding speaker verification, like those reported in <cit.>, DNNs have been investigated for text-independent setup. However, none of these efforts investigated 3D-CNN architectures. In some research efforts, such as <cit.>, CNNs and Locally Connected Networks (LCNs) have been investigated for SV. However, they only consider the text-dependent setup. In some other works, such as <cit.>, DNNs have been utilized as feature extractors which are then used to create speaker models for the SV process. In <cit.>, <cit.> and <cit.>, the pre-trained DNN is used as the feature extractor to create a speaker model based on averaging the representative feature from enrollment utterances by the same speakers, known as a d-vector system for SV. We propose to employ the intrinsic characteristics of a CNN to capture a cohort various speaker utterances that can be used for creating the speaker models. To the best of our knowledge, this is the first research effort in which the 3D-CNNs are used for simultaneous feature extraction and speaker model creation for both the development and enrollment stages. The proposed method creates identical speaker representation frameworks for both the stages, which has practical and computational advantages.§ SPEAKER VERIFICATION USING DNN The speaker verification protocol should be addressed by using DNN. The general process has been explained in Section <ref>. In this section, we described the three phases of development, enrollment and evaluation as follows:Development: In the development phase, a background model must be built for speaker representation extracted from the speakers' utterances. The representation is generated by the model. In the case of a DNN, the input data representation can be built using the extracted speech feature maps of the speaker utterances. Ideally, during the training, the model loss (e.g., Softmax) directs the ultimate representations to be speaker discriminative. This phase has been under investigation by several research efforts, using approaches such as i-vectors <cit.> and d-vectors <cit.>, which are the state-of-the-art. The main idea is to use a DNN architecture as a speaker feature extractor operating at frame- and utterance-level for speaker classification. Enrollment: In the enrollment phase, for each speaker, a distinct model will be built. Each speaker-specific model will be built upon the utterances provided by the targeted speaker. In this stage, each utterance (or frame, depending on the representation level) will be fed to the supervised trained network in the development phase and the final output (the output of one of the layers prior to the softmax layer, whichever provides better representation) will be accumulated for all utterances (or frames). The final representation of the utterance projected by the outputs of the DNN is called the d-vector. For speaker model creation, all d-vectors of the utterances of the targeted speaker can be averaged to generate the speaker model. However, instead of the averaging typically used in a d-vector system, we propose an approach in which the architecture generates the speaker model in one shot by capturing speaker utterances from the same speaker (Section <ref>). Evaluation: During the model evaluation stage, each test utterance will be fed to the network and its representation will be extracted. The main setup for verification is the one-vs-all setup where the test utterance representation will be compared to all speaker models and the decision will be made based on the similarity score. In this setup, false rejection and false acceptance rates are investigated as the main error indicators. The false rejection/acceptance rates depend on the predefined threshold. The Equal Error Rate (EER) metric projects the error when the two aforementioned rates are equal.§ BASELINE APPROACH In this section, we describe the baseline method. The architecture that we used as the baseline is a Locally-Connected Network (LCN) as used in <cit.>. This network uses locally-connected layers <cit.> for low-level features extraction and fully-connected layers as the high-level feature generators. We used PReLU activation instead of the ReLU and it demonstrated more stability is training and improving the performance <cit.>. The locally connected layer is utilized to enforce sparsity in the first hidden layer. The cross-entropy loss has been used as the criterion for the training. After the training stage, the network parameters will be fixed. Utterance d-vectors are extracted by averaging the output vectors of the last layer (prior to Softmax and without the PReLU non-linearity elimination). For enrollment, the speaker model is generated using the averaged d-vectors of the utterances belonging to the speaker. Ultimately, during the evaluation phase, the similarity score is obtained by computing the cosine similarity between the speaker model and the test utterance.To operate the DNN-based SV at the utterance level rather than the frame level, the stacked frames of the audio stream are fed to the DNN architecture and one d-vector will be directly generated for each utterance. The baseline architecture is a locally-connected layer, followed by three fully-connected layers and a softmax layer at the end. The output is a softmax layer and its cardinality is the number of speakers present in the development set. Each fully connected layer has 256 hidden units and the locally connected layer uses 8×8 local patches in which each of the hidden units' activations is obtained by processing a patch, rather than the whole visible features as in conventional DNNs.§ PROPOSED ARCHITECTURE Different issues may arise for the utilized baseline method. The frame level representation may not extract enough context of speaker-related information. Even the utterance level representation, achieved by simple stacking of the frames, can be highly affected by the non-speaker related information, such as the variety of the spoken words in the text-independent setup. Additionally, the Softmax layer, along with cross-entropy loss, requires abundant samples per speaker to optimally generate the speaker-discriminative model. To tackle the aforementioned issues, we propose a 3D CNN architecture which is aimed to simultaneously capture the spatial and temporal information. Our proposed approach for softmax criterion issue is to generate highly overlapped utterances of each speaker to transform the problem to a semi text-dependent problem such that the neighbor utterances from a spoken sentence be highly overlapped.The general framework which is used for training, enrollment, and evaluation with the utterance level as input, is shown in Fig. <ref>, and the 3D-CNN architecture is described in Table <ref>. The spatial size of the kernels is reported as D × H × W where H and W are the kernel sizes in height (temporal) and width (frequency) dimensions, respectively. The parameter D is the kernel dimension alongside the depth, which determines in how many utterances information is captured for the specific convolutional operation. The variety of spoken words can become a major challenge in this scenario, as one can claim that the different spoken words can be inferred differently by softmax, even when being spoken by the same speaker. This leads to an obstacle when generalization of the background model is desired. To handle this problem, we proposed to capture different within-speaker utterances simultaneously. By doing so, ideally, we expect the network to be able to extract the speaker-discriminative features, and yet be able to capture the within-speaker variations. Our proposed method is to stack the feature maps for several different utterances spoken by the same speaker when used as the input to the CNN. So, instead of utilizing single utterance (in the development phase) and building speaker model based on the averaged representative features of different utterances from the same speaker (d-vector system), for both stages, we use the same number of utterances, all of which are concurrently fed into the proposed 3D-CNN architecture. In our architecture, pooling operations are only applied in the frequency axis (domain) to keep the useful temporal information within the time frames. This approach is inspired by the discussions in [5] in which downsampling in time is avoided. We use stride 2 for low-level convolutional layers to perform a simple reduction in capturing highly overlapped features. To create a more computationally efficient architecture, instead of cubic kernels, successive 2D kernels are used <cit.>. However, we are effectively using 3D kernels. § EXPERIMENTS In the training phase, the variance scaling initializer that has been recently developed for weight initialization <cit.>, is used in our architecture. Batch normalization <cit.> has also been used for improving the training convergence and better generalization. The output of the last layer (FC5) will be fed to the softmax layer which has the cardinality of N=511, where N is the number of speakers in the development phase. For the enrollment and evaluation stages, 100 subjects have been used and the speaker utterances are split into two equal parts for two aforementioned phases. All layers except the last one are followed by PReLU activation.§.§ Evaluation and verification metricIn this paper, we evaluate the experimental results using the Receiver Operating Characteristic (ROC) and Precision-Recall (PR) curves characteristics. The ROC curve consists of the Validation Rate (VR) and False Acceptance Rate (FAR). All match pairs (X_P_1,X_P_2), i.e., the ones of the same identity are denoted with 𝒫_gen whereas all pairs non-match are denoted as 𝒫_imp. Assume D_W is the Euclidean distance between the outputs of the network with (X_P_1,X_P_2) as the input. So true positive and false acceptance can be defined as below: TP(τ) = { (X_P_1,X_P_2)∈𝒫_gen, D_W≤τ}. FA(τ) = { (X_P_1,X_P_2)∈𝒫_imp, D_W≤τ}. Here, TP(τ) demonstrate the test samples that are classified as match pairs and FA(τ) are non-match pairs which are incorrectly classified as positive pairs. So the True Positive Rate (TPR) and the False acceptance rate (FAR) will be calculated as follows: TPR(τ)=TP(τ)/𝒫_gen,FAR(τ)=FA(τ)/𝒫_imp. The metric employed for performance evaluation is the Equal Error Rate (EER) which is the point that the False acceptance Rate and False Rejection Rate[1-TPR] become equal. Moreover, Area Under the Curve (AUC) has been utilized as well as an indication of accuracy, which is the area under the ROC curve.§.§ Dataset The dataset that has been used for our experiments is the WVU-Multimodal 2013 dataset <cit.>. The audio part of WVU-Multimodal dataset consists of up to 4 sessions of interviews for each of the 1083 different speakers. The WVU-Multimodal dataset includes different modalities of data collected over a period from 2013 to 2015. The audio part of data consists of both scripted and unscripted voice samples. For the scripted samples, the participants read a fixed sample of text. For the unscripted samples, the participants answer interview questions that require conversational responses. We only use the scripted audio samples, as only the voice of the subject of interest is present in the sample. Voice Activity Detection (VAD) has been performed on all audio samples to eliminate the silent parts of speech <cit.>. §.§ Data representationThe MFCC[Mel-frequency cepstral coefficients] features can be used as the data representation of the spoken utterances at the frame level. However, a drawback is their non-local characteristics due to the last DCT[Discrete Cosine Transform] operation for generating MFCCs. This operation disturbs the locality property and is in contrast with the local characteristics of the convolutional operations. The employed approach in this paper is to use the log-energies, which we call MFECs[Mel-frequency energy coefficients]. The extraction of MFECs is similar to MFCCs by discarding the DCT operation. SpeechPy package has been used for speech feature extraction <cit.>.The temporal features are overlapping 20ms windows with the stride of 10ms, which are used for the generation of spectrum features. From a 0.8-second sound sample, 80 temporal feature sets (each forms a 40 MFEC features) can be obtained which form the input speech feature map. Each input feature map has the dimensionality of ζ×80×40 which is formed from 80 input frames and their corresponding spectral features, where ζ is the number of utterances used in modeling the speaker during the development and enrollment stages. By default we set ζ=20. The data input architecture is shown in Fig. <ref>.For the evaluation phase, since we need ζ utterances for utterance representation[The CNN architecture has been train in such a way to take ζnumber of channels], and we only have a single utterance, we copy each test utterance feature map ζ times, alongside its depth, to have the desired input representation. It is equivalent to artificially provide ζ highly correlated representations of an utterance to capture the speaker information. §.§ Effect of the number of utterances The enrollment representations are provided by feeding-forward the utterances for a speaker through the trained network in the development stage to generate the speaker model. The number of utterances per speaker (ζ) can affect the model that is built based upon the speaker utterances' representation. Here, we investigate the effect of the number of speaker-specific enrollment utterances on the evaluation phase. The results are demonstrated in Table <ref>.It is worth noting that the ζ parameter must be the same for development and enrollment stages. As it can be observed from Table <ref>, increasing the number of speaker utterances does not necessarily create a better speaker model, although intuitively the opposite is more acceptable to common sense. One possible reason is that as the number of speaker utterances increases, a deeper feature cube represents the speaker in the development phase and distinguishing between the speaker and non-speaker related information becomes more complex due to possible over-fitting. Moreover, due to memory problem increasing the number of speaker utterances is not possible and fewer speaker utterances is desired computationally. §.§ Proposed architecture vs other methods For this experimental setup, we investigate the effect of frame-level or utterance-level representation. For the utterance-level, the entire input feature map will be fed to the network, but in the frame-level, the weight update will be performed per frame of input, which can belong to any speaker with the available class label. Moreover, we compare our results with the traditional i-vector system <cit.> as well as the state-of-the-art in text-dependent speaker verification <cit.>. The method presented by <cit.>, trains the system in an end-to-end fashion using Long Short-Term Memory (LSTM) recurrent neural networks in which no enrollment stage is required. In our experiments in the text-independent setting, the proposed method outperforms the end-to-end training fashion. As can be observed from Table <ref>, our proposed 3D-CNN architecture significantly outperforms all the other methods. Our proposed method is, in essence, a one-shot representation method for which the background speaker model is created simultaneously with learning speaker characteristics.In general, an end-to-end system is expected to learn the verifier (or classifier) and features simultaneously in which usually a cost function in consistent with the evaluation criterion is utilized. However, our experiment for the text-independent scenario in which non-speaker related components are more dominant to speaker information compared to the text-dependent mode, adaptive feature learning without end-to-end training is empirically proven to be more effective. The reason that we call our feature learning adaptive is that our proposed feature learning method is customized for the specific SV tasks with feeding an ensemble of speaker utterances directly. § ACKNOWLEDGEMENT This work is based upon a work supported by the Center for Identification Technology Research and the National Science Foundation under Grant #1650474.§ CONCLUSION In this paper, for text-independent speaker verification, we have proposed a novel 3D-CNN-based speaker and utterance representative model. A 3D-CNN architecture has been trained as a feature extractor for direct modeling of the speakers. Experimental results demonstrated that the proposed method can outperform the d-vector SV system significantly by simultaneously capturing the speaker-related information and the within-speaker variation. The proposed architecture, outperformed the d-vector method by 6% in Equal Error Rate (EER) for our default experimental settings. IEEEbib | http://arxiv.org/abs/1705.09422v7 | {
"authors": [
"Amirsina Torfi",
"Jeremy Dawson",
"Nasser M. Nasrabadi"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170526031908",
"title": "Text-Independent Speaker Verification Using 3D Convolutional Neural Networks"
} |
Convergence of the Population Dynamics algorithm in the Wasserstein metric Mariana Olvera-Cravioto December 30, 2023 ============================================================================We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the special endogenous solution to a stochastic fixed-point equation of the form:R𝒟=Φ( Q, N, { C_i }, {R_i}),where (Q, N, {C_i}) is a real-valued random vector with N ∈ℕ, and {R_i}_i ∈ℕ is a sequence of i.i.d. copies of R, independent of (Q, N, {C_i}); the symbol 𝒟= denotes equality in distribution. Specifically, we show its convergence in the Wasserstein metric of order p (p ≥ 1) and prove the consistency of estimators based on the sample pool produced by the algorithm.Keywords: Population dynamics; iterative bootstrap; Wasserstein metric; distributional fixed-point equations. § INTRODUCTIONWe study an iterative bootstrap algorithm, known as the “population dynamics" algorithm, that can be used to efficiently generate samples of random variables whose distribution closely approximates that of the so-called special endogenous solution to a stochastic fixed-point equation (SFPE) of the form:R𝒟=Φ( Q, N, { C_i }, {R_i}) ,where (Q, N, {C_i}) is a real-valued random vector with N ∈ℕ = {0, 1, 2, …}, and {R_i}_i ∈ℕ is a sequence of i.i.d. copies of R, independent of (Q, N, {C_i}). These equations appear in a variety of problems, ranging from computer science to statistical physics, e.g.: in the analysis of divide and conquer algorithms such as Quicksort <cit.> and FIND<cit.>, the analysis of Google's PageRank algorithm <cit.>, the study of queueing networks with synchronization requirements <cit.>, and the analysis of the Ising model <cit.>, to name a few. In general, SPFEs of the form in (<ref>) can have multiple solutions, but in most cases we are interested in computing those that can be explicitly constructed on a weighted branching process, known as endogenous solutions. In some cases, even the endogenous solution is not unique <cit.>, but characterizing all endogenous solutions can be done using the special endogenous solution, which is the only attracting solution, and can be constructed by iterating (<ref>) starting from some well-behaved initial distribution. This work focuses on the analysis of a simulation algorithm that can be used to generate samples from a distribution that closely approximates that of the special endogenous solution to a variety of SFPEs. The need for such an approximate algorithm lies on the numerical complexity of simulating even a few generations of a weighted branching process using naive Monte Carlo methods. The population dynamics algorithm, described in 14.6.4 in <cit.> and 8.1 in <cit.>, circumvents this problem by resampling with replacement from previously computed iterations of (<ref>), i.e., by using an iterative bootstrap technique. However, as is the case with the standard bootstrap algorithm, the samples obtained are neither independent nor exactly distributed according to the target distribution, which raises the need to study the convergence properties of the algorithm.Before presenting the algorithm and stating our main results, it may be helpful to describe in more detail some of the examples mentioned above. Throughout the paper, we use x ∨ y = max{x, y}and x ∧ y = min{x,y} to denote the maximum and the minimum, respectively, of x and y.* The linear SFPE or “smoothing transform": R𝒟= Q + ∑_i=1^N C_i R_i,appears in the analysis of the number of comparisons required by the sorting algorithm Quicksort <cit.>, and can also be used to describe the distribution of the ranks computed by Google's PageRank algorithm on directed complex networks <cit.>. * The maximum SFPE or “high-order Lindley equation": R𝒟= Q ∨⋁_i=1^N C_i R_i, equivalently, X 𝒟= T ∨⋁_i=1^N ( ξ_i + X_i),arises as the limiting waiting time distribution on queueing networks with parallel servers and synchronization requirements <cit.> and in the analysis of the branching random walk <cit.>. * The discounted tree-sum SFPE: R 𝒟= Q + ⋁_i=1^N C_i R_iappears in theworst-case analysis of the FIND algorithm <cit.> and the analysis of the “discounted branching random walk" <cit.>. * The “free-entropy" SFPE:R 𝒟= Q + ∑_i=1^Narctanh ( tanh(β) tanh( R_i))characterizes the asymptotic free-entropy density in the ferromagnetic Ising model on locally tree-like graphs <cit.>. In this case, C_i ≡tanh(β) for all i ≥ 1,β≥ 0 represents the “inverse temperature", and Qthe magnetic field.* Although the analysis presented here does not directly apply to this case, we mention that the population dynamics algorithm can also be used to simulate the fixed points of the belief propagation equations on random graphical models <cit.>: R 𝒟=Φ( Q, N, { C_i}, {R̃_i}) andR̃𝒟=Ψ( Q̃, Ñ, {C̃_i}, { R_i}),where the {R̃_i} are i.i.d. copies of R̃ independent of the vector (Q, N, {C_i}) and the {R_i} are i.i.d. copies of R independent of the vector (Q̃, Ñ, {C̃_i }), with Φ and Ψ potentially different. We refer the reader to<cit.> for even more examples, including some involving minimums. The existence and uniqueness of solutions to any of these SFPEs is in itself a non-trivial problem. We refer the reader again to <cit.> for a broad survey of known results and open problems on this topic. The most well-studied equations are the linear (<ref>) and maximum (<ref>) SFPEs, which have been extensively studied in <cit.> and <cit.>, respectively.However, to provide some context to where the population dynamics algorithm fits in, we briefly mention that the existence of solutions is often established by showing that the transformation T that maps the distribution μ on ℝ to the distribution of Φ( Q, N, { C_i}, { X_i}),where the {X_i} are i.i.d. random variables distributed according to μ, independent of the vector (Q, N, {C_i}), is strictly contracting under some suitable metric. Note that in this case, we have that the sequence of probability measures μ_n+1 = T(μ_n) converges as n →∞ to a fixed point of (<ref>).Moreover, as long as the initial distribution μ_0 has sufficiently light tails, one can show that {μ_n} converges to the special endogenous solution to (<ref>), and the contracting nature of T provides an upper bound of the form d(μ_n, μ) ≤ d(T(μ_n-1), T(μ)) ≤ c d(μ_n-1, μ) ≤ c^n d(μ_0,μ),n = 1, 2, …,for some constant 0 < c < 1, where d is the distance under which T is a contraction.As will be discussed in more detail later (see Examples <ref>), all the examples provided earlier define contractions under d_p, the Wasserstein metric of order p, for some p ≥ 1. For completeness, we also include a result (Theorem <ref>) that gives easy to verify conditions guaranteeing that E[ | R^(k) - R |^β] ≤ c^kfor some 0 < c < 1, where R^(k) and R have distributions μ_k and μ, respectively. It follows that from a computational point of view, it suffices to have an algorithm for computing μ_k for a fixed number of iterations k ∈ℕ.The population dynamics algorithm produces a sample of observations approximately distributed according to μ_k, which can also be helpful in searching for the existence of endogenous solutions, as stated in <cit.>. We now describe how to obtain an exact sample of μ_k, which will also make clear the need for a computationally efficient method. §.§ Constructing endogenous solutions on weighted branching processes As mentioned earlier, the attracting endogenous solution to (<ref>), provided it exists, can be constructed on a structure known as a weighted branching process <cit.>. We now elaborate on this point.Let ℕ_+ = {1, 2, 3, …} be the set of positive integers and let U = ⋃_k=0^∞ (ℕ_+)^k be the set of all finite sequences i = (i_1, i_2, …, i_n), n≥ 0, where by convention ℕ_+^0 = {∅} contains the null sequence ∅. To ease the exposition, we will use ( i, j) = (i_1,…, i_n, j) to denote the index concatenation operation. Next, let (Q, N, {C_i}_i≥ 1) be a real-valued vector with N ∈ℕ. We will refer to this vector as the generic branching vector. Now let { (Q_ i, N_ i, {C_( i, j)}_j≥ 1}_ i∈ U be a sequence of i.i.d. copies of the generic branching vector. To construct a weighted branching process we start by defining a tree as follows: let A_0 = {∅} denote the root of the tree, and define the nth generation according to the recursionA_n = { ( i, i_n) ∈ U:i∈ A_n-1, 1 ≤ i_n ≤ N_ i},n ≥ 1.Now, assign to each node i in the tree a weight Π_ i according to the recursionΠ_∅≡ 1, Π_( i, i_n) = C_( i, i_n)Π_ i,n ≥ 1,see Figure <ref>. If P(N< ∞)=1 and C_i ≡ 1 for all i ≥ 1, the weighted branching process reduces to a Galton-Watson process.To generate a sample from μ_k we first need to fix the initial distribution μ_0, e.g., by letting μ_0 be the probability measure of a constant, say zero or one. Now construct a weighted branching process with k generations, and let {R^(0)_ i}_ i∈ A_k be i.i.d. random variables having distribution μ_0. Next, define recursively for each i∈ A_k-r, 1 ≤ r ≤ k, R^(r)_ i = Φ( Q_ i, N_ i, { C_( i,j)}_j ≥ 1, { R^(r-1)_( i,j)}_j ≥ 1).The random variable R^(k)_∅ is distributed according to μ_k, and its generation requires on average (E[N])^k i.i.d. copies of the generic branching vector (Q, N, {C_i}_i≥ 1). It follows that if the goal was to obtain an i.i.d. sample of size m from distribution μ_k, one would need to generate on average m (E[N])^k copies of the generic branching vector. However, in applications one typically has E[N] > 1, e.g., N ≡ 2 for Quicksort, E[N] ≈ 30 in the analysis of PageRank on the WWW graph, and E[N] can be in the hundreds for MapReduce implementations related to the maximum SFPE. This makes the exact simulation of R^(k) using a weighted branching process impractical. The population dynamics algorithm, described below, uses a bootstrap approach to produce a sample of size m of random variables that are approximately distributed according to μ_k, and that although not independent, can be used to obtain consistent estimators for moments, quantiles and other functions of μ_k.§.§ The population dynamics algorithm The population dynamics algorithm is based on the bootstrap, i.e., in the idea of sampling with replacement random variables from a common pool.As described above, the algorithm starts by generating a sample of i.i.d. random variables having distribution μ_0, with the difference that when computing the next level of the recursion, it samples with replacement from this pool as needed by the map Φ. In other words,to obtain a pool of approximate copies of R^(j) we bootstrap from the pool previously obtained of approximate copies of R^(j-1). The approximation lies in the fact that we are not sampling from R^(j-1) itself, but from a finite sample of conditionally independent observations that are only approximately distributed as R^(j-1). The algorithm is described below. Let (Q,N, {C_r}) denote the generic branching vector defining the weighted branching process. Let k be the depth of the recursion that we want to simulate, i.e., the algorithm will produce a sample of random variables approximately distributed according to μ_k. Choose m∈ℕ_+ to be the bootstrap sample size. For each 0 ≤ j ≤ k, the algorithm outputs 𝒫^(j,m)≜(R̂^(j,m)_1, R̂^(j,m)_2,…,R̂^(j,m)_m), which we refer to as the sample pool at level j.* Initialize: Set j=0. Simulate a sequence {R^(0)_i}_i = 1^m of i.i.d. random variables distributed according to some initial distribution μ_0. Let R̂^(0,m)_i = R^(0)_i for i = 1, …, m.Output𝒫^(0,m)=(R̂^(0,m)_1, R̂^(0,m)_2,…,R̂^(0,m)_m) and update j = 1. * While j ≤ k:* Simulate a sequence { (Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1 ) }_i =1^m of i.i.d. copies of the generic branching vector, independent of everything else. * LetR̂^(j,m)_i = Φ( Q_i^(j), N_i^(j), { C_(i,r)^(j)}, {R̂^(j-1,m)_(i,r)}),i=1,…,m,where the R̂^(j-1,m)_(i,r) are sampled uniformly with replacement from the pool 𝒫^(j-1,m). * Output 𝒫^(j,m)=(R̂^(j,m)_1, R̂^(j,m)_2,…,R̂^(j,m)_m) and update j = j+1.We conclude this section by pointing out that the complexity of the algorithm described above is of order k m, while the naive Monte Carlo approach described earlier, which consists on sampling m i.i.d. copies of a weighted branching process up to the kth generation, has order (E[N])^k m.Our main results establish the convergence of the algorithm in the Wasserstein metric of order p (p ≥ 1), as well as the consistency of estimators constructed using the pool 𝒫^(k,m). The following section contains all the statements, and the proofs are given in Section <ref>. § MAIN RESULTSWe start by defining the Wasserstein metric of order p. Let M(μ, ν) denote the set of joint probability measures on ℝ×ℝ with marginals μ and ν. Then, the Wasserstein metric of order p (1 ≤ p < ∞) between μ and ν is given byd_p(μ, ν) = inf_π∈ M(μ, ν)(∫_ℝ×ℝ | x - y |^p d π(x, y) )^1/p.An important advantage of working with the Wasserstein metrics is that on the real line they admit the explicit representation d_p(μ, ν) = ( ∫_0^1 | F^-1(u) - G^-1(u) |^p du )^1/p ,where F and G are the cumulative distribution functions of μ and ν, respectively, and f^-1(t) = inf{ x ∈ℝ: f(x) ≥ t} denotes the generalized inverse of f. It follows that the optimal coupling of two real random variables X and Y is given by (X, Y) = (F^-1(U), G^-1(U)), where U is uniformly distributed in [0, 1].With some abuse of notation, we use d_p(F,G) to denote the Wasserstein distance of order p between the probability measures μ and ν, where F(x) = μ((-∞, x]) and G(x) = ν((-∞, x]) are their corresponding cumulative distribution functions.Our main results establish the convergence of d_p(F̂_k,m, F_k) as m →∞, both in mean and almost surely, where F̂_k,m(x) = 1/m∑_i=1^m 1(R̂_i^(k,m)≤ x) and F_k(x) = μ_k((-∞, x]),k ∈ℕ,and 𝒫^(k,m) = ( R̂_1^(k,m), …, R̂_m^(k,m)) is the pool generated by the population dynamics algorithm.The theorems are proven under two different assumptions, the first one imposing a Lipschitz condition on the mean of Φ, and the second one requiring Φ to be Lipschitz continuous almost surely.For some p ≥ 1 there exist a constant 0 < H_p < ∞ such that if { (X_i, Y_i) : i ≥ 1 } is a sequence of i.i.d. random vectors, independent of (Q, N, {C_r}), thenE[ | Φ(Q, N, {C_r}, {X_r}) - Φ(Q, N, {C_r}, {Y_r}) |^p ] ≤ H_p E[ |X_1 - Y_1|^p] .[linear0] For the linear SFPE (<ref>), it suffices that the inequality holds for { X_i } and { Y_i} having the same mean. Suppose that for any vector (q, n, {c_r}), with n ∈ℕ∪{∞}, and any sequences of numbers { x_r} and {y_r} for which Φ(q, n, {c_r}, {x_r}) and Φ(q, n, {c_r}, {x_r}) are well defined, there exists a function φ:ℝ→ℝ_+ such that| Φ(q, n, {c_r}, {x_r}) - Φ(q, n, {c_r}, {y_r}) | ≤∑_r=1^n φ(c_r) |x_r - y_r|.* To see that Assumption <ref> implies Assumption <ref>, note that Lemma 4.1 in <cit.> gives thatE[ ( ∑_r=1^N φ(C_r) |X_r - Y_r| )^p ]≤E[ ∑_r=1^N φ(C_r)^p | X_r - Y_r|^p ] + E[ ( ∑_r=1^N φ(C_r))^p ] ( E[ |X_1 - Y_1|^⌈ p ⌉ -1] )^p/(⌈ p ⌉-1)≤ 2 E[ ( ∑_r=1^N φ(C_r))^p ] E[ |X_1 - Y_1|^p ],and therefore Assumption <ref> holds with H_p = 2E[ ( ∑_r=1^N φ(C_r))^p ], provided the expectation is finite. However, much tighter bounds can be obtained for specific examples, and we can usually find p ≥ 1 such that H_p < 1. * The existence of a p ≥ 1 for which H_p < 1 is important for obtaining estimates for the rate of convergence of the algorithm that are uniform in k, and has also important implications for the convergence of R^(k)→ R as k →∞, as the next result shows. Suppose Assumption <ref> holds for some p ≥ 1, H_p < 1, and any i.i.d. sequence {(X_i, Y_i): i ≥ 1} independent of (Q,N, {C_r}). Then, provided E[ |R^(0)|^p + | Φ(Q, N, {C_r}, { 0})|^p ] < ∞, there exists a random variable R and constants 0 ≤ c_p < 1 and A_p <∞ such thatE[ | R^(k) - R |^p ] ≤ A_p c_p^k → 0,k →∞,where R^(k) and R are distributed according to μ_k and μ, respectively. For the linear SFPE (<ref>), we have that (<ref>) also holds under either of the following conditions: i) If Assumption <ref> [linear0] holds and E[Q] = E[R^(0)] = 0.ii) If E[ ( ∑_i=1^N |C_i| )^p + |R^(0) |^p + |Q|^p ] < ∞ and ρ_1 ∨ρ_p < 1, where ρ_β≜ E[ ∑_i=1^N |C_i|^β]. As the proof of Theorem <ref> shows, one can take c_p = H_p under the main set of conditions as well as under conditions (i), whereas for (ii) we have c_p = ρ_1 ∨ρ_p.As a consequence of the proof of Theorem <ref> we also obtain the following explicit bound for the moments of R^(k).Suppose Assumption <ref> holds for some p ≥ 1. In the linear case, if only Assumption <ref> [linear0] holds, suppose further that E[ R^(0)] = E[Q] = 0. Then, for any k ≥ 0,( E[ |R^(k)|^p ] )^1/p≤ A_p ∑_i=0^k-1 (H_p^1/p)^i ,where A_p = (H_p^1/p+1) ( E[ |R^(0)|^p] )^1/p + (E[|Φ(Q, N, {C_r}, { 0 })|^p ] )^1/p. Before stating the main theorems establishing the convergence of the algorithm in the Wasserstein metric, we point out how Assumptions <ref> and <ref> are satisfied by all the examples mentioned in the introduction.* The linear SFPE (<ref>) clearly satisfies Assumption <ref> with φ(t) = |t|. Moreover, for the Quicksort algorithm studied in <cit.> we have N ≡ 2, C_1 = U = 1- C_2 and Q =2Uln U + 2(1-U) ln(1-U) + 1, with U uniformly distributed on [0,1] and E[Q] = 0, in which case we can take any p ∈ℕ_+ and H_p = 1 - 2p E[ U^p-1 (1-U)] = (p-1)/(p+1) < 1 in Assumption <ref> [linear0]. Lemma <ref> also gives that E[|R^(k)|^p] is uniformly bounded in k for all p ≥ 1. For the PageRank algorithm studied in<cit.> we have { C_i }_1 ≤ i ≤ N i.i.d. and independent of N, |C_i| ≤ c < 1 a.s., and E[ |C_1|^p ] ≤ c^p/E[N] for any p ≥ 1. Hence, we can take p = 1 and H_1 = E[N] E[|C_1|] ≤ c < 1 in Assumption <ref>. Furthermore, Theorem <ref>(ii) gives that E[ |R^(k) - R|^q ] = O( γ^k) for some 0 < γ < 1 provided E[ |Q|^q + N^q] < ∞, which in turn gives the uniform boundedness of E[|R^(k)|^q].* Using the inequality | max_1≤ i ≤ n{x_i} - max_1 ≤ i ≤ n{ y_i }| ≤max_1 ≤ i ≤ n |x_i - y_i| ≤∑_i=1^n |x_i - y_i| for any real numbers {x_i, y_i} and any n ≥ 1, we obtain that the maximum SFPE (<ref>) satisfies Assumption <ref> with φ(t) = |t| as well. Furthermore, in the analysis of queueing networks with parallel servers and synchronization requirements from <cit.>, where T ≡ 0 (equivalently, Q ≡ 1), the stability condition of the system implies that H_p < 1 for any p ≥ 1 whenever the system is stable. Lemma <ref> then implies that E[|R^(k)|^p] is uniformly bounded in k for all p ≥ 1. * In the case of the discountedtree sum SFPE (<ref>), inequality (<ref>) implies that we can also take φ(t) = |t| in Assumption <ref>.For the analysis of the FIND algorithm in <cit.> in particular, we have N ≡ 2, C_1 = U = 1 - C_2 and Q ≡ 1, with U uniformly distributed on [0,1], and we can takeH_p = 2 E[U^p] =2/(p+1) < 1 for any p > 1 in Assumption <ref>. Lemma <ref> then gives that E[|R^(k)|^p] is uniformly bounded in k for all p > 1 * To see that (<ref>) also satisfies Assumption <ref> with φ(t) = |t| (in this case C_i ≡tanh(β) for all i≥ 1), let c = tanh(β) ∈ [0, 1) (since β≥ 0) and note that the function f(x)= arctanh (c tanh(x)) = 1/2ln( 1+c (e^2x-1)/(e^2x+1) /1-c (e^2x-1)/(e^2x+1)) = 1/2ln( e^2x(1+c) + 1-c /e^2x(1-c)+ 1+c )has derivativef'(x)= 4c/2(1 + c^2) + (e^2x + e^-2x) (1 - c^2) = 2c/1+c^2 + cosh(2x) (1-c^2),and therefore satisfies |f(x) - f(y)| = | f'(ξ) | |x-y| ≤ c | x - y| , for some ξ between x and y.Assumption <ref> is then satisfied for p = 1 and H_1 ≤ E[N] tanh(β), with H_p < 1 at high temperatures (β < 1/E[N]). Moreover,since |f(x)| ≤ c|x|, R^(k) in the“free entropy" SFPE (<ref>) is smaller or equal than R̃^(k), where R̃^(k) = |Q| + ∑_i=1^N tanh(β) R̃_i^(k-1).Hence, provided β < 1/E[N], Theorem <ref>(ii) gives that for any p ≥ 1 for which E[|Q|^p + N^p] < ∞, E[|R^(k)|^p] is uniformly bounded in k.Our first result establishes the convergence in mean of d_p(F̂_k,m, F_k) under the “optimal" moment conditions, that is, assuming only thatmax_0 ≤ j ≤ k E[|R^(j)|^p] < ∞. In view of Remark <ref>(ii), this is implied in all our examples by E[ ( ∑_i=1^N φ(C_i) )^p ] < ∞.This result was previously proven in<cit.> for the linear SFPE (<ref>) for p = 1.Fix 1 ≤ p < ∞ and suppose that Φ satisfies Assumption <ref> ,or Assumption <ref> [linear0], for p. Assume further that for any fixed k ∈ℕ,max_0 ≤ j ≤ k E[|R^(j)|^p] < ∞. Let { R^(j)_1, …, R^(j)_m} be an i.i.d. sample from distribution F_j, and let F_j,m denote their corresponding empirical distribution function. Then, E[ d_p(F̂_k,m, F_k)^p ]≤( ∑_r=0^k (H_p^1/p)^r)^p-1∑_j=0^k (H_p^1/p)^k-j E[d_p(F_j,m, F_j)^p],where 0 < H_p < ∞ is the same from Assumption <ref>. Moreover, if max_0 ≤ j ≤ k E[ |R^(j)|^q] < ∞ for q > p ≥ 1, q ≠ 2p, thenE[ d_p(F̂_k,m, F_k)^p ]≤K ( ∑_r=0^k (H_p^1/p)^r)^p-1∑_j=0^k (H_p^1/p)^k-j (E[|R^(j)|^q])^p/q· m^-min{(q-p)/q,1/2},where K = K(p,q) is a constant that only depends on p and q. * Note that Assumption <ref> does not require that H_p < 1, i.e., it is not necessary for Φ to define a contraction for the algorithm to work. However, when H_p < 1 the bound provided by Theorem <ref> becomes independent of k, ensuring that the complexity of the population dynamics algorithm remains linear in k, rather than exponential, i.e., (E[N])^k, as the naive algorithm. When H_p ≥ 1 for all p ≥ 1 the bound given above may grow with the level of the recursion, i.e., the value of k, and the convergence of the sequence {μ_k} as k →∞ may not be guaranteed.* Even in the case when H_p ≥ 1 for all p ≥ 1, the explicit bounds provided by Theorem <ref> may be useful for determining whether endogenous solutions exist, since they guarantee that we can accurately approximate R^(k).* We also point out that the first inequality in Theorem <ref> implies that the rate at which E[ d_p(F̂_k,m, F_k)^p ] converges to zero is determined by max_0 ≤ j ≤ k E[ d_p(F_j,m, F_j)]. Since d_p( F_j,m, F_j) corresponds to implementing the population dynamics algorithm by sampling without replacement from a “perfect" i.i.d. pool of observations from μ_j-1, this convergence rate is in some sense optimal. * For all the examples given in Examples <ref>, we have H_p < 1 and sup_k ≥ 0 E[|R^(k)|^q] < ∞ for some q > p, making the bound provided by Theorem <ref> independent of k. Moreover, for the Quicksort and FIND algorithms, as well as for the queuing networks with parallel servers and synchronization requirements, the best possible rate of convergence is achieved, i.e., E[d_p(F̂_k,m, F_k)^p] = O( m^-1/2) uniformly in k. We now turn our attention to the almost sure convergence of d_p(F̂_k,m, F_k), for which we provide two different results. The first one holds under Assumption <ref> as above, but under rather strong moment conditions. Note that for the linear case Assumption <ref>, in its general form, holds for any p ≥ 1 for which E[ ( ∑_i=1^N |C_i| )^p ] < ∞ by Remark <ref>(i).Allowing Assumption <ref> to hold for only E[ X_i - Y_i] = 0 is important for guaranteeing that we can choose H_p < 1 in Theorem <ref>, but is unimportant for the almost sure convergence of the algorithm.Fix 1 ≤ p < ∞ and suppose that Φ satisfies Assumption <ref> for both p and 2p. Assume further that for any fixed k ∈ℕ, max _0 ≤ j ≤ k E[ (R^(j))^2p (log |R^(j)|)^+] < ∞. Then,lim_m →∞ d_p(F̂_k,m, F_k) = 0 a.s.The moment condition requiring the finiteness of the 2p absolute moment also appears in some related (stronger) results for the convergence of the Wasserstein distance between a distribution function and its empirical measure, specifically, concentration inequalities <cit.> and a central limit theorem <cit.>. In our case, where we seek only to establish the almost sure convergence of the algorithm, this condition is too strong, so we provide below an improved result under the finer Assumption <ref>.Fix 1 ≤ p < ∞ and suppose that Φ satisfies Assumption <ref>. Assume further that E[ |R^(0)|^p+δ + Z^p+δ] < ∞ for some δ > 0, where Z = ∑_i=1^N φ(C_i). Then, for any fixed k ∈ℕ, lim_m →∞ d_p(F̂_k,m, F_k) = 0 a.s. Our last result relates the convergence of d_p(F̂_k,m, F_k) to the consistency of estimators based on the pool 𝒫^(k,m). More precisely, the value of the algorithm lies in the fact that it efficiently produces a sample of identically distributed random variables whose distribution is approximately F_k. A natural estimator for quantities of the form E[ h(R^(k))] is then given by1/m∑_i=1^m h (R̂_i^(k,m)) = ∫_ℝ h(x) d F̂_k,m(x).However, the random variables in 𝒫^(k,m) are not independent of each other, and the consistency of such estimators requires proof. In the sequel, the symbol P→ denotes convergence in probability. We say that Θ_n is a weakly consistent estimator for θ if Θ_n P→θ as n →∞. We say that it is a strongly consistent estimator for θ if Θ_n →θ a.s.Our last result shows the consistency of estimators of the form in (<ref>) for a broad class of functions. Fix 1 ≤ p < ∞ and suppose that h: ℝ→ℝ satisfies |h(x)| ≤ C (1 + |x|^p) for all x ∈ℝ and some constant C > 0. Then, the following hold: * If E[ d_p(F̂_k,m, F_k)^p ] → 0 as m →∞, then (<ref>) is a weakly consistent estimator for E[ h(R^(k))] for each fixed k ∈ℕ.* If d_p(F̂_k,m, F_k) → 0 a.s., as m →∞, then (<ref>) is a strongly consistent estimator for E[ h(R^(k))] for each fixed k ∈ℕ. We conclude that the population dynamics algorithm can be used to efficiently generate sample pools of random variables having a distribution that closely approximates that of the special endogenous solution to SFPEs of the form in (<ref>). Furthermore, these sample pools can be used to produce consistent estimators for a broad class of functions. The gain of efficiency of the algorithm compared to a naive Monte Carlo approach, combined with the consistency guarantees proved in this paper, make it extremely useful for the numerical analysis of many problems where SFPEs appear.§ PROOFSThis section includes the proofs of Theorems <ref>, <ref>, <ref>, Proposition <ref>, Theorem <ref>, and of Lemma <ref>, in that order. The last two appear at the end since they are not directly related to the Population Dynamics algorithm. The first four proofs are based on a construction of the pools {𝒫^(j,m): 0 ≤ j ≤ k} where we carefully couple the random variables {R̂_i^(j,m)} with i.i.d. observations from their limiting distribution F_j. To start, for any k ∈ℕ let ℰ_k = {(Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1, { U_(i,r)^(j)}_r ≥ 1): i ≥ 1, 0 ≤ j ≤ k }be a collection of i.i.d. random vectors where (Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1) has the same distribution as the generic branching vector (Q, N, {C_r}_i ≥ 1) and the {U_(i,r)^(j)}_r ≥ 1 are i.i.d. random variables uniformly distributed in [0,1], independent of (Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1). Next, we recursively construct a sequence of random variables { (R̂_i^(j,m), R_i^(j)): 1 ≤ i ≤ m, 0 ≤ j ≤ k} as follows: * Set R̂_i^(0) = F_0^-1(U_(i,1)^(0)) = R_i^(0,m), for 1 ≤ i ≤ m; define F̂_0,m(x) = 1/m∑_i=1^m 1(R̂_i^(0,m)≤ x) = F_0,m(x). * For 1 ≤ j ≤ k and each 1 ≤ i ≤ m, R̂_i^(j,m) = Φ( Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1, {F̂^-1_j-1,m (U_(i,r)^(j)) }_r ≥ 1) andR_i^(j) = Φ( Q_i^(j), N_i^(j), { C_(i,r)^(j)}_r ≥ 1, { F^-1_j-1 (U_(i,r)^(j)) }_r ≥ 1);defineF̂_j,m(x) = 1/m∑_i=1^m 1(R̂_i^(j,m)≤ x) and F_j,m(x) = 1/m∑_i=1^m 1(R_i^(j)≤ x). Note that the random variables { R_i^(j)}_i=1^m are i.i.d. and have distribution F_j, and therefore, F_j,m is an empirical distribution function for F_j. The distribution functions F̂_j,m are those obtained through the population dynamics algorithm. Throughout the proofs we will also use repeatedly the sigma-algebra ℱ_k = σ (ℰ_k) for k ∈ℕ. We point out that all the random variables { ( R̂_i^(k,m), R_i^(k)) : i ≥ 1} are measurable with respect to ℱ_k for all m ≥ 1. We are now ready to prove Theorem <ref>.Let { (R̂_i^(j,m), R_i^(j)): 1 ≤ i ≤ m, 0 ≤ j ≤ k} be a sequence of random vectors constructed as explained above. Next, note that from the triangle inequality we obtaind_p(F̂_j,m, F_j) ≤ d_p(F̂_j,m, F_j,m) + d_p(F_j,m, F_j). Now let χ be a Uniform(0,1) random variable independent of everything else, and define the random variables R̂^(j,m) = ∑_i=1^m R̂_i^(j,m) 1((i-1)/m < χ≤ i/m) andR^(j) = ∑_i=1^m R_i^(j) 1((i-1)/m < χ≤ i/m),which conditionally on ℱ_j are distributed according to F̂_j,m and F_j,m, respectively. Then, from the definition of d_p we haved_p(F̂_j,m, F_j,m)^p= inf_X∼F̂_j,m, Y∼ F_j,m E[ . |X - Y|^p | ℱ_j ] ≤ E[ . | R̂^(j,m) - R^(j)|^p | ℱ_j ] = 1/m∑_i=1^m | R̂_i^(j,m) - R_i^(j)|^p.It follows from the observation that the random variables X_i^(j) = R̂_i^(j,m) - R_i^(j) are identically distributed, thatE[ d_p(F̂_j,m, F_j,m)^p ] ≤ E[ | R̂_1^(j,m) - R_1^(j)|^p ] . Next, suppose first that Assumption <ref> for any { X_i } and {Y_i}, and note thatE[ | R̂_1^(j,m) - R_1^(j)|^p ]= E[| Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, {F̂^-1_j-1,m (U_(1,r)^(j)) }_r ≥ 1) . . . .- Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, { F^-1_j-1 (U_(1,r)^(j)) }_r ≥ 1) |^p] ≤H_p E[ |F̂^-1_j-1,m (U_(1,1)^(j)) - F^-1_j-1 (U_(1,1)^(j)) |^p ] = H_pE[ d_p(F̂_j-1,m, F_j-1)^p ] .For the linear case when only Assumption <ref> [linear0] holds,note that E[ F̂_j-1,m^-1(U) - F_j-1^-1(U) ]= E[ ∑_i=1^N C_i ] E[R̂_1^(j-2,m) - R^(j-2) ]= ( E[ ∑_i=1^N C_i ] )^j-1 E[ R̂_1^(0,m) - R^(0) ]= 0,and therefore, E[ | R̂_1^(j,m) - R_1^(j)|^p ]= E[| ∑_r=1^N_1^(j) C_(1,r)^(j)( F̂^-1_j-1,m (U_(1,r)^(j)) - F^-1_j-1 (U_(1,r)^(j)) )|^p] ≤ H_p E[ |F̂^-1_j-1,m (U_(1,1)^(j)) - F^-1_j-1 (U_(1,1)^(j)) |^p ] = H_pE[ d_p(F̂_j-1,m, F_j-1)^p ] .It now follows from(<ref>) and Minkowski's inequality,that( E[ d_p(F̂_j,m, F_j)^p ] )^1/p ≤( E[ ( d_p(F̂_j,m, F_j,m) + d_p(F_j,m, F_j) )^p ] )^1/p≤( E[ d_p(F̂_j,m, F_j,m)^p])^1/p +( E[d_p(F_j,m, F_j)^p])^1/p≤( H_p E[ d_p(F̂_j-1,m, F_j-1)^p ] )^1/p + ( E[d_p(F_j,m, F_j)^p])^1/p. Iterating the recursion above we obtain( E[ d_p(F̂_j,m, F_j)^p ] )^1/p ≤∑_r=1^j (H_p^1/p)^j-r( E[d_p(F_r,m, F_r)^p])^1/p+ (H_p^1/p)^j (E[ d_p(F̂_0,m, F_0)^p ] )^1/p= ∑_r=0^j (H_p^1/p)^j-r( E[d_p(F_r,m, F_r)^p])^1/p.Now let λ_j,r = (H_p^1/p)^j-r( ∑_r=0^j (H_p^1/p)^j-r)^-1 and use the fact that g(x) = x^1/p is concave to obtain( ∑_r=0^j (H_p^1/p)^j-r)^-1( E[ d_p(F̂_j,m, F_j)^p ] )^1/p ≤∑_r=0^j λ_j,r( E[d_p(F_r,m, F_r)^p])^1/p≤( ∑_r=0^j λ_j,r E[d_p(F_r,m, F_r)^p] )^1/p,or equivalently,E[ d_p(F̂_j,m, F_j)^p ]≤( ∑_s=0^j (H_p^1/p)^s)^p-1∑_r=0^j (H_p^1/p)^j-r E[d_p(F_r,m, F_r)^p].This completes the first part of the proof.Next, assume that max_0 ≤ r ≤ k E[ |R^(r)|^q] < ∞ for q > p ≥ 1, q ≠ 2p, and use Theorem 1 in <cit.> to obtain thatE[ d_p(F_r,m, F_r)^p ] ≤ C (E[|R^(r)|^q])^p/q( m^-1/2 + m^-(q-p)/q),where C = C(p,q) is a constant that does not depend on F_r. The second statement of the theorem now follows.We now turn to the proof of Theorem <ref>. To simplify its exposition we first provide a preliminary result for the mean Wasserstein distance between a distribution and its empirical distribution function. Let G be a distribution on ℝ and let { X_i}_i ≥ 1 be i.i.d. random variables distributed according to G. Suppose E[ |X_1|^q (log |X_1|)^+] < ∞ for some q ≥ 2,and let G_m(x) = m^-1∑_i=1^m 1(X_i ≤ x) denote the empirical distribution function of the {X_i}. Then,∑_m = 1^∞1/m E[ d_q(G_m, G)^q] < ∞.Fix ϵ > 0 and define for x ≥ 0 the functionsa(x) = min{ 1/G(x), x^q+ϵ}and b(x) = min{ 1/G(-x), x^q+ϵ}.Next, use Proposition 7.14 in <cit.> followed by the monotonicity of the L_p norm, to see that∑_m=1^∞1/m E[ d_q(G_m, G)^q]≤ q 2^q-1∑_m=1^∞1/m∫_-∞^∞ |x|^q-1 E[ | G_m(x) - G(x) | ] dx ≤ q 2^q-1∑_m=1^∞1/m∫_-b^-1(m)^a^-1(m) |x|^q-1( E[ ( G_m(x) - G(x) )^2 ] )^1/2 dx+ q 2^q-1∑_m=1^∞1/m∫_a^-1(m)^∞ x^q-1 E[G_m(x) + G(x) ] dx+ q 2^q-1∑_m=1^∞1/m∫_-∞^-b^-1(m) |x|^q-1 E[ G_m(x) + G(x)] dx = q 2^q-1∑_m=1^∞1/m∫_-b^-1(m)^a^-1(m) |x|^q-1√(G(x) G(x)/m) dx+ q 2^q∑_m=1^∞1/m∫_a^-1(m)^∞ x^q-1G(x)dx+ q 2^q∑_m=1^∞1/m∫_-∞^-b^-1(m) |x|^q-1 G(x) dx,where g^-1(t) = inf{ x ∈ℝ: g(x) ≥ t} is the generalized inverse of function g. Next, to bound (<ref>) note that ∑_m=1^∞1/m^3/2∫_-b^-1(m)^a^-1(m) |x|^q-1√( G(x) G(x))dx ≤∑_m=1^∞1/m^3/2∫_0^a^-1(m) x^q-1√(G(x))dx + ∑_m=1^∞1/m^3/2∫_-b^-1(m)^0 (-x)^2p-1√( G(x))dx = ∫_0^∞∑_m=⌊ a(x) ⌋ +1^∞x^q-1/m^3/2√(G(x))dx + ∫_0^∞∑_m=⌊ b(x) ⌋+1^∞x^q-1/m^3/2√( G(-x))dx,where in the last equality we used the observation that { x < a^-1(m)} = {a(x) < m}, respectively, { x < b^-1(m)} = {b(x) < m}. Now note that for any n ≥ 0 we have∑_m=n+1^∞1/m^3/2 ≤∑_m=n+1^∞( m+1/m)^3/2∫_m^m+11/t^3/2dt ≤( 1 + 1/n+1)^3/2∫_n+1^∞ t^-3/2dt ≤ 2^5/2(n+1)^-1/2.Hence, (<ref>) is bounded from above by a constant times∫_0^∞ x^q-1√(G(x))(⌊ a(x) ⌋+1)^-1/2dx + ∫_0^∞ x^q-1√( G(-x))(⌊ b(x) ⌋+1)^-1/2dx≤ 2 + ∫_1^∞ x^q-1√(G(x)/a(x))dx + ∫_1^∞ x^q-1√(G(-x)/b(x))dx= 2+ ∫_{x ≥ 1: 1/G(x) ≤ x^q+ϵ}x^q-1G(x)dx + ∫_{x ≥ 1: 1/G(x) > x^q+ϵ} x^q/2-1-ϵ/2√(G(x) )dx + ∫_{x ≥ 1: 1/G(-x) ≤ x^q+ϵ}x^q-1 G(-x)dx+ ∫_{x ≥ 1: 1/G(-x) > x^q+ϵ} x^q/2-1-ϵ/2√( G(-x))dx≤ 2+∫_1^∞ x^q-1G(x) dx + ∫_1^∞ x^q-1 G(-x) dx + 2 ∫_1^∞ x^-1-ϵdx ≤ 2 + 1/q∫_1^∞ x^q G(dx)+ 1/q∫_-∞^-1 (-x)^q G(dx)+ 2/ϵ≤ 2 + 1/q E[ |X_1|^q ] + 2/ϵ < ∞. To analyze (<ref>) use the observation that { x ≥ a^-1(m)} = { a(x) ≥ m} to obtain that∑_m=1^∞1/m∫_a^-1(m)^∞ x^q-1G(x)dx= ∫_a^-1(1)^∞∑_m=1^⌊ a(x) ⌋1/m x^q-1G(x)dx ≤∫_a^-1(1)^∞ x^q-1G(x) ∑_m=1^⌊ a(x) ⌋m+1/m∫_m^m+11/tdt dx ≤ 2 ∫_a^-1(1)^∞ x^q-1G(x) ∫_1^⌊ a(x) ⌋ +1 t^-1dt dx ≤ 2 ∫_a^-1(1)^∞ x^q-1G(x) log( x^q+ϵ+1) dx ≤ 2 log 2 + 2 (q+ϵ) sup_t ≥ 1log (t+1)/log t∫_1^∞ x^q-1 (log x) G(x) dx.Since sup_t ≥ 1log(t+1)/log t < ∞ and ∫_1^∞ x^q-1 (log x) G(x) dx= .x^q ( log x - 1)/qG(x) |_1^∞ + ∫_1^∞x^q (log x - 1)/q G(dx) = G(1)/q + E[ |X_1|^q (log X_1 - 1) 1(X_1 ≥ 1) ]/q≤E[ |X_1|^q log X_1 1(X_1 ≥ 1)]/q < ∞,we obtain that (<ref>) is finite. Finally, the same steps used to bound (<ref>) give that (<ref>) is bounded by q2^q(2 log 2 + 2(q+ϵ) E[ |X_1|^qlog |X_1| 1(X_1 ≤ -1)]/qsup_t ≥ 1log(t+1)/log t) < ∞.We now give the proof for the first result on the almost sure convergence of the algorithm. The idea of the proof is to first identify a recursive formula for the Wasserstein distance d_p(F̂_k,m, F_k) as it was done for the convergence in mean theorem. Once we do this, the main difficulty lies in ensuring that the errors in the bound converge sufficiently fast to satisfy the criterion for almost sure convergence in the Borel-Cantelli lemma. In the case when we have a bit more than 2p finite moments this can be done using Chebyshev's inequality, similarly to the proof of the strong law of large numbers under finite fourth moment conditions. We start with this case below.We will start the proof by deriving an upper bound for d_p(F̂_k,m, F_k). To this end, we construct the random variables { (R̂_i^(j,m), R_i^(j)) : 1 ≤ i ≤ m, 0 ≤ j ≤ k} according to the construction given at the beginning of the section. Recall that ℱ_j = σ(ℰ_j), where ℰ_j is given by (<ref>), and that Assumption <ref> holds for both p and 2p. We start by noting that the triangle inequality followed by (<ref>) gived_p(F̂_k,m, F_k)≤ d_p(F̂_k,m, F_k,m) + d_p(F_k,m, F_k) ≤( 1/m∑_i=1^m | R̂_i^(k,m) - R_i^(k)|^p )^1/p + d_p(F_k,m, F_k) . Next, define for j ≥ 1, X_i^(j,m) =| R̂_i^(j,m) - R_i^(j)|^p and note that by construction, the random variables { X_i^(j,m)}_i ≥ 1 are identically distributed and conditionally independent given ℱ_j-1. Now set Z_i^(j,m) = X_i^(j,m)- E[ X_1^(j,m)| ℱ_j-1 ] and note that E[ X_1^(j,m) | ℱ_j-1 ]= E[ | Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, {F̂^-1_j-1,m (U_(1,r)^(j)) }_r ≥ 1) . . . . .- Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, { F^-1_j-1 (U_(1,r)^(j)) }_r ≥ 1) |^p | ℱ_j-1] ≤ H_p E[ .|F̂^-1_j-1,m (U_(1,1)^(j)) - F^-1_j-1 (U_(1,1)^(j)) |^p| ℱ_j-1]= H_p d_p(F̂_j-1,m, F_j-1)^p .It follows that1/m∑_i=1^m | R̂_i^(k,m) - R_i^(k)|^p ≤1/m∑_i=1^m Z_i^(k,m) + H_p d_p(F̂_k-1,m, F_k-1)^p,which in turn implies thatd_p(F̂_k,m, F_k)≤ d_p(F_k,m, F_k) + (1/m∑_i=1^m Z_i^(k,m) + H_p d_p(F̂_k-1,m, F_k-1)^p )^1/p≤ d_p(F_k,m, F_k) + |1/m∑_i=1^m Z_i^(k,m)|^1/p + H_p^1/p d_p(F̂_k-1,m, F_k-1),where in the last step we used the inequality ( x+y )^β≤ x^β + y^β for 0 < β≤ 1 and x,y ≥ 0. Iterating (<ref>) k-1 more times we obtaind_p(F̂_k,m, F_k)≤∑_j=1^k ( d_p(F_j,m, F_j) + | 1/m∑_i=1^m Z_i^(j,m)|^1/p) (H_p^1/p)^k-j + (H_p^1/p)^k d_p(F̂_0,m, F_0) = ∑_j=0^k (H_p^1/p)^k-j d_p(F_j,m, F_j) + ∑_j=1^k (H_p^1/p)^k-j| 1/m∑_i=1^m Z_i^(j,m)|^1/p . Now note that by the Glivenko-Cantelli lemma and the strong law of large numbers, sup_x ∈ℝ| F_j,m(x) - F_j(x) |→ 0 a.s. and1/m∑_i=1^m |R^(j)_i |^p = ∫_-∞^∞ |x|^p dF_j,m(x)→∫_-∞^∞ |x|^p dF_j(x) a.s., as m →∞, and therefore, by Definition 6.8 and Theorem 6.9 in <cit.>, d_p(F_j,m, F_j) → 0 a.s. for each j ≥ 1. It suffices then to show that for each 1 ≤ j ≤ k the sums m^-1∑_i=1^mZ_i^(j,m)→ 0 a.s. as well. To see this note that for any ϵ > 0,∑_m=1^∞ P( 1/m∑_i=1^m Z_i^(j,m) > ϵ)≤∑_m=1^∞1/ϵ^2 m^2 E[ ( ∑_i=1^m Z_i^(j,m))^2 ] = 1/ϵ^2∑_m=1^∞1/m( E[ ( Z_1^(j,m))^2 ] + (m-1) E[Z_1^(j,m)Z_2^(j,m)] ) = 1/ϵ^2∑_m=1^∞1/m E[ ( X_1^(j,m)| ℱ_j-1)] . Moreover, using the same arguments we used in the proof of Theorem <ref>, we obtain that( X_1^(j,m) | ℱ_j-1)≤ E[ . ( X_1^(j,m))^2 | ℱ_j-1] = E[ ( Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, {F̂^-1_j-1,m (U_(1,r)^(j)) }_r ≥ 1) . . . . .- Φ( Q_1^(j), N_1^(j), { C_(1,r)^(j)}_r ≥ 1, { F^-1_j-1 (U_(1,r)^(j)) }_r ≥ 1) )^2p| ℱ_j-1] ≤ H_2pE[ .(F̂^-1_j-1,m (U_(1,1)^(j)) - F^-1_j-1 (U_(1,1)^(j)) )^2p| ℱ_j-1] (by Assumption <ref>)= H_2pd_2p( F̂_j-1,m, F_j-1)^2p . Next, note that by Theorem <ref> we haveE[ d_2p( F̂_j-1,m, F_j-1)^2p]≤( ∑_s=0^j-1 H_2p^s )^2p-1∑_r=0^j-1 H_2p^j-1-r E[ d_2p(F_r,m, F_r)^2p].It follows that for any 1 ≤ j ≤ k, ∑_m=1^∞ P( 1/m∑_i=1^m Z_i^(j,m) > ϵ) ≤H_2p/ϵ^2∑_m=1^∞1/m E[ d_2p( F̂_j-1,m, F_j-1)^2p] ≤H_2p/ϵ^2( ∑_s=0^j-1 H_2p^s )^2p-1∑_r=0^j-1 H_2p^j-1-r∑_m=1^∞1/m E[ d_2p(F_r,m, F_r)^2p]. Finally, since by Lemma <ref> we have that ∑_m=1^∞1/m E[ d_2p(F_r,m, F_r)^2p] < ∞for each 0 ≤ r ≤ j-1, the Borel-Cantelli Lemma gives that lim_m →∞ m^-1∑_i=1^mZ_i^(j,m) = 0 a.s. This completes the proof.We now move on to the proof of Theorem <ref>, where we only have a bit more than p finite moments. In this case, we cannot use Chebyshev's inequality to verify the condition for the Borel-Cantelli lemma, and a finer analysis of the errors is required. In particular, our proof uses the Lipschitz condition from Assumption <ref> to derive a large-deviations bound for the sum of independent random variables appearing in the recursive analysis of d_p(F̂_k,m, F_k).Before proceeding to the main proof, we give three preliminary results. The first one provides an upper bound for the generalized inverse of any distribution function having finite q absolute moments. Let G be a distribution function on ℝ, and let G^-1 be its generalized inverse. Suppose that G has finite absolute moments of order q > 0. Then, for any u ∈ (0,1), |G^-1(u)| ≤ || X^+ ||_q (1-u)^-1/q + ||X^-||_q u^-1/q . Let X be a random variable having distribution G, and define G_+(x) = P(X^+ ≤ x) = G(x) 1(x ≥ 0) and G_-(x) = P(X^- ≤ x) = P(X ≥ -x) 1(x ≥ 0). Then, G_+^-1(u) = inf{ x ∈ℝ: G_+(x) ≥ u } = inf{ x≥ 0: G(x) ≥ u } = G^-1(u)^+, while if we define G_-^* to be the right-continuous generalized inverse of G_-, thenG_-^*(1-u)= inf{ x ∈ℝ: G_-(x) > 1- u }= inf{ x ≥ 0: 1 - G(-x) + P(X = -x) > 1-u }= inf{ x ≥ 0: G(-x) - P(X = -x) < u }= -inf{ x ≤ 0: G(x) ≥ u } = G^-1(u)^-.Now use Markov's inequality to obtain that for all x > 0, 1 - G_+(x) ≤min{ 1, E[ (X^+)^q] } x^-q≜ 1 - H_+(x)and 1 - G_-(x) ≤min{ 1, E[ (X^-)^q ] } x^-q≜ 1 - H_-(x).The first inequality implies that for any u ∈ (0,1), G_+^-1(u)= inf{ x ∈ℝ: G_+(x) ≥ u }≤inf{ x ∈ℝ: H_+(x) ≥ u } = H_+^-1(u) = || X^+ ||_q (1-u)^-1/q,while the second one plus the continuity of H_- givesG^-1(u)^-= G_-^*(1-u) = inf{ x ∈ℝ: G_-(x) > 1- u }≤inf{ x ∈ℝ: H_-(x) > 1- u }= inf{ x ∈ℝ: H_-(x) ≥ 1- u } =H_-^-1(1-u) = ||X^-||_q u^-1/q.It follows that|G^-1(u)| = G^-1(u)^+ + G^-1(u)^-≤ || X^+ ||_q (1-u)^-1/q + ||X^-||_q u^-1/q .The next two preliminary results provide key steps for the proof of Theorem <ref>, which essentially consist on giving a large-deviations bound (uniform in m) for the sample mean of (conditionally) i.i.d. random variables. The random variables { Y_i^(j,m)} defined below will be used as upper bounds for d_p+δ_j+1(F̂_j,m, F_j) in the proof of Theorem <ref>, and the estimates we need have to be very tight considering that we no longer have finite second moments, so the rate of convergence to their mean can be very slow. The lemma below gives an upper bound for the truncated summands.Fix 1 ≤ p < ∞ and ϵ > 0. Suppose Assumption <ref> holds and E[ |R^(0)|^p+δ + Z^p+δ] < ∞ for some δ > 0, where Z = ∑_i=1^N φ(C_i).Let ℱ_j = σ( ℰ_j), where ℰ_j is defined by (<ref>), set δ_j = δ (k-j)/k, 0 ≤ j ≤ k, η = ( ϵ^-1 4 e^2/ϵmax{ 1, E[Z^p+δ]})^-(p+δ_j)/(p+δ_j+1), andY_i^(j,m) = ( ∑_r=1^N_i^(j+1)φ(C_(i,r)^(j+1)) | F̂_j,m^-1(U_(i,r)^(j+1)) - F_j^-1(U_(i,r)^(j+1)) | )^p+δ_j+1,for i = 1, …, m. Then, on the event {sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η},we haveP( .sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j ) ≤ 2(n-1)^-1/2. We start by noting thatP( . sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1( Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j ) ≤∑_m=n^∞ P( . 1/m∑_i=1^m Y_i^(j,m) 1( Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j ). To bound each of the probabilities in (<ref>) use Chernoff's bound to obtain thatP( . 1/m∑_i=1^m Y_i^(j,m) 1( Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j )≤min_θ≥ 0 e^-θϵ m ( E [ . e^θ Y_1^(j,m) 1(Y_1^(j,m)≤ m/log m)| ℱ_j ] )^m. Note that by Remark <ref>(i), we have that on the event{sup_m ≥ n d_p+δ_j (F̂_j,m,F_j)^p+δ_j≤η}, E[ . Y_1^(j,m)| ℱ_j ]≤ 2 E[ Z^p+δ_j+1] E[ . | F̂_j,m^-1(U_1) - F_j^-1(U_1) |^p+δ_j+1| ℱ_j ] = || Z ||_p+δ_j+1^p+δ_j+1 d_p+δ_j+1(F̂_j,m, F_j)^p+δ_j+1≤|| Z ||_p+δ^p+δ_j+1 d_p+δ_j(F̂_j,m, F_j)^p+δ_j+1≤max{1, E[ Z^p+δ] }η^(p+δ_j+1)/(p+δ_j)= ϵ/4 e^2/ϵ. Next, use the inequality e^x ≤ 1 + x e^x for x ≥ 0to obtain thatE [ . e^θ Y_1^(j,m) 1(Y_1^(j,m)≤ m/log m)| ℱ_j ] ≤ 1 + θ E [ . Y_1^(j,m) 1(Y_1^(j,m)≤ m/log m) e^θ Y_1^(j,m) 1(Y_1^(j,m)≤ m/log m)| ℱ_j ] ≤ 1 + θ E [ . Y_1^(j,m)| ℱ_j ] e^θm/log m≤ 1 + θ e^θ m/log mϵ/4 e^2/ϵ .Now use the inequality 1 + x ≤ e^x to see that( E [ . e^θ Y_1^(j,m) 1(Y_1^(j,m)≤ m/log m)| ℱ_j ] )^m ≤ e^θϵ m e^θ m/log m /(4e^2/ϵ).It follows that by choosing θ = (2/ϵ) log m/m we obtainP( . 1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m ) > ϵ| ℱ_j )≤min_θ≥ 0 e^-θϵ m +θϵ m e^θ m/log m /(4e^2/ϵ) = min_θ≥ 0 e^-θϵ m (1 - e^θ m/log m/4 e^2/ϵ)≤ e^-2log m ( 1 - 1/4 ) ,which in turn implies that (<ref>) is bounded from above by∑_m=n^∞ e^-(3/2) log m =∑_m=n^∞ m^-3/2≤∑_m=n^∞∫_m-1^m 1/x^3/2dx =∫_n-1^∞ x^-3/2dx = 2(n-1)^-1/2. This completes the proof. The next lemma gives the complementary estimate for the probability that any of the {Y_i^(j,m)} exceeds the truncation value in Lemma <ref>. The challenge here is the uniformity in m of the result.Fix 1 ≤ p < ∞. Suppose Assumption <ref> holds and E[Z^p+δ] < ∞ for some δ > 0, where Z = ∑_i=1^N φ(C_i). Let δ_j = δ (k-j)/k and q_j = p + δ_j for 0 ≤ j < k, fix η > 0, and let Y_1^(j,m) be defined according to (<ref>). Then, for any q_j+1 < r_j < q_j and all t ≥ n,P(sup_m ≥ tlog m/m Y_1^(j,m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤3^r_j ||Z||_r_j^r_j{4 (η^1/q_j+ || R^(j) ||_q_j )^r_j/1- r_j/q_j + 2 || R^(j) ||_r_j^r_j}( log t/t)^r_j/q_j+1.To simplify the notation, let ( Q, N, {C_r}_r ≥ 1, { U_r}_r ≥ 1) = ( Q_1^(j+1), N_1^(j+1), { C_(1,r)^(j+1)}_r ≥ 1, { U_(1,r)^(j+1)}_r ≥ 1) . Next, note that sup_m ≥ tlog m/m Y_1^(j,m)= sup_m ≥ tlog m/m( ∑_r=1^N φ(C_r) | F̂_j,m^-1(U_r) - F_j^-1(U_r) | )^p+δ_j+1≤( ∑_r=1^N φ(C_r)sup_m ≥ t( log m/m)^1/(p+δ_j+1)| F̂_j,m^-1(U_r) - F_j^-1(U_r) | )^p+δ_j+1= ( ∑_r=1^N φ(C_r)W_r^(j,t))^p+δ_j+1,whereW_r^(j,t) = sup_m ≥ t( log m/m)^1/(p+δ_j+1)| F̂_j,m^-1(U_r) - F_j^-1(U_r) |.Now, let ℱ_j = σ(ℰ_j), where ℰ_j is given by (<ref>), and note thatP(sup_m ≥ tlog m/m Y_1^(j,m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤ P(∑_r=1^N φ(C_r)W_r^(j,t)> 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) = E[ P( . ∑_r=1^N φ(C_r)W_r^(j,t)> 1 | ℱ_j ) 1( sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ].Moreover, if we let q_j = p+δ_j and use Lemma <ref>, we obtain that, conditionally on ℱ_j, W_r^(j,t) ≤sup_m ≥ t(log m/m)^1/q_j+1| F̂_j,m^-1(U_r)| + sup_m ≥ t( log m/m)^1/q_j+1 | F_j^-1(U_r) | ≤sup_m ≥ t(log m/m)^1/q_j+1( E[ . | F̂_j,m^-1(U_r) |^q_j| ℱ_j ] )^1/q_j{ U_r^-1/q_j + (1-U_r)^-1/q_j} + (log t/t)^1/q_j+1| F_j^-1(U_r) |.Furthermore, by Minkowski's inequality, we have that on the event {sup_m ≥ n d_q_j (F̂_j,m, F_j)^q_j≤η}, sup_m ≥ t(log m/m)^1/q_j+1( E[ . | F̂_j,m^-1(U_r) |^q_j| ℱ_j ] )^1/q_j≤sup_m ≥ t(log m/m)^1/q_j+1{( E[ . | F̂_j,m^-1(U_r) - F_j^-1(U_r) |^q_j| ℱ_j ] )^1/q_j+ || F_j^-1(U_r) ||_q_j}= sup_m ≥ t(log m/m)^1/q_j+1{ d_q_j(F̂_j,m, F_j)+ || R^(j) ||_q_j}≤(log t/t)^1/q_j+1{η^1/q_j+ || R^(j) ||_q_j} .It follows that conditionally on ℱ_j, we have that on the event {sup_m ≥ n d_q_j (F̂_j,m, F_j)^q_j≤η},W_r^(j,t)≤( log t/t)^1/q_j+1{ K_j ( U_r^-1/q_j + (1-U_r)^-1/q_j) + |F_j^-1(U_r)| },where K_j ≜η^1/q_j+ || R^(j) ||_q_j < ∞ by Remark <ref>(ii). Thus, we have that on the event {sup_m ≥ n d_q_j (F̂_j,m, F_j)^q_j≤η}, the union bound and Markov's inequality yieldP( . ∑_r=1^N φ(C_r)W_r^(j,t)> 1 | ℱ_j ) ≤ P( ∑_r=1^N φ(C_r) { K_j ( U_r^-1/q_j + (1-U_r)^-1/q_j) + |F_j^-1(U_r)| }> ( t/log t)^1/q_j+1) ≤ P(∑_r=1^N φ(C_r) K_j U_r^-1/q_j > 1/3( t/log t)^1/q_j+1)+ P(∑_r=1^N φ(C_r) K_j (1-U_r)^-1/q_j > 1/3( t/log t)^1/q_j+1)+ P(∑_r=1^N φ(C_r) |F_j^-1(U_r)| > 1/3( t/log t)^1/q_j+1) ≤ 3^r_j( log t/t)^r_j/q_j+1{ 2 E[ ( ∑_i=1^N φ(C_i) K_j U_i^-1/q_j)^r_j] . . + E[ ( ∑_i=1^N φ(C_i) R_i^(j))^r_j] } ,where by assumption q_j+1 < r_j < q_j, and we have used the observation that U_i 𝒟= 1- U_i. Finally, note that by Remark <ref>(i), we haveE[ ( ∑_i=1^N φ(C_i) K_j U_i^-1/q_j)^r_j] ≤ 2 E[ Z^r_j ] K_j^r_j E[ U_1^-r_j/q_j]= 2 K_j^r_j || Z ||_r_j^r_j/1 - r_j/q_jandE[ ( ∑_i=1^N φ(C_i) R_i^(j))^r_j] ≤ 2 E[Z^r_j] E[ |R^(j)|^r_j] = 2 || Z ||_r_j^r_j || R^(j) ||_r_j^r_j. We conclude thatP(sup_m ≥ tlog m/m X_1^(m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤ 3^r_j( log t/t)^r_j/q_j+1{4 K_j^r_j ||Z||_r_j^r_j/1- r_j/q_j + 2 || Z ||_r_j^r_j || R^(j) ||_r_j^r_j}. We are now ready to prove Theorem <ref>, which proves by induction that d_p+δ(F̂_k,m, F_k) → 0 a.s. as m →∞. Define δ_j = δ (k-j)/k for 0 ≤ j ≤ k. We will prove by induction in j that lim_m →∞d_p+δ_j(F̂_j,m, F_j) = 0 a.sfor 0 ≤ j ≤ k. Since F̂_0,m(x) ≡ F_0,m(x) for all x ∈ℝ and E[|R_0|^p+δ] < ∞,the Glivenko-Cantelli lemma and the strong law of large numbers yieldsup_x ∈ℝ| F_0,m(x) - F_0(x) |→ 0 a.s. as m →∞ and1/m∑_i=1^m |R^(0)_i |^p+δ = ∫_-∞^∞ |x|^p+δ dF_0,m(x)→∫_-∞^∞ |x|^p+δ dF_0(x) a.s. as m →∞.Therefore, by Definition 6.8 and Theorem 6.9 in <cit.>, lim_m →∞ d_p+δ_0 (F̂_0,m, F_0) = lim_m →∞ d_p+δ(F_0,m, F_0) = 0 a.s.Suppose now that (<ref>) holds for 0 ≤ j < k.To prove that d_p+δ_j+1(F̂_j+1,m, F_j+1) → 0 a.s. as m →∞, we start by constructing the random variables { (R̂_i^(t,m), R_i^(t)) : 1 ≤ i ≤ m, 0 ≤ t ≤ k } as explained at the beginning of this section. Now note that for any ϵ, η > 0, P(sup_m ≥ nd_p+δ_j+1 (F̂_j+1,m, F_j+1)^p+δ_j+1 > 2^p+δ_j+1ϵ) ≤ P( sup_m ≥ n{ d_p+δ_j+1 (F̂_j+1,m, F_j+1,m) + d_p+δ_j+1 (F_j+1,m, F_j+1)} > 2 ϵ^1/(p+δ_j+1))≤ P( sup_m ≥ n d_p+δ_j+1 (F̂_j+1,m, F_j+1,m) > ϵ^1/(p+δ_j+1))+ P( sup_m ≥ n d_p+δ_j+1 ( F_j+1,m, F_j+1) > ϵ^1/(p+δ_j+1))≤ P( sup_m ≥ n d_p+δ_j+1 ( F̂_j+1,m, F_j+1,m)^p+δ_j+1 > ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η)+ P( sup_m ≥ nd_p+δ_j (F̂_j,m,F_j)^p+δ_j > η)+ P( sup_m ≥ nd_p+δ_j+1 (F_j+1,m, F_j+1)^p+δ_j+1 > ϵ) . To analyze (<ref>) note that its convergence to zero as n →∞ is equivalent to the a.s. convergence of d_p+δ_j (F̂_j,m,F_j) to zero as m →∞, which corresponds to the induction hypothesis (<ref>).To show that (<ref>) converges to zero as n →∞, note that by Remark <ref>(ii) we have E[ |R^(j+1)|^p+δ ] < ∞, which implies that E[ |R^(j+1)|^p+δ_j+1 ] < ∞. Hence, the Glivenko-Cantelli lemma, the strong law of large numbers, and Definition 6.8 and Theorem 6.9 in <cit.> give that lim_m →∞ d_p+δ_j+1(F_j+1,m, F_j+1) = 0 a.s., which is equivalent tolim_n →∞ P( sup_m ≥ nd_p+δ_j+1 (F_j+1,m, F_j+1)^p+ δ_j+1 > ϵ) = 0. Next, to prove that (<ref>)converges to zero we first define the random variables { Y_i^(j,m): 1 ≤ i ≤ m} according to (<ref>), and define the eventsA_i,n = {sup_m ≥ n ∨ ilog m/mY_i^(j,m)≤ 1 }.Now use (<ref>) and Assumption <ref> to obtainP( sup_m ≥ n d_p+δ_j+1 ( F̂_j+1,m, F_j+1,m)^p+δ_j+1 > ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤ P( sup_m ≥ n1/m∑_i=1^m | R̂_i^(j+1,m) - R_i^(j+1)|^p+δ_j+1> ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤ P( sup_m ≥ n1/m∑_i=1^m Y_i^(j,m)> ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η,⋂_i=1^∞ A_n,i)+ P(sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η,⋃_i=1^∞ A_n,i^c ) ≤ P( sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m)> ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η)+ ∑_i=1^∞ P(A_n,i^c,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η).To analyze (<ref>), choose η = ( ϵ^-1 4 e^2/ϵmax{ 1, E[Z^p+δ]})^-(p+δ_j)/(p+δ_j+1) and let ℱ_j = σ( ℰ_j) denote the sigma-algebra generated by ℰ_j, as given by (<ref>). Note thatP( sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m)> ϵ,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) = E[ P( .sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j ) . .·1( sup_m ≥ nd_p+δ_j(F̂_j,m, F_j)^p+δ_j≤η) ] . By Lemma <ref>, we obtain that on the event {sup_m ≥ nd_p+δ_j(F̂_j,m, F_j)^p+δ_j≤η}, we haveP( .sup_m ≥ n1/m∑_i=1^m Y_i^(j,m) 1(Y_i^(j,m)≤ m/log m)> ϵ| ℱ_j )≤ 2 (n-1)^-1/2,which implies that (<ref>) is bounded from above by 2(n-1)^-1/2. To analyze (<ref>) note that∑_i=1^∞ P(A_n,i^c,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) = n P(sup_m ≥ nlog m/m Y_1^(j,m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η)+ ∑_t=n+1^∞ P(sup_m ≥ tlog m/m Y_1^(j,m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η).Now set q_j = p + δ_j and r_j = q_j+1 + δ/(2k), and note that q_j+1 < r_j < q_j ≤ p+δ. Then, by Lemma <ref>, P(sup_m ≥ tlog m/m Y_1^(j,m) > 1,sup_m ≥ nd_p+δ_j (F̂_j,m, F_j)^p+δ_j≤η) ≤K̃_j ( log t/t)^r_j/q_j+1for any t ≥ n, whereK̃_j =3^r_j||∑_i =1^N φ(C_i) ||_r_j^r_j{4 (η^1/q_j+ || R^(j) ||_q_j )^r_j/1- r_j/q_j + 2 || R^(j) ||_r_j^r_j} < ∞by Remark <ref>(ii). It follows that (<ref>) is bounded from above byK̃_j n ( log n/n)^r_j/q_j+1 + K̃_j ∑_t=n+1^∞( log t/t)^r_j/q_j+1≤K̃_j n ( log n/n)^r_j/q_j+1 + K̃_j ∑_t=n+1^∞∫_t-1^t ( log x/x)^r_j/q_j+1dx = K̃_j n ( log n/n)^r_j/q_j+1 + K̃_j∫_n^∞( log x/x)^r_j/q_j+1dxfor all n ≥ 3. Since r_j/q_j+1 > 1 and∫_n^∞( log x/x)^r_j/q_j+1 = (log n)^r_j/q_j+1/(r_j/q_j+1 - 1) n^r_j/q_j+1 - 1 (1 + o(1))as n →∞, we conclude that (<ref>)is bounded from above by2(n-1)^-1/2 + K̃_j ( 1 + 1/r_j/q_j+1 - 1 + o(1) ) (log n)^r_j/q_j+1/n^r_j/q_j+1-1,which converges to zero as n →∞. This completes the proof.We now give below the proof of Proposition <ref>. The second statement of the proposition, regarding the almost sure convergence, follows directly from Definition 6.8 and Theorem 6.9 in <cit.>. For the convergence in probability we argue as follows.Define Θ_k,m = 1/m∑_i=1^m h(R̂_i^(k,m)) and θ_k = E[h(R^(k))]. By assumption, we have that d_p(F̂_k,m, F_k) → 0 in L_p and therefore in probability, as m →∞. Hence, for every subsequence {m_i}_i ≥ 1 there is a further subsequence { m_i_j}_j ≥ 1 such that d_p(F̂_k,m_i_j, F_k) → 0 a.s. as j →∞. Definition 6.8 and Theorem 6.9 in <cit.> now give thatΘ_k,m_i_j→θ_k a.s. asj →∞.We conclude that for any subsequence {m_i}_i ≥ 1 we can find a further subsequence {m_i_j}_j ≥ 1 such that (<ref>) holds, and therefore,Θ_k,mθ_kasm →∞.The remaining two proofs in the paper correspond to Theorem <ref> and Lemma <ref>, which although not directly related to the Population Dynamics algorithm, may be of independent interest. Suppose first that Assumption <ref> holds for any i.i.d. {(X_i, Y_i): i ≥ 1} independent of (Q, N, {C_i}).Recall that F_k(x) = P(R^(k)≤ x). Then, for any j ∈ℕ_+ we haved_p(F_j, F_j-1)≤(E[| Φ(Q, N, {C_r}, {F_j-1^-1(U_r) }) - Φ(Q, N, {C_r}, {F_j-2^-1(U_r) })|^p ] )^1/p≤ H_p^1/p( E[ | F_j-1^-1(U_1) - F_j-2^-1(U_1) |^p] )^1/p= H_p^1/p d_p(F_j-1, F_j-2) ≤ (H_p^1/p)^j-1 d_p(F_1, F_0).Moreover, d_p(F_1, F_0)≤(E[| Φ(Q, N, {C_r}, {F_0^-1(U_r) }) - Φ(Q, N, {C_r}, { 0 })|^p ] )^1/p + (E[|Φ(Q, N, {C_r}, { 0 })|^p ] )^1/p≤ H_p^1/p( E[ |R^(0) |^p] )^1/p + (E[|Φ(Q, N, {C_r}, { 0 })|^p ] )^1/p.It follows that for any m ∈ℕ_+ we haved_p(F_k+m, F_k)≤∑_j=1^m d_p(F_k+j, F_k+j-1) ≤∑_j=1^m (H_p^1/p)^k+j-1 d_p(F_1, F_0) ≤ (H_p^1/p)^k d_p(F_1, F_0) ∑_j=0^m-1 (H_p^1/p)^j,which converges to zero as k →∞ uniformly in m whenever H_p < 1 and E[ |R_0|^p + | Φ(Q, N, {C_r}, { 0})|^p ]. Therefore, the sequence { R^(k): k ≥ 0} is Cauchy, and since the Wasserstein space P_p(ℝ) metrized by d_p (see Definition 6.4 in <cit.>) is complete by Theorem 6.18 in <cit.>, we have that there exists a random variable R having distribution F_*(x) = P(R ≤ x) such thatlim_k →∞ d_p(F_k, F_*) = 0.Equation (<ref>) now follows by taking m →∞ to obtain:d_p(F_k, F_*)^p = lim_m →∞ d_p(F_k, F_k+m)^p ≤ d_p(F_1, F_0)^p H_p^k/(1 - H_p^1/p)^pand using the optimal coupling (R^(k), R) = (F_k^-1(U), F_*^-1(U)).We now move to the linear SFPE (<ref>), for which it is known (see <cit.>) that R admits the explicit representationR = ∑_k=0^∞∑_ i∈ A_kΠ_ i Q_ i,as described in Section <ref>. When conditions (i) hold wehave E[ R^(k) ] = 0 for all k ≥ 0 and the arguments used above remain valid. Suppose now that conditions (ii) hold, in which case we can take R^(k) = ∑_j=0^k-1∑_ i∈ A_jΠ_ i Q_ i + ∑_ i∈ A_kΠ_ i R^(0)_ i, where the { R^(0)_ i:i∈ U} are i.i.d. copies of R^(0). Therefore, Minkowski's inequality givesE[ | R^(k) - R |^p ]≤ E[ ( ∑_j=k^∞∑_ i∈ A_j|Π_ i| |Q_ i| + ∑_ i∈ A_j|Π_ i| |R^(0)_ i| )^p ] ≤( ∑_j=k^∞( E[ (W_j )^p ] )^1/p +( E[ (W_k(R^(0)) )^p ] )^1/p)^p,where W_j ≜∑_ i∈ A_j|Π_ i| |Q_ i| and W_k(R^(0)) ≜∑_ i∈ A_k|Π_ i| |R^(0)_ i|. Now use Lemma 4.4 in <cit.> to obtain that under conditions (ii) there exist a constants K_p, K_p' < ∞ such thatE[ |W_j|^p ] ≤ K_p ( ρ_1 ∨ρ_p )^j and E[|W_k(R^(0) )|^p ] ≤ K_p (ρ_1 ∨ρ_p)^k,where ρ_β≜ E[ ∑_i=1^N |C_i|^β]. Hence,E[ | R^(k) - R |^p ]≤( (K_p + K_p') ∑_j=k-1^∞ (ρ_1 ∨ρ_p)^j/p)^p ≤( K_p+K_p'/1 - (ρ_1 ∨ρ_p)^1/p)^p (ρ_1 ∨ρ_p)^k-1.This completes the proof. Finally, we provide the proof of Lemma <ref>. By (<ref>) we have for any j ∈ℕ_+, d_p(F_j, F_j-1) ≤ (H_p^1/p)^j-1 d_p(F_1, F_0),and by (<ref>), d_p(F_1, F_0) ≤ H_p^1/p( E[ |R^(0)|^p] )^1/p + (E[|Φ(Q, N, {C_r}, { 0 })|^p ] )^1/p≜ A_p'.Hence,d_p(F_k, F_0) ≤∑_i=1^k d_p(F_i, F_i-1) ≤ A_p' ∑_i=1^k (H_p^1/p)^i-1,and we obtain that( E[ |R^(k)|^p ] )^1/p ≤( E[ |F_k^-1(U) - F_0^-1(U) |^p ] )^1/p + ( E[ |R^(0)|^p ] )^1/p= d_p(F_k, F_0) + ( E[ |R^(0)|^p ] )^1/p≤ A_p' ∑_i=1^k (H_p^1/p)^i-1+ ( E[ |R^(0)|^p ] )^1/p≤(A_p' + ( E[ |R^(0)|^p ] )^1/p) ∑_i=0^k-1 (H_p^1/p)^i. plain | http://arxiv.org/abs/1705.09747v2 | {
"authors": [
"Mariana Olvera-Cravioto"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170527005551",
"title": "Convergence of the Population Dynamics algorithm in the Wasserstein metric"
} |
http://arxiv.org/abs/1705.09825v2 | {
"authors": [
"Arus Harutyunyan",
"Armen Sedrakian"
],
"categories": [
"hep-ph",
"astro-ph.HE",
"astro-ph.SR",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170527145353",
"title": "Bulk viscosity of two-flavor quark matter from the Kubo formalism"
} |
|
Adversarial Learning:A Critical Reviewand Active Learning StudyThis research is supported in part by a Cisco Systems URP gift and AFOSR DDDAS grant. David J. Miller^1Xinyi Hu ^1Zhicong Qiu^1 George Kesidis^1 [email protected] [email protected] [email protected] [email protected]^1School of Electrical Engineering and Computer Science, Pennsylvania State University, University Park, PA 16802 USA December 30, 2023 ======================================================================================================================================================================================================================================================================================emptyplain This papers consists of two parts.The first is a critical review of prior art on adversarial learning, identifying some significant limitations of previous works.The second part is an experimental study considering adversarial active learning and an investigation of the efficacy of a mixed sample selection strategy for combating an adversary who attempts to disrupt the classifier learning.adversarial learning, active learning, reverse engineering a classifier, sample selection, mixed strategy§ INTRODUCTIONWhile there is still skepticism concerning the value of machine learning (ML) for network security <cit.>, there has been growing interest in the “dual” problem of investigating the security ofML systems, as applied both to security-sensitive applications (network intrusion detection systems (NIDS), biometric authentication, email), as well as more generally (image, character, and speech recognition, document classification), <cit.>. Much of the focus is on attempting to degrade or foil supervised classifiers, as well as anomaly detectors (ADs). <cit.> provide a useful taxonomy for various attacks on classifiers, including whether they affect training (what we will call tampering) or just testing/use (foiling).Moreover, if the attack is on training, one can distinguish attacks that involve mislabeling from those that add correctly labeled examples, but ones with contrived features chosen to bias learning.<cit.> demonstrated naive Bayes spam filters can easily be degraded by labeled spam examples (from known spam sources)which use many tokens that commonly appear in normal (ham) email (an “indiscriminate dictionary attack”).This attack on training destroys discrimination power of most tokens in the “dictionary”.Huge false positive rates ensued when as little as 5% of training data consisted of these contrived emails. <cit.> considered active learning (AL), a promising framework for security applications, asthe classifier can adapt to track evolving threats and also because oracle labeling may discovernovel classes <cit.>, <cit.> that may be zero-day threats. <cit.> demonstrated, using SVMs, thatif an adversary “salts” the unlabeled data batch in a biased fashionnear the current decision boundary (where AL seeks to choose samples for labeling), one can induce classifier degradation – each adversarial sample was chosen such that, if labeled, it will decrease accuracy the most. We will discuss in the sequel that the approach in <cit.> appears to rely on (oracle) mislabeling, even though frequent mislabeling is not too realistic in practice.We will also demonstrate experimentally that such mislabeling is not necessary in order for the attacker to degrade classification accuracy.<cit.> considers classifier testing/operation, constructing examples that will be classified differently by a classifier than by a human being.This is an attack, in the context of human and autonomous drivers, where one sees a STOP sign and the other does not, or in future man-machine interactions where robot servants may misunderstand their master's commands.<cit.> showed that one could make relatively small perturbationsto images of digits (presumably below human visual detectability, although we discuss this further in the sequel) that alter a deep neural net's decision.Effectively, <cit.> minimally “pushes” patterns acrossthe decision boundary.While not recognized in <cit.>, their perturbation approach is related to boundary-finding algorithms (neural network inversion) <cit.>.Foiling has also been performed in other domains, against detectingmalware in PDF documents: synthetic “mimicry" attacks <cit.> ornatural documents with embedded malware<cit.> (“reverse" mimicry, as a trojan).In the related study <cit.>,the attacker generates “obfuscated” voice commands typically perceived as background noiseby any human being who happens to be listening, but which are recognized as valid commands by a speech recognition system. Such commands could be used for financial fraud or to perpetrate a terrorist attack (controlling a crane, a train, etc.). To defend against this, they suggest, : a challenge (CAPTCHA, reactive two-factor authentication);supervised (speaker-dependent) training to recognize only authorized individuals;password[Recently, a child used Amazon Echo to place an order - the parents had not set up controls (a password) to prevent this <cit.>.], speaker/voice authentication, or some other kind of proactive two-factor authentication; training a classifier to discriminate between computer-generated obfuscated commands and nominalhuman commands; increasing command specificity.Also, disadvantages of thesedefenses are discussed, latency and additional human effort associated with a CAPTCHAfor each utterance[This is completely reasonable in some cases (accessing bank accounts or entering passwords), but may be considered very inconvenient in others (casual web surfing).].Regarding the latter defenses, theidea is to reduce the allowed variation of utterances classified to each word.Thus, effectively, an “unrecognized" (anomalous) class surrounds each word in feature (cepstral or wavelet coefficient) space. Note that such AD in speech recognition is not new, <cit.>, <cit.>. Also, iPhone's Siri voice interface added “individualized speech"(voice/speaker) recognition in 2015 <cit.> (before <cit.> but, again, this is very old technology, and easily configured upon purchase of the phone). Still, the threats posed by the attacks like <cit.> need to be addressed[As do other security concerns involving Siri,<cit.>,though it is not clear that one could use Siri to input data into web pages via Safari on the iPhone.].Supervised learning to discriminate the attack from “valid" speech, which assumes either that labeled examples of the attack (and a sufficient number of these) have been captured or that the attack strategy is known (so that labeled examples can be synthesized) may not be realistic[The attacker may have great degrees of freedom he can apply to create inobtrusive examples – it may be both difficult and impractical to try to create representative supervising examples “covering” all of these possibilities.].Alternatively, we will advocate for an AD defense, which requires neither detailed knowledge of the attack nor (up front) labeled examples of same.All of the above research works purport to demonstrate “security holes” in ML techniques.Theseexamples are provocative and they motivate further research. However, these studies also make convenient assumptions, about the attacker's knowledge, which may be grossly unrealistic.Moreover, these studies ignore existing ML techniques that are much more resilient to adversaries, even without considering explicit (and well known) defenses, which may detect (and thus foil) the attack.We next identify key limitations of some prior adversarial learning works. Unknown Classes and Authentication: <cit.> assume each pattern must be classified to one of a known set of categories. In many systems, in fact, there is an augmented class space with an “unknown”(unrecognized) category. Patterns not confidently assigned to any known category may be assigned “unknown” – the attack examples in <cit.> could beassigned thus[For example, Siri responds “don't know" to speech it does not recognize as an english word, where every english word (or word-tense combination) here corresponds to a known category Siri has been trained to recognize.]. Moreover, in an AL setting, human oracles who cannot confidently classify selected samples may reject the sample (and thus the attack). Likewise, emails that are part of an indiscriminate dictionary attack could easily be detected as anomalous even relative to the existing “spam” class, could thus be labeled as “unknown”, rather than “spam”, and then would not be used to help learn (corrupt) the “spam” model. Also, particularly in IDS settings with unknown unknowns, the full complement of classes is a priori unknown and one may discover new classes in an unlabeled or semisupervised data batch <cit.>. This is especially true in an AL context, starting from few labeled examples <cit.>. Finally, many security applications in fact involve authentication, not classification per se – <cit.> assume the problem is classification.The difference between these problems is well-known – the latter assumes that a datum originates from one of the N known classes whereas the former allows for the possibility that the datum is not associated with a known class – assignment to the “unknown unknown" class amounts to AD. As an obvious example, consider a perimeter authenticator (access controller) based on a challenge for a userid and password; here the number of correct responses (N authorized individuals) is miniscule compared to the total number of possible inputs, the vast majority of which result in failed access. Moreover, authentication (at least to enter the perimeter) may require an exact (password) match for access. This is essentially an extreme example of giving emphasis to class specificity. Biometric based challenges (with well-known “replay attack" protections) can be used instead to explore usability/security trade-offs, or to further secure access based on passwords.Unrealistic Assumptions: <cit.>, <cit.>, and <cit.> assume the classifier, both its structureand its learned parameters, are known to the attacker. <cit.> further assumes that the true joint (feature vector, class) distribution is also known to the attacker. Knowledge of this distribution is usually the “holy grail”in ML (it is never assumed known) – given it, one can form the Bayes-optimal classifier. In practice, at best, one can only imperfectly estimate this joint distribution, given a finite training set. So this latter assumption is wholly unrealistic even for an inside attacker. Even ignoring this latter assumption, the former one –full knowledge of the classifier – is really only reasonable under two cases:1) a non-security setting, where public-domain classifiers might be used; 2) in a “security" setting, if the attacker is in fact an insider,if they work for the company that designed the classifier. Otherwise, inbiometric authentication,where one seeks to gain access to sensitive or restricted resources, there is zero incentive to publicize many details of theauthentication system.<cit.> further assumes that the classifier is precisely knownafter every round of AL oracle labeling. Note that such knowledge might only be obtainable by intensive probing of the system (to “relearn” the decision boundary) in between active labelings.Such probing may bevery resource-intensive and must complete within the limited time window between labelings.Moreover, <cit.> considers just a 2-D example. The number of probes needed to accurately learn the decision boundary will grow with the feature dimensionality –consider,document or image domains, where one may easily work with tens to hundreds of thousands of features.Asymmetry: “Straw Man” vs. A Robust System: While <cit.> discuss possible defenses,the example attacks in<cit.> are on completely defenseless systems, as well as inherently vulnerable ones. Even without explicitly building defenses, there are ML techniques that are widely used and which also (as a side benefit) make the system robust to exploits.Consider the naive Bayes (NB) spam filter attacks <cit.>, including the “indiscriminate dictionary” attack and the red herring attack, applicable both to email and to NIDS.Here, spurious tokens areintroduced into samples that come from a known spam source (an email address recognized as a source for spam). Once the NB classifier “takes the bait” and adapts its classifier to focus on these (apparently) discriminating tokens, subsequent spam emails (from various addresses) omit these tokens and thus avoid detection. The reason such attacks are successful is because NB is a weak classifier, building only a single model to represent a spam class that may (in a time-varying fashion) exhibit great diversity – NB effectively puts all its eggs in one basket.Suppose, rather than a single NB model, that one uses a mixture of NB models, with new mixture components introduced in a time-varying fashion, as needed, to well-model patterns that are not well-fit by the existing model.Such a mixture can capture and isolate a red herring attack within a single (new) NB component.Thus, the main (legitimate) NB components representing the spam class will not be corrupted by the attack.Moreover, once the attack is over, the probability mass of this component will dwindle and this component can eventually be removed.Thus, we suggest a dynamic mixture, responsive to attacks (as well as time-varying classes), which should be highly robust to red herring and indiscriminate dictionary attacks.Other ML techniques that may help achieve attack robustness includeensembleclassification– here, uncompromised classifiers may compensate for compromised ones, helping to achieve robust decisions via ensemble decision fusion.The attacks in <cit.>could likely be defeated by relatively simple AD defenses. For example, the AL attack in <cit.> “tricks” the classifier to select for labeling biased, attacker-generated samples near the current decision boundary. To maximize the success of the attack, these synthetic examples should be chosen for active labelingwith high prevalence. Moreover, as elaborated in the next section, this attack appears to rely on the assumption that the oracle may mislabel samples near the true (optimal) decision boundary. As demonstrated in the next section: 1) a successful attack does not require oracle mislabeling; 2) irrespective of whether there is such mislabeling, attacks can be defeated by using a mixed strategy for AL sample selection (not always selecting the sample nearest the current decision boundary), and with no significant loss in the efficiency and accuracyof classifier learning.Likewise, <cit.> effectively assumes that if the number of altered features (image pixels, for character recognition) is below a fixed threshold, the attack will not be detectable by a human being. First, this is questionable, as the resulting salt and pepper noise isquite visible (see Fig. 1 in <cit.>) and is not typical of the original (clean) images in the database[Human subject testing was used in <cit.>.However, the authors did not ask respondents whether they thought images had been tampered with –they only asked them to classify the images.]. Second, whether or not the attack is perceptible to a human, it may be easily detected by an AD defense.Peculiarly, <cit.> limits the number of features (pixels) one is allowed to alter.In so doing, to induce a classification error, the magnitudes of the perturbations of the chosen features must be substantial[Even though the authors impose the minimum norm perturbations needed to induce classification errors (note again that the introduced salt and pepper noise is quite visible).].Accordingly, an AD may easily detect salt and pepper noise pixels as anomalous, relative to intensity values of pixels in a surrounding local spatial neighborhood. Finally, we note that if a human and machine do disagree on an example – but if they can share their decisions – then the introduction of these “attack” examples becomes an opportunity – to actively learn – so as to rectify the machine's decision on such examples[This assumes that the human is more accurate than the machine.In some application domains, this is certainly the case.In domains where this is not the case, such disagreement may by the same token (appropriately) cause the human to reconsider their judgement.]. Tampering Power: In <cit.>,while the authors limit the power of the attacker to corrupt training data (5% of email training data), it is unclear that this level of tampering power is plausible/corresponds to a realistic scenario.In principle, if the attacker is an insider, she may have unlimited capability to corrupt training data, in which case there may be no adequate defense.Otherwise, the chosenpower of the attackmay intricately depend on knowledge of the particular defense system in play, on the attacker's tradeoff between achieving high attack potency and minimizing the probability of the attack's detection and thwarting. Even if only a few training samples are altered, this tradeoff may exist.For example,in <cit.>, tampering with a single support vector is show to dramatically degrade classification accuracy.However, to achieve such effect, the tampered sample may become an extreme outlier of its class, and thus may either be ignored (via use of margin slackness) or may be detectable as a suspicious sample. Reverse Engineering: <cit.> (strongly) assumed that the classifier structure and its parameter values are known to the attacker. Recent works <cit.>, <cit.> have proposed techniques to reverse-engineer a (black box) classifier without necessarily even knowing its structure.In <cit.>, theauthors consider black box machine learning services, offered by companies such as Google, where, for a given (presumably big data, big model) domain, a user pays for class decisions on individual samples (queries) submitted to the ML service.<cit.> demonstrates that, with a relatively modest number of queries (perhaps as many as ten thousand or more), one can learn a classifier on the given domain that closely mimics the black box ML service decisions.Once the black box has been reverse-engineered, the attacker need no longer subscribe to the ML service.One weakness of <cit.> is that it neither considers very large (feature space) classification domains nor very large networks (deep neural networks (DNNs)) – orders of magnitude more queries may be needed to reverse-engineer a DNN on a large-scale domain.However, a much more critical weakness of of <cit.> stems from one of its (purported) greatest advantages – the authors tout that their reverse-engineering does not require any labeled training samples from the domain[For certain sensitive domains, or ones where obtaining real examples is expensive, the user may in fact have no realistic means of obtaining a significant number of real data examples from the domain.This is one main reason why the ML service is needed in the first place – the company or its client for this domain are the (exclusive) owners of this (labeled, precious) data resource.].In fact, in <cit.>, the attacker's queries to the black box are randomly drawn, uniformly, over the given feature space.While such random queryingis demonstrated to achieve reverse-engineering, what was not recognized in <cit.> is that this random queryinig makes the attack easily detectable by the ML service – randomly selected query patterns will typically look nothing like legitimate examples from any of the classes – they are very likely to be extreme outliers, of all the classes.Each such query is thus individually highly suspicious by itself – thus, even tens, let alone thousands of such queries will be trivially detected as jointly improbable under a null distribution (estimable from the training set defined over all the classes from the domain).Even if the attacker employed bots, each of which makes a small number of queries (even as few as ten), each bot's random queries should be easily detected as anomalous, likely associated with a reverse-engineering attack.We next perform an experimental study involving adversarial active learning. § ACTIVE LEARNING EXPERIMENTAL SETUPIn <cit.>, as noted earlier, the attacker can perfectlyestimate the current decision rule after every AL round (may require lots of probing between rounds) and knows the true joint density on the feature vector and class label p(Y,X) (wholly unrealistic). It was furthered assumed that the attacker knows the AL sample selection strategy (uncertainty sampling) and injects one new (high decision uncertainty) sampleinto the unlabeled batch at each AL round, crafted so that it will be chosen by the oracle for labeling. Moreover, from the description given in <cit.>, the authors do not assume that the oracle labels the attacker's sample consistent either with the Bayes-optimal decison rule or randomly, according to the true class posteriors.That is, in <cit.>, although it is not very clearly stated, the oracle may mislabel the samples crafted by the attacker.This is crucial to the success of the attack in <cit.>. In this paper, we propose a more realistic framework, under which the attacker still possesses the ability to degrade classification accuracy even without the unrealistic oracle mislabeling assumption.However, we also demonstrate that attacks on AL can be defeated by a mixed sample selection strategy, through which the attacker's injected samples are not so frequently chosen for oracle labeling.Moreover, this mixed strategy does not make a significant sacrifice either in classifier accuracy or learning convergence (number of queries needed to achieve good accuracy). In fact, this strategy is also suitable for defeating the attack even in the presence of oracle mislabeling (though we focus on more realistic oracle labeling in the sequel). §.§ Preliminaries As in <cit.>, we assume a two-class problem and use a two-class linear support vector machine. We first select a (large) (unlabeled) training pool (T_r), then randomly select a relatively small number of samples from both classes in T_r and assign (ground-truth) class labels to them[For our synthetic data experiment, consistent with the ground-truth class distributions from <cit.>, these labels are assigned according to the Bayes-optimal rule.For our real-world digits experiment, we use the labels provided with the given data set.].One half of this labeled subset is used as a labeled training set (T_l) to train the initial SVM classifier.The other half of this labeled subset is used as a “validation” set (V), to estimate an approximate class posterior for the SVM, using the approach described in <cit.>. The (large) remainder in T_r is taken as the pool of unlabeled samples available to the active learner (T_u), from which the active learner selects for Oracle labeling (and into which the attacker inserts adversarial samples). In addition, there is a labeled test set to evaluate classifier accuracy. §.§ Sample Selection Criteria for Active Learning The active learner selects samples from T_u, one by one, for labeling by the oracle, with the SVM classifier retrained after each oracle labeling. We investigate the following AL sample selection strategies:Uncertainty sampling: Uncertainty sampling is the simplest and most commonly used sample selection strategy <cit.>. In this strategy, the AL chooses the nearest sample in T_u to the current boundary (the most uncertain sample).Max-Expected-Utility (MEU): Sampling to maximize expected utility (classifier gain) <cit.>. For data x_i∈ T_u, i^*=argmax_i Ũ_̃ĩ(θ), where Ũ_̃ĩ(θ)=∑_y_ip_θ (y_i|x_i)1/N( ∑_j∈ L∪ i p_θ_+i(y_j|x_j)+ ∑_j∈ U ∖ i∑_y_j p_θ (y_j|x_j)p_θ_+i(y_j|x_j)). Here, θ is the current set of class posterior parameters, and θ_+i reflects the updated parameters after adding x_i to T_l with its putative label y_i. Using Platt probabilistic outputs for SVMs <cit.>, the posterior probability p_θ (y_i|x_i) is based on a logistic regression approximation to the SVM (hard) decision function. Finally, N=|T_l|+ |T_u|.Random sampling: Select from T_u according to a uniform distribution. This sample selection acts as a baseline. Mixed strategies: Choose the sample by MEU (or random sampling) with probability p; otherwise by uncertainty sampling with probability 1-p.Note that a non-zero proportion for uncertainty sampling is warranted becauseuncertainty sampling is a very good mechanism for discovering unknown classes that may be latently present in T_u <cit.>. At the same time, using uncertainty sampling plays into the hands of the attacker.§.§ The Attacker We assume the attacker has the same knowledge as the active learner (the attacker knows T_l, T_u, and the current SVM classifier boundary). The attacker adds one sample to T_u at each active sample selection round, chosen as follows: 1) he projects all training pool samples (T_u and T_l) onto the current decision boundary (hence creating a (rich) candidate pool of high uncertainty samples), as shown in the figure below; 2) the attacker injects into T_u the candidate from this pool with minimum expected utility, based on the expected utility objective given above. § SYNTHETIC DATA EXPERIMENTS§.§ DatasetWe first consider the two-dimensional feature space problem from <cit.>. The data X is generated such thatthe class 1 instances have a bivariate normal distribution centered at (2,0), with the class 2 instances bivariate normal centered at (-2,0); both classes use an isotropic (identity) covariance matrix, and the classes are equally likely. We generated 105 instances from each of the two classes as T_r initially, from which we drew 5 samples at random from each class to form T_l, andanother 5 samples from each class to form V. The remaining 190 instances were taken as T_u. We also generated 200 instances from each class as test set. This dataset is (obviously) not linearly separable. The optimal decision boundary is the Y axis.Our oracle deterministically assigns labels consistent with this optimal decision boundary. §.§ ResultsFirst, we conducted a single experimental trial, for which AL selects samples strictly using uncertainty sampling; thus, the attacker's adversarial samples are selected and labeled at each round[Note, though, that, unlike<cit.>, the oracle does not perform any mislabeling.]. Fig. <ref> shows that the labeled adversarial samples induce a decision function that deviates from the optimal rule (the boundary becomes tilted from vertical). Hence, the attacker does have the ability to degrade AL classification accuracy (even without any oracle mislabeling). In the following experiment, we performed 10 random trials. In each trial, we used the same T_r, but randomly chose the initial T_l and V from T_r.We computed average performance for different AL strategies, over these 10 trials. Fig. <ref> and Fig. <ref> show performance both in the presence of and in the absence of the attack, respectively. Different sample selection strategies show different abilities to defeat the attack, as shown in Fig. <ref>. We have the following observations: * When AL uses strict uncertainty sampling (p=0), adversarial samples degrade classification accuracy successfully for the first 15 queries, which is consistent with single-trial results. However, the fabricated samples the attacker inserts do not degrade performance in the long run. Thus, the attack effectivelyonly delays convergence to a good decision boundary[Mislabeling would likely allow perpetuated accuracy degradation.]. When the current boundary is very close to the optimal one, if the attacker still adds fabricated samples to the current boundary, these samples may even be helpful to refine the boundary.As shown in the sequel, however, the attack is more successful on a real-world, high-dimensional digit recognition domain. * Fig. <ref> indicates MEU (p=1) is not substantially affected by adversarial samples (as one would expect, since the MEU sample selection criterion is the antithesis of the adversary's sample generation strategy), and makes the test error decrease the most at the beginning. However, we also noticed that MEU selects samplesin a class-biased fashion (many samples from one class), leading to a biased active learner after multiple queries. It is also clear from Fig.<ref> and Fig. <ref> that MEU converges to a suboptimal decision boundary in the long run.* With the attack present, the mixed strategies shows improved accuracy for increasing p, p ∈ [0.25,0.75].Moreover, in the absence of the attack, there is little accuracy difference for different choices of p.* Note also that the random strategy is robust to the attack, but does not converge to a solution as accurate as the mixed strategies.§ HANDWRITTEN DIGIT EXPERIMENTS§.§ DatasetWe used the MNIST dataset, consisting of 28×28 pixel grayscale images, learning a linear SVM to discriminate between the digits “5" and“6", that is, we have 784 (pixel) features and a two-class problem. We initially chose 105 “5" digits and 105 “6" digits as T_r and again labeled 5 samples from each class to form T_l, and 5 samples from each class to form V. The remainder of T_r was taken as T_u. Also, we randomly chose another (distinct set of) 456 “5" and 462 “6" samples to form a test set. Our oracle is assumed to bean SVM trained on the entire data set.Since the entire data set is linearly separable, this is a plausible choice for the oracle.Note, also, that because the data set is linearly separable, this oracle assigns ground-truth labels to all original data samples that are chosen for oracle labeling – the oracle only manufactures labels for the adversarial samples that are selected for labeling (and in this case it does so objectively, consistent with maximum margin linear separation of the entire data set). §.§ ResultsAs described in <ref>, candidate adversarial samples are the projections of all samples in T_l and T_u onto the current boundary. Some candidates shown in Fig. <ref> involve superposition of the two digits – labeling such samples and subsequent classifier retraining is expected to have a (negative) impact on the classifier decision boundary.Again, we performed 10 random trials to get the average performance with/without attack in Fig. <ref> and Fig. <ref>, respectively.To summarize the results:* Using uncertainty sampling (p=0), AL always selects the attacker's injected samples, and is the best strategy without attack, but worst with attack.Note also that the attack much more substantially delays learning progress for this high-dimensional domain, compared with the 2-D example.* MEU (p=1) makes the test error decrease the most at the beginning, but it converges to a suboptimal decision boundary, both with and without the attack.* The mixed strategies have a good defensive capability against the attack (by sometimes using MEU), and they also alleviate the unbalanced selection problem of MEU by sometimes using uncertainty sampling. With the attack, there is monotonically improving performance with p, for p ∈ [0.25,0.75] – using the mixed strategy with p=0.75, Fig.<ref> shows the test error has a fast and steady decrease.All the mixed strategies perform similarly without the attack.* The random strategy fares well with the attack, but not when the attack is absent.§ FUTURE WORKIn future work, we will investigate some of the other adversarial learning defenses suggested in our review section.We may also investigate alternative AL sample selection strategies – , a modification of the MEU strategy that does not suffer from the biased sampling we observed in our experiments (This may be achieved by estimating class proportions and modifying the MEU strategy to maximize expected utility, but while also sampling consistently with these class prior estimates.). Further, we will continue to study mixed strategies to discover the unknown classes. plain | http://arxiv.org/abs/1705.09823v1 | {
"authors": [
"David J. Miller",
"Xinyi Hu",
"Zhicong Qiu",
"George Kesidis"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170527143527",
"title": "Adversarial Learning: A Critical Review and Active Learning Study"
} |
http://arxiv.org/abs/1705.09264v1 | {
"authors": [
"Michał Tomza"
],
"categories": [
"physics.atom-ph",
"cond-mat.quant-gas",
"physics.chem-ph"
],
"primary_category": "physics.atom-ph",
"published": "20170525171159",
"title": "Cold interactions and chemical reactions of linear polyatomic anions with alkali-metal and alkaline-earth-metal atoms"
} |
|
Radar sounding of Lucus Planum, Mars, by MARSIS [=============================================== We examine the frequency shifts in low-degree helioseismic modes from the Birmingham Solar-Oscillations Network (BiSON) covering the period from 1985 – 2016, and compare them with a number of global activity proxies well asa latitudinally-resolved magnetic index. As well as looking at frequency shifts in different frequency bands, we look at a parametrization of the shift as a cubic function of frequency. While the shifts in the medium- and high-frequency bands are very well correlated with all of the activity indices (with the best correlation being with the 10.7 cm radio flux), we confirm earlier findings that there appears to have been a change in the frequency response to activity during solar cycle 23, and the low-frequency shifts are less correlated with activity in the last two cycles than they were in Cycle 22. At the same time, the more recent cycles show a slight increase in their sensitivity to activity levels at medium and higher frequencies, perhaps because a greater proportion of activity is composed of weaker or more ephemeral regions. This lends weight to the speculation that a fundamental change in the nature of the solar dynamo may be in progress.methods: data analysis – methods: statistical – Sun: helioseismology § INTRODUCTIONThe current solar activity Cycle 24 has been significantly weaker than the previous few cycles <cit.>. These changes were signposted by the unusually extended and deep solar minimum at the boundary of Cycles 23 and 24.Very few of the predictions collated by the Solar Cycle 24 Prediction Panel <cit.> forecast the extent of the minimum or the low levels of activity that followed.One must go back around one-hundred years to find cycles that show levels of activity as low as those observed in Cycle 24, e.g., Cycles 14 and 15 both provide very good matches in traditional proxies such as the International Sunspot Number (ISN). Tellingly, this earlier epoch pre-dates both the modern Grand Maximum period and the satellite era. The wide range of contemporary observations and data products was therefore not available to characterize and study the Sun during that era.Some activity indicators dropped to remarkably low values during the Cycle 23/24 minimum (e.g. the geomagnetic aa-index and the ISN). Solar wind turbulence, as captured by measures of interplanetary scintillation, had been declining since the early part of Cycle 23. There have been results suggesting a decline – from the Cycle 23/24 boundary through the rise of Cycle 24 – in the average strength of magnetic fields in sunspots (; see also)and others pointing to a change in the size distribution of spots between Cycles 22 and 23<cit.>. Helioseismic studies of the internal solar dynamics showed that the characteristics of the meridional flow altered between Cycles 23 and 24 <cit.>.Changes to the meridional flow have potential consequences for flux-transport dynamo models. Differences have also been seen in the east-west zonal flows<cit.> and in the frequency shifts of globally coherent p modes, which have been weaker than in preceding cycles <cit.>.<cit.> have suggested that it was actually a weak Cycle 23 that was responsible for the following, extended minimum and weak Cycle 24. <cit.> have proposed that observed weak polar magnetic fields, and as a result the weak Cycle 24, may have resulted from the emergence of low-latitude flux having the opposite polarity to that expected (which then hindered growth of the polar fields). Predictions for Cycle 25 are now beginning to appear <cit.>.Around the time of the early stages of Cycle 24, we used helioseismic data collected by the Birmingham Solar-Oscillations Network (BiSON) to uncover clear signs of unusual behaviour in the near-surface layers that appeared as far back as the latter stages of Cycle 22 <cit.>. We analysed the cycle-induced frequency shifts shown by modes in three different frequency bands of the p-mode spectrum. The relationship of the frequency shifts to the ISN and the 10.7-cm radio flux – the latter another commonly used proxy of global solar activity – changed noticeably, and the close correlation with the indices was lost during a period stretching from the tail end of Cycle 22 through the Cycle 23 maximum (with the lowest-frequency modes losing the correlation first). Whilst the correlation recovered for modes above ≃ 2400 μ Hz, it failed to do so for the lower-frequency modes. The results imply an underlying change in structure very close to the surface, where the lower-frequency cohort is less sensitive to perturbations. We showed that one may interpret the structural change in terms of a thinning of the layer of near-surface magnetic field, post-Cycle 22.These helioseismic markers were sufficiently robust that, with the benefit of hindsight, they arguably may have provided an early warning of the changes that would be seen in other proxies as the Sun headed into Cycle 24. At the time of writing, Cycle 24 is about half-way down its declining phase. Our goal in this paper is, therefore, to use the up-to-date BiSON data to test whether the unusual behaviour uncovered by our previous analysis has persisted through the declining phase of the cycle, and what the results might mean for Cycle 25. The rest of the paper is laid out as follows. Details on the data used, and the analysis performed, are given in Section <ref>. We discuss results on the extracted frequency shifts and activity proxies in Section <ref>, including a new way of presenting the information encapsulated in the frequency shifts. We finish the paper in Section <ref> by discussing the implications of the results, not only for Cycle 25, but also in the wider context of the activity behaviour of Sun-like stars. § DATA AND ANALYSIS The six telescopes comprising BiSON make unresolved “Sun-as-a-star” observations of the visible solar disc <cit.>, and thereby provide data that are sensitive to the globally coherent, low angular-degree (low-l) solar p modes. Whilst these modes are formed by acoustic waves that penetrate the solar core, they are very sensitive to perturbations in the near-surface layers and hence provide a useful diagnostic and probe of the global response of the Sun to changing levels of solar activity and the resulting near-surface structural changes.The BiSON observations constitute a unique database that now stretches over four 11-year solar activity cycles, i.e., from Cycle 21 through to the falling phase of the current Cycle 24. The observations we use here span the period 1985 July 2 through 2016 December 31, or in other words Cycles 22, 23, and most of 24; the data from Cycle 21 are too sparse to be suitable for the current analysis. The raw Doppler velocity data were prepared for analysis using the procedures described by <cit.>, and then they were divided into overlapping subsets of length 365 days, offset by 91.25 days. The frequency-power spectrum of each subset was fitted to a multi-parameter model to extract estimates of the low-l mode frequencies.Details can be found in<cit.>, but while that paper used two different codes to extract the frequencies, for this work we used only frequencies estimated using the more sophisticated method, a Markov-Chain Monte-Carlo sampler.We deliberately re-analyzed the entire database in order to verify, using independent analysis codes, the results presented by <cit.>. Having extracted the individual frequencies, we then followed the procedure outlined by <cit.>to extract averaged frequency shifts for each 365-day segment, in three frequency bands. The bands cover low (1860 < ν_nl≤ 2400 μ Hz), medium (2400 < ν_nl≤ 2920 μ Hz) and high frequencies (2920 < ν_nl≤ 3450 μ Hz). The reference frequencies used for computing the variations came from averages of the frequencies for the 13 subsets straddling the Cycle 22 maximum (spanning the dates 1988 October 1 to 1992 April 30).In addition to using averaged frequency shifts, we also used results from parametrizing the frequency shifts as a function of frequency(;see also ). The frequency shifts of individual modes, δν_nl(t), were first scaled by the mode inertia, E_nl, of model “S” of<cit.>. This removes the dependence of the shifts on inertia, leaving signatures due to perturbations in the near-surface layers.The scaled shifts of each 365-day subset were then fitted to a parametrized one or two-term function in frequency ν, i.e., ℱ(ν) = a_ inv( ν/ν_ ac)^-1 + a_ cub( ν/ν_ ac)^3,or in the one-term case simply ℱ(ν) =a_ cub( ν/ν_ ac)^3, where ν_ac is the acoustic cut-off frequency of 5 mHz. The fit yields best-fitting coefficientsa_ cub, and optionally a_ inv, for each subset. The rationale given by <cit.> for this choice of terms is, briefly, that the cubic term corresponds to amodification of the propagation speed of the waves by a fibril magnetic field close to the surface, as also suggested by <cit.>, while the inverse term would correspond to a change in scale height in the superadiabatic boundary layer.For the low-degree data we found that the a_ inv term was not statistically significant, so here we work with the cubic term only.The averaged frequency shifts for each band, and the best-fitting coefficients from the cubic-function parametrization of the shifts, constitute our core helioseismic diagnostic data.§ RESULTS Fig. <ref> shows the extracted averaged frequency shifts in each band as a function of time. Also plotted are data on the 10.7-cm radio flux <cit.> [available from the National Geophysical Data Center, http://www.ngdc.noaa.gov], the revised Brussels-Locarno Sunspot Number <cit.>[available fromhttp://www.sidc.be], a merged CaK index<cit.>[http://solis.nso.edu/0/iss/sp_iss.dat] and a global magnetic field-strength index based on Kitt Peak synoptic magnetogram data <cit.>, each averaged over the same epochs as the frequencies, and scaled by a linear fit to the frequency shifts overthe maximum and descending phase of Cycle 22, (i.e, the date range from 1990.0 to 1996.5). Fig. <ref> shows the extracted coefficients a_ cub, as a function of time, with the radio-flux, sunspot, CaK, and magnetic data overplotted, again scaled by a linear fit to the cubic coefficients in the latter part of Cycle 22.Signatures of the 11-year solar activity cycle are the dominant feature of both sets of diagnostics. However, both frequencies and activity proxies also show some shorter-term variability with an average period of around 2 years. This periodicity is a known feature of several global proxies of solar activity <cit.> and has been detected in a number of helioseismic datasets<cit.>. Following <cit.>, we have removed shorter-period variations, including the 2-year signal, by smoothing the respective sets of averaged frequency shifts and extracted coefficients over nine samples or 2.25 years.The resulting smoothed diagnostic data are plotted in Fig. <ref> for the average frequency shifts and Fig. <ref> for the cubic coefficients, together with smoothed and scaled versions of the activity proxies. Figures <ref> and <ref> correspond to Figures 2 and 3 of<cit.>, with the slight difference that we have scaled the activity proxies to the frequencies over a wider range in time, while the cubic-fit coefficients in Figures <ref> and <ref> combine data from all three bands but are more heavily weighted towards the higher-frequency modes. The first important point to make is that our re-analysis of the BiSON data confirms the results presented in <cit.>, although there are some differences in detail that may be due to the improved frequencyestimation. Prior to ≃ 1994, during the latter stages of Cycle 22, changes in the frequencies followed reasonably closely the variations shown by the global activity proxies. However, at epochs thereafter, the behaviour of the frequency shifts in the low-frequency band departed strongly from the proxies, with detected variations in the frequencies being much weaker than expected (based on the behaviour seen in Cycles 21 and 22); in particular, these frequencies stayed higher than expected during the very low-activity period of the minimum following Cycle 23.We also note that the relationship between the radio flux and sunspot data temporarily changed during the declining phase of Cycle 23 (as pointed out by; see alsofor comments on the comparison with the newly recalibrated sunspot data). Our new results extend the observations into the declining phase of Cycle 24, and they show clearly that the significant departures seen at low frequencies have persisted as we head towards the onset of the next cycle. Put another way, the acoustic properties of the near-surface layers have failed to re-set to their pre-1994 state.While for the mid- and high-frequency band shifts and the cubic parametrization coefficient the correlation with the activity indices does appear to recover in the rising phase of Cycle 24, the shifts again deviate from the extrapolated fit to Cycle 22 in the most recent data corresponding to the declining phase of Cycle 24, at least for the RF, magnetic, and sunspot indices. We note that the scaled RF and sunspot proxies show very similar behaviour except in the declining phase of Cycle 24, while the CaK and magnetic proxies are fairly close to one another. While the magnetic and CaK proxies fall to noticeably lower levels during the Cycle 23/24 minimum than during the previous minimum, the difference between the minima is less pronounced for the RF and sunspot number. Interestingly, the frequency shifts in the high-frequency band, as well as the cubic-fit coefficients, appear to follow the magnetic and CaK pattern, while the frequency shifts in the middle band follow the RF and sunspot number and the low-frequency band is not a good match to any of the proxies in this period. During the maximum epoch of Cycle 24 the frequencies in the high and middle bands seem to follow the extrapscaled RF and CaK proxies while the sunspot and magnetic proxies show poorer agreement, and in the declining phase so far only the CaK looks like a good match to the frequency shifts. We emphasise that all of the proxies have been scaled to match the relationship to the frequency shifts that was seen in the 1990 – 1996.5 epoch; better fits could be obtained by fitting to each cycle separately. The difference in behaviour at the Cycle 23/4 minimum is striking, however, and it is not an artefact of the scaling. <cit.> also found that the medium-degree frequencies from the Global Oscillation Network Group were systematically lower during this minimum than during the previous one, as would be expected when the fields are weaker.Because the BiSON observations are primarily sensitive to the sectoral, or m=l modes, we can expect the frequency shifts for l≥ 1 to show some hysteresis with global activity measures. To examine the impact this may have on our results, we carried out the frequency-cubed parametrization for each l separately. Figure <ref> shows the unsmoothed cubic coefficient derived from fits to thel=0 shifts only, plotted against each of the three unresolved activity indices and colour-coded by cycle. Also shown is a straight line representing the extrapolation of a linear fit to the Cycle 22 data after 1990. In each case, it can be seen that the data for the Cycle 23 and 24 maxima lie generally above the line. This suggests that in the two most recent cycles we are seeing a larger shift overall for the same amount of activity than we did in Cycle 22, in the middle and higher frequency bands that dominate the cubic fit. This might make sense if there has been a shift towards a greater proportion of weak, ephemeral activity that does not register in the sunspot number or on the synoptic magnetic charts but could still have an influence on the modes. The numerical results of fits betweenthe l=0 cubic coefficient and the activity proxies, for each cycle individually and for the entire data sets, are given in Table <ref>.In the case of the magnetic proxy, we do have information on the latitudinal distribution of activity in the synoptic magnetograms, and we can use this to derive an index appropriate to the sectoral mode at each value of l <cit.>. In Figure <ref> weshow the cubic coefficientsfor each l separately as a function of the projection of the latitudinal field-strength distribution on the corresponding m=l spherical harmonic. The results of linear fits to the magnetic proxy for each cycle separately and for the whole dataset are shown in Table <ref>. Again, we can see that the points corresponding to the maxima of Cycles 23 and 24 lie above the curve indicating the trend in Cycle 22, so we cannot attribute the hysteresis we observe in the averaged frequencies simply to the latitudinal distribution of activity. The l=3 points from early Cycle 22 that lie well below the line are probably due to poor fits of these modes (to which a Sun-as-a-star instrument like BiSON is not very sensitive) in the low duty cycle of the early observations. § DISCUSSION We have used the latest BiSON helioseismic data to show that previously uncovered changes in the structure of the near-surface layers of the Sun, which date back to the latter stages of Cycle 22 (around 1994), have persisted through the declining phase of the current, weak Cycle 24. The acoustic properties have as such failed to re-set to their pre-1994 state. While the agreement at higher frequencies did appear to recover in the rising phase of Cycle 24, when the most recentdata are added we can see that there are again differences from the proxies extrapolated from Cycle 22. This supports the suggestion of <cit.> that the magnetic changes affecting the Cycle 23 (and later) oscillation frequencies were confined to a thinner layer than those in Cycle 22. We also find that the sensitivity of the frequencies in the higher-frequency bands to the magnetic proxy is slightly higher in the two most recent cycles than in Cycle 22, which could be due to a higher proportion of weaker, more ephemeral active regions that are not accounted for in the synoptic magnetic data. This observation still holds when we separate out the frequency shifts by degree and compare with a latitudinally resolved magnetic proxy. It is tempting to speculate whether these results, and the multitude of other unusual signatures relating to Cycle 24, might be indicative of a longer-lasting transition in solar activity behaviour, and the operation of the solar dynamo. The existence of the Maunder Minimum, and other similar minima suggested by proxy data relevant to millennal timescales, indicate that there have likely been periods when the action of the dynamo has been altered significantly. We finish by speculating whether these events might presage a radical transition suggested by data on other stars. Results on activity cycle periods shown by other stars hint at a change in cycle behaviour – a possible transition from one type of dynamo action to another – at a surface rotation period of around 20 days <cit.>. There is also more recent intriguing evidence from asteroseismic results on solar-type stars <cit.> that shows that the spin-down behaviour of cool stars changes markedly once they reach a critical epoch, with the corresponding surface rotation period depending on stellar mass. For solar-mass stars, the results suggest a change in behaviour at about the solar age (and solar rotation period). §.§ ACKNOWLEDGMENTS We would like to thank al those who are, or have been, associated with BiSON, in particular P. Pallé and T. Roca-Cortes in Tenerife and E. Rhodes Jr. and S. Pinkerton at Mt. Wilson.BiSON is funded by the Science and Technology Facilities Council (STFC), under grant ST/M00077X/1.SJH, GRD, YPE, and RH acknowledge the support of the UK Science and Technology Facilities Council (STFC). Funding for the Stellar Astrophysics Centre (SAC) is provided by The Danish National Research Foundation (Grant DNRF106).NSO/Kitt Peak data used here were produced cooperatively by NSF/NOAO, NASA/GSFC, and NOAA/SEL; SOLIS data are produced cooperatively by NSF/NSO and NASA/LWS. RH thanks the National Solar Observatory for computing support. SB acknowledges National Science Foundation (NSF) grant AST-1514676. | http://arxiv.org/abs/1705.09099v1 | {
"authors": [
"R. Howe",
"G. R. Davies",
"W. J. Chaplin",
"Y. Elsworth",
"S. Basu",
"S. J. Hale",
"W. H. Ball",
"R. W. Komm"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170525091540",
"title": "The Sun in transition? Persistence of near-surface structural changes through Cycle 24"
} |
firstpage–lastpage Holomorphic foliations tangent to Levi-flat subsets Jane Bretas & Arturo Fernández-Pérez & Rogério Mol Received: date / Accepted: date ======================================================== One longstanding problem for the potential habitability of planets within M dwarf systems is their likelihood to be tidally locked in a synchronously rotating spin state. This problem thus far has largely been addressed only by considering two objects: the star and the planet itself. However, many systems have been found to harbor multiple planets, with some in or very near to mean-motion resonances. The presence of a planetary companion near a mean-motion resonance can induce oscillatory variations in the mean-motion of the planet, which we demonstrate can have significant effects on the spin-state of an otherwise synchronously rotating planet. In particular, we find that a planetary companion near a mean-motion resonance can excite the spin states of planets in the habitable zone of small, cool stars, pushing otherwise synchronously rotating planets into higher amplitude librations of the spin state, or even complete circulation resulting in effective stellar days with full surface coverageon the order of years or decades. This increase in illuminated area can have potentially dramatic influences on climate, and thus on habitability. We also find that the resultant spin state can be very sensitive to initial conditions due to the chaotic nature of the spin state at early times within certain regimes. We apply our model to two hypothetical planetary systems inspired by the K00255 and TRAPPIST-1 systems, which both have Earth-sized planets in mean-motion resonances orbiting cool stars. planets and satellites: dynamical evolution and stability – stars: low-mass § INTRODUCTION The discovery of many exoplanetary systems in recent years has accelerated the long-standinginterest in the prospect of discovering and characterising habitable planets around other stars. Because of their ubiquity and their longevity, there has been particular interest in determining the frequency with which habitable planets exist around M dwarf stars. Due to the low luminosity of M dwarfs, the habitable zone in such systems is closer to the central star than in direct analogs of our own Solar System. The transit detection method is thus very well suited to detecting exoplanets within the habitable zone of M dwarfs, with larger light curve transit depths and a higher likelihood of the favorable geometry necessary for a transit detection. The overall advantages that M dwarfs present for harboring and discovering habitable exoplanets are such that it is often called the “small star advantage”. Data from the Kepler mission <cit.> suggest a probability of 95% confidence that an Earth sized planet within the habitable zone of an M dwarf can be detected through the transit method within 10 pc. Such a detection wouldrepresent an attractive opportunity for detailed atmospheric study and characterisation. However, much of this optimism is predicated on the assumption that a similar level of irradiation to that of Earth correlates with habitable conditions and the existence of liquid water at the surface. Indeed, a long known cautionary note holds that such planets would be tidally locked in a synchronous spin state, raising the possibility of extreme temperature differences and strong winds across the planetary surface, which may lessen the similarity to Earth-like conditions <cit.>. This problem may be mitigated by the behaviour of the planetary atmosphere itself – such as if it has sufficiently strong atmospheric tides <cit.> or cloud formation near the substellar point <cit.>. It is also possible for a planet to get trapped into a non-synchronous spin-orbit resonance <cit.>, depending on the degree to which the planet deviates from spherical and the functional dependence of the tidal dissipation. This is believed to be the case for Mercury in our own Solar System. In this paper, we investigate the possibility that the tidal locking can also be weakened by purely gravitational effects. The salient point to note is that most prior analyses of this problem have addressed the case of a single planet orbiting the star. Yet, we know that many of the recently discovered planetary systems are (often highly) multiple <cit.>), even when restricting the sample to only M dwarfs <cit.>. Thus, gravitational interactions between neighbouring planets may also influence planetary spin states. One proposed way for a neighbouring planet to influence the spin of another is through spin-orbit resonances of the second kind <cit.>, where interactions cause the same face of the concerned planet to be oriented towards the neighbouring planet at conjunction, given that the concerned planet is sufficiently deformed. However, this requires that the tidal torque contains a significant dependance on the synodic alignment of the planets, which is doubtful. Instead, we demonstrate that, if a system contains two planets near a mean-motion resonance, then interactions can be sufficiently strong (and sufficiently rapid) to have a significant effect on the spin state. Although secular interactions can also drive changes in planetary properties, these usually occur on longer timescales and suggest that the planetary spin would simply represent a long-term average over the secular oscillations. As we shall show, resonant interactions can drive evolution on timescales sufficiently fast to lead to interactions with the climate system, opening up the possibility of interesting feedback between the spin evolution and climate. However, a mean motion resonance is a special configuration, and thus the model is only applicable to a fraction of the observedand postulated planetary systems. Nevertheless, such systems are known to exist. Examples include K2-21 <cit.>, which features two sub-Jovian planets near a 5:3 resonance orbiting around an M0 dwarf; Kepler-32 <cit.>, an M0 star which features 5 planets with low order commensurabilities between neighbours, and TRAPPIST-1 <cit.>, with seven planets orbiting a star 0.08 M_⊙, including several that appear to be close to first order commensurabilities. Even if such systems represent only a fraction of the M dwarf planetary population, their special properties may make them preferred candidates for detailed study. Consequently, in this paper, we model the spin state of an Earth-sized planet near a first order mean-motion resonance with another planetary mass companion. We introduce our model and define our parameter space of interest in <ref>. In <ref> we then present a case study of two hypothetical systems inspired by the K00255and the TRAPPIST-1 systems. These serve to illustrate the kind of behavior we might expect, and we draw more general conclusions in <ref>, including the potential for interaction with the planetary climate. § MODEL We are principally interested in systems containing rocky, terrestrial-type worlds, orbiting within the habitable zone of small, cool stars. With the expectation that obliquity should quickly erode for planets within such systems <cit.>, we will consider a system with a planet of mass m orbiting a star of mass M_* with zero obliquity. We must also allow for potential deviations from a spherical figure, and therefore let A, B, and C be the principal moments of inertia of the planet, with C being the moment about the spin axis, and choose A to be the moment about the long axis of the planet (axis in the plane of the orbit) such that B > A. We let θ be the angle formed between the long axis of the planet and a stationary line in the inertial frame. In anticipation that the final equilibrium is one close tosynchronous rotation, we define an angle γ≡θ - M, with M being the mean anomaly of the planet's orbit. Thus, γ can be said to represent deviations from a perfectly synchronous spin state, and roughly corresponds to the longitude of the substellar point on the planet (if the orbit is nearly circular). If we assume that the system is near a 1:1 spin-orbit resonance, such that θ̇≈ n, then we obtain the following equation of motion, as shown by <cit.> (hereafter MD99) γ̈ + sign[H(e)]1/2ω_S^2 sin 2γ +ṅ = 0 where H(e) is a power series in orbital eccentricity e, explained further by MD99, and ω_S^2 = 3 n^2 (B-A/C) |H(e)|. Equation <ref> describes the standard model of a single, spinning, planet in a frame rotating with the orbit. This model, along with a component to describetidal damping, is the standard way of modelling the spin evolution of such planets, and can incorporate both synchronous and asynchronous equilibria, depending on the form of the function H. The most notable application of this model is to describe the trapping of Mercury in the 3:2 spin-orbit resonance <cit.>. This model is also the basis for our analysis, with one crucial distinction. One traditionally sets ṅ = 0 at this point and describes the libration of the spin state about various possible equilibria defined by the properties of H(e). This is appropriate for a single planet on a Keplerian orbit, and even for planets in multiple systems when the interactions are predominantly secular. However, the interaction of planets near mean motion resonances can yield small variations in n that prove important for our purposes. In this paper, we allow mean-motion n to vary by considering the consequences of adding a second planet to the system such that the pair lies near a first order mean-motion resonance. We can describe any fluctuations in n due to a planetary companion via the “Pendulum Model”, also described in detail by MD99. This model describes variations of a resonant angle ϕ = (j + 1) n' - j n - ω̅, where n and n' are the orbital frequencies of the planet and perturber respectively, and ω̅ is the planets longitude of periastron. The constant j is the index which denotes the specific choice of resonance.With this formulation, then, we describe the evolution of the planet in the context of the circular restricted-three body problem, where the outer satellite is on a perfectly circular orbit in the same plane as the inner satellite and does not react to the presence of the inner body. This model is thus most directly applicable when the perturber is substantially larger than the planet itself, but captures the qualitative behaviour in the case of approximately equal mass pairs as well. Thus, the evolution of n is given by ṅ = - 1/jω_M^2 sinϕ and that the motion of ϕ is given by ϕ̈ = -ω_M^2 sinϕ where ω_M^2 = - 3 j^2 C_r n e is, for first-order resonances, the angular frequency of oscillations of the resonant argument. C_r is a constant depending on the nature of the perturber and the resonance, given as C_r = (m'/m_c)(a/a')nf_d, with a and a' being the semi-major axis of the orbit of the primary and outer perturber, and f_d being a function of the ratio a/a', tabulated in MD99 as a/a' f_d = -0.7500 and -1.5455 for the 2:1 and 3:2 resonant cases, respectively. We can now use the pendulum model to calculate ṅ in Equation <ref>, from which we obtain the following equation of motion to describe the spin of the planet of interest γ̈ + sign[H(e)]1/2ω_S^2 sin 2γ + 1/jω_M^2 sinϕ = 0 We note that Equation <ref> describes the behaviour ofa forced pendulum, a model that arises in a variety of contexts. Indeed, this is one of the favorite subjects of dynamical systems studies (e.g. <cit.>), because it can show a range of interesting chaotic behaviors when the two frequencies ω_S and ω_M are of similar order. The only difference here relative to the more commonly analysed harmonic forcing case is that our forcing term ṅ is itself a pendulum, with ϕ being allowed to circulate or librate. Note also that the behaviour of ϕ does not depend on γ, so there is no feedback into the orbit from the spin evolution. §.§ Parameter Choices Clearly, in order for this new driving force to exert a significant effect on the planetary spin state, the coefficient of the second pendulum term should have a value of similar order to the first, or larger. Therefore, the behaviour of the system will be characterised by the ratio 2 ω_M^2/ j ω_S^2 = 2 f_d (j^5/2/j+1)^2/3( m'/M_*) ( e/H(e) /(B-A)/C) where j is an integer which characterises the particular resonance, along with f_d, which is a dimensionless value arising from the expansion of the direct part of the disturbing function, as shown in MD99. Let us also make the simplifying assumption that H(e) ∼ 1, as would be the case for the analogue of the synchronous equilibrium to lowest-order in e. In principle, this system could be trapped into higher order spin states just like Mercury is, but our goal is to examine the consequences of resonant perturbations for the synchronous state. If we set equation (<ref>) equal to unity, we can derive an estimate of the kind of perturber mass that is likely to yield deviations from synchronism around a prototypical M dwarf (assuming a 2:1 mean motion resonance) m'≈0.33/ eM_⊙(B-A/C) (M_*/0.5 M_⊙) . It is clear that, the more triaxial a planet is, the bigger is the perturber required to have a significant effecton the planetary spin, and that a smaller perturber is required if eccentricity is large (because this will amplify the oscillations in ṅ resulting from the resonant forcing). Thus, an important question is what values of these parameters are likely to be appropriate. We take a value of (B-A)/C ∼ 2 × 10^-5 as our default value, based on the available empirical evidence determined for the Earth, Mars and Mercury <cit.>, which implies a characteristic perturber mass m' ∼ 2.2/e M_⊕. Estimates of the maximum possible triaxiality <cit.> suggest that a value of order ∼ 10 times higher is possible, depending on mass and ocean depth. This would require a correspondingly higher value of m'. The optimal perturber mass also depends on eccentricity. The eccentricities for low mass planetary systems are quite difficult to measure, but estimates based on transit timing variations <cit.> suggest that the majority have non-zero but small eccentricities, with estimates ranging from e ∼ 0.01-0.1. Such values are also consistent with the level of eccentricity expected from in situ assembly models <cit.> or the requirement that many escape capture into resonant chains during an extended period of inward migration <cit.>. Thus, we will adopt a characteristic value of e ∼ 0.05, suggesting an optimal perturber mass m' ∼ 44 M_⊕ for a 0.5 M_⊙ stellar host. Thus, it appears as though, by simple numerical coincidence, the susceptibility of a terrestrial planet to spin perturbations is such that Neptune-mass companions, which are observed in these systems, can drive interesting spin dynamics. As an example of an observed system to which this model might apply, consider the Kepler candidate system K00255, which contains a star of mass M_* = (0.53 ± 0.06) M_⊙, a confirmed planet K00255.01 of radius R=(2.51 ± 0.3) R_⊕ and orbital period T = 27.52 days, and a candidate planet K00255.02 of radius R = (0.68 ± 0.08) R_⊕ and orbital period T=13.60 days. This would imply a system near a 2:1 mean-motion resonance. We note that this system is almost perfect for our needs except that the outer planet is closer to the habitable zone than the inner planet, with the inner planet likely being too hot for life. Even so, the system can serve as a useful illustration of the kinds of behavior we may observe for more Earth-like planets. To estimate planetary mass, <cit.> and <cit.> suggest a mass-radius relation of M_p = M_⊕(R_p/R_⊕ )^2.06 for R_p > R_⊕, which fits well within the solar system.Adopting this scaling relation for the outer planet of the K00255 system yields a mass of m' = 7 M_⊕. Based on the values described above, equation (<ref>) implies 2 ω_M^2/j ω_S^2 ∼ 0.11. Although this is less than unity, it is not negligible, implying the possibility of a substantial interaction between the two terms in equation <ref>. Another recent system of interest in this regard is the TRAPPIST-1 system <cit.>, which contains seven transitting planets orbiting a very low mass (∼ 0.08 M_⊙) M dwarf. Several of the resulting planets form a chain of near resonant neighbouring pairs, and three lie within, or close to, the calculated habitable zone for a star of this mass. Radii and masses (constrained by transit timing variations) suggest that these planets are also rocky, making them an excellent case study for our model. The above discussion assumes a constant eccentricity. In principle this is a dynamical quantity, which varies under our formalism adopted from MD99 as ė = j_4 C_r sinϕ We can see that e and ϕ then feed back onto each other, with ω_M being dependent on e. We note, however, that to conserve angular momentum, the variations of eccentricity will depend on the variations of the semi-major axis a. If δ a_max is small enough, and the mean eccentricity large enough, then δ e_max will also be small.If we choose our fiducial e = 0.05, and then allow e to evolve with ϕ over time, we find that e varies by ∼ 20% at most for both our fiducial K00255 and TRAPPIST-1 analogue systems, with semi-major axis varying by ∼ 1%. With ω_M^2 ∝ e, we find that the strength of the forcing term in Equation <ref> can vary by as much as ∼ 20% and the driving frequency by ∼ 4% due to these variations in eccentricity. Though this can give us slightly different quantitative results for any individual simulation, we find that behavior remains qualitatively similar if we force e to be constant in our simulations. Thus, for our calculations to follow, we will assume a constant e=0.05 for clarity. §.§.§ Tidal Damping In order for the driving force in our model to affect the spin-state of the primary, eccentricity must have a non-zero value. Although we noted above that there are potential reasons to believe formation may leave small remnant eccentricities, the long term operation of tides can act to damp and circularise eccentricity. This is especially true in compact multiple planet systems, where the effects of secular interactions can extend the reach of tidal circularisation significantly <cit.>. The empirical estimates based on TTV measurements suggest that many systems have not yet been completely circularised. Nevertheless, it is important to confirm that there is a reasonable expectation of finite eccentricity in these systems. Furthermore, the effect of tides on spin will add another dimension to equation (<ref>), acting to synchronise the spin and thus enforcing the dynamics of a damped, forced pendulum. The effects of tides on stellar and planetary orbits is a subject of longstanding study (see <cit.> for a review), and the strength of the effects can be calibrated for giant planets (e.g. <cit.>). However, the data for lower mass planets is not good enough for a direct calibration, and we will draw on the traditional calibration relative to the strength of tides on Earth. This is traditionally expressed in terms of the `tidal Q' parameter, but we wish to retain the functional form of our prior formulation based on the work of <cit.> and <cit.>, because it has the attractive property that γ̈→ 0 as γ̇→ 0. Thus, using equations (4) and (14) of <cit.> we may describe the evolution of γ under the action of tides with γ̈ = -15/2γ̇M_*/m( R/a)^6 M_* R^2 σ where we express the strength of dissipation in terms of a bulk dissipation constant σ, and then calibrate that by calculating the equivalent value of Q under the specific forcing applied to Earth. In this expression, a is the semi-major axis of the orbit and R is the planetary radius. Under an assumption that the initial spin rateθ̇≫ n, then for Earth-sized planets in the habitable zone of an M dwarf (e.g. M_* = 0.5 M_⊙), this yields a spin-down (i.e. synchronization) timescale of τ_sync∼ 2× 10^6 years(a/0.15 AU)^6 (R/R_⊕)^-6m/M_⊕(M_*/0.5 M_⊙)^-2Q_⊕/10. We can calculate the circularization timescale in the same formalism, under the assumption that synchronization has happened quickly and θ̇≃ n, we get τ_circ =10^11years(a/0.15 AU)^13/2(R/R_⊕)^-5m/M_⊕(M_*/0.5 M_⊙)^-3/2Q_⊕/10. This demonstrates that our formulation is consistent, in the sense that planetary spins should be rapidly damped to the system equilibrium, but that orbital eccentricities can persist for the length of system age except for the very shortest period planets. In fact, if we use this formalism to describethe K00255 system, we get synchronization timescales of 6× 10^5 years for the inner planet and 10^6 years for the outer, while circularization timescales become 2× 10^11 years and 5.6× 10^9 years respectively. At first glance, this is a curious result, as it implies the more distant planet is circularised but the closer one is not. This is because of the strong sensitivity of the tidal strength to planet radius, and serves to further bolster our adoption of the circular restricted problem as a description of the system. Equation (<ref>) provides a torque on the spin that will act to damp this libration of γ. There is a growing literature on the mechanisms of tidal dissipation in earth-like planets <cit.>, but this simple formalism is sufficient to illustrate the necessary behaviour. In particular, we note that the torque goes to zero as γ̇ goes to zero, thereby avoiding unphysical discontinuities in the torque at synchronism. Adding this to Equation <ref>, we find γ̈ + 1/2ω_S^2 sin 2γ + ω_M^2/jsinϕ - ϵγ̇= 0. where ϵ = 15/2M_*/m(R/a)^6 M_*R^2σ. We also define a parameter β = 2ω_M^2/jω_S^2, so that β and ϵ will together characterise the behaviour of γ. It is the evolution of the spin under the influence of this equation that will determine the final spin state of the planet and the pattern of irradiation that it experiences. Our model is constructed to consider the evolution of γ when near the synchronously rotating resonant case, when γ̇ is small compared to the mean-motion n. However, equilibria potentially exist for γ = θ - pM, where p is any integer multiple of 1/2. One can then perform a similar analysis, with H(e) now also a function of p. One example is the Mercury 3:2 spin-orbit case, where the evolution of γ is studied choosing p=3/2. Thus, we note that the synchronously rotating case (p=1) is not the only spin-state that a planet can be captured into, and other resonant states would in fact result in stable asynchronous rotations. This potentially provides yet another solution to the problem of tidal-locking within the habitable zone of cool stars. The criteria for capture into one of these resonant states depends on the properties of the tidal dissipation<cit.>. Translated into our formalism, this requires the inequality | ϵγ̇_max | < 1/2ω_S^2 be satisfied. If this criterion is satisfied for a particular choice in p, then one may hope to determine the probability of capture into the corresponding spin-orbit resonance. Our study focusses on just the p=1 case, where the planet is very near the synchronously-rotating case, as the default configuration that will result if capture into higher order resonances is not possible. § RESULTS Equation (<ref>) will apply to the spin of any terrestrial planet in orbit about a low mass star, with a companion in an orbit close to a mean motion resonance. However, as we have noted, the effects are likely to be strongest when ω_S ∼ω_M (i.e. β∼ 1). As such, we focus here on the hypothetical system based on the properties of K00255. This system contains both an Earth-like interior planet and an external perturber of mass ∼ 7 M_⊕ near the 2:1 commensurability, in orbit around a 0.53M_⊙ star. However, the inner planet is too close to the star and lies interior to the nominal habitable zone for this star. For our illustrative simulations, we therefore choose to place the spinning 1 M_⊕ planet at a location farther from the star (0.15 AU), such that it receives the same stellar flux as Earth, with M_* = 0.5 M_⊙, e=0.05, and (B-A)/C = 2× 10^-5. We then place the companion of mass m'=7 M_⊕ on a perfectly circular outer orbit such that the system forms a 2:1 mean-motion resonance with the primary planet, and examine the spin dynamics of this system. These parameters yield a value of β = 0.11. For all following simulations in this paper, we solve Equation (<ref>) for γ using the Runge-Kutta fourth-order method of integration. Time steps adapt to the highest value of ω_M and ω_S, taking 70 steps in the period corresponding to the highest frequency. To start with, let us examine the evolution of our K00255 analogue system in the absence of dissipation (i.e. ϵ=0). The dynamics of a pendulum also requires the specifications of initial conditions. In this case, we not only require γ and γ̇, but also ϕ and ϕ̇. Starting with the latter pair, we choose ϕ=0 initially for all runs, but different ϕ̇. The value of ϕ̇(0) will then characterise the dynamics of the resonant pair – small enough values will yield libration of ϕ and larger values will result in circulation. It is also important to note that, when the driving force is of pendulum form, ω_M is a characteristic value, but the actual forcing frequency will be a function of ϕ̇(0). We must also choose initial conditions in γ and ϕ. We set γ(0) = 0,and ϕ(0) = 0. The choices of γ̇(0) and ϕ̇(0) will then determine the initial spin state andthe nature of the driving force. The separatrix of the γ – γ̇ system occurs at γ̇ = 0.33 yr^-1 when γ = 0 for its natural oscillation (no driving force). We therefore consider three reprsentative states of the γ–γ̇ spin system consisting of small amplitude libration (γ̇(0) = 0.02 yr^-1), large amplitude libration (γ̇(0)=0.25 yr^-1), and circulation (γ̇(0)=0.38 yr^-1), and examine how each responds to different levels of forcing as determined by the amplitude of ϕ̇(0). For all these cases, we let the simulations run for ∼ 10^5 years to allow sufficient time to approach any final spin-state. Figure <ref> shows the evolution of the spin for the large amplitude libration case (γ̇(0)=0.25 yr^-1) in response to four different levels of forcing. In panel a), ϕ̇(0)=0.09 yr^-1, which has little effect. The behaviour seen here is qualitatively the same as if there was no forcing. To better characterise the behaviour, we highlight in red the values of γ and γ̇ that occur when ϕ=0 and ϕ̇>0, known as a Poincare section (e.g. <cit.>). In panel b), we show the same evolution, but with ϕ̇(0)=0.21 yr^-1. We see here that the spin is now chaotic, switching between circulation and libration about both γ=0 and γ=π. It is worth noting that ϕ is circulating here, since the seperatrix between libration and circulation occurs for ϕ̇(0)=0.11 yr^-1 in this case. Thus, the most dramatic effects occur for ϕ near resonance, but not actually in it. In panel (c), we see the behaviour for ϕ̇(0)=0.29 yr^-1. Here γ is no longer chaotic, but shows a slight broadening of the libration trajectory. We note also that the Poincare section in red produces a closed curve for γ̇ < 0 yr^-1. This suggests an approximate commensurability between the period of libration for γ and period of circulation for ϕ. As we continue to increase ϕ̇(0) this commensurability gets weaker and the surface of section expands, eventually enclosing the origin again, as seen in panel (d), for ϕ̇(0)=0.39 yr^-1. We can summarise this behaviour in Figure <ref>, which shows the values of γ corresponding to the surface of section ϕ=0, ϕ̇>0, plotted against initial ϕ̇(0). The large amplitude libration may be especially susceptible to showing chaotic behaviour, since the natural oscillations (in the absence of forcing) are still a non-negligible amplitude relative to the seperatrix value. Thus, in Figure <ref> we consider the effect of a similar range of forcing, but now our initial γ̇(0)=0.02 yr^-1, so that the natural libration is of a much smaller amplitude. Once again, the effect of a librating ϕ is minimal, with the libration in γ not significantly affected, but with a circulating ϕ, with ϕ̇(0) ∼± 0.3 yr^-1, we again see stronger effects, with larger-amplitude librations. Thus, the character of the forced solution for a librating γ is similar regardless of amplitude – the effects are strongest for ϕ circulating at about twice the separatrix distance from exact resonance. The character of the behaviour for an initially circulating γ is different. This is shown in Figure <ref> by depicting values of γ̇ against ϕ̇(0), which presents a more useful picture than the Poincare projection in this specific case. Here, the effect of a librating ϕ is to enforce intermittent libration of γ as well, but only if γ̇(0) and ϕ̇(0) have the same sign. This is evident from the fact that Figure <ref> shows far less symmetric behaviour about the origin than either of Figure <ref> or Figure <ref>. In this case the influence of the perturber is stronger if the system is closer to resonance. Overall, these estimates suggest that a system with the parameters of the K00255 system could experience episodes of chaotic spin evolution if close to a mean-motion resonance. We can infer what a likely ϕ̇ is from observation, with ϕ̇ = (j+1) n - j n' assuming ω̅ = 0, and let ϕ̇(0) equal this. For the observed K00255 period ratio of 2.024, we infer ϕ̇(0)=2.0 yr^-1, which is probably too distant from resonance to produce large effects. However, a less triaxial planet, with (B-A)/C ∼ 10^-6, would lie within the chaotic zone at these periods. Given the uncertainty in the eccentricity, perturber mass, and especially (B-A)/C in the K00255 system, it is also worth considering the sensitivity of the system to changes in β. If we fix ϕ̇(0)=2.0 yr^-1, consistent with the currently observed value based on period ratio, we can characterise the evolution of the spin for different values of β. For the case where γ̇(0)=0.25 yr^-1, Figure <ref> shows how the spin-state varies with forcing amplitude β. We see that this system doesn't begin to undergo circulation, or chaotic motions, until β > 12.5. The character of the spin evolution is the same for larger values of β, except that the rotation gets progressively more rapid. From the threshold value of β∼ 12.5, we can estimate that a perturber mass of (55 /e M_⊕) |H(e)|, with our fiducial (B-A)/C = 2× 10^-5, is required for K00255.01 in order to get circulation. If we set a limit of e<0.6 to avoid orbit crossing, this gives us a strict lower limit of 17 M_⊕, although it is likely larger than this. For our nominal value of e=0.05, this would imply a larger mass of 555 M_⊕ to yield circulation in γ. These masses are much larger than the radius of the companion would suggest, and show that the K00255 system is likely too far from a mean-motion resonance for these effects to be very significant. If the planet is less triaxial, with (B-A)/C = 2× 10^-6, these values change to a lower limit of m' = 1.7M_⊕, andan implied mass of m' = 56M_⊕, still somewhat larger than the radius of the companion would imply. Nevertheless, for a similar system closer to a mean-motion resonance, circulation or larger amplitude oscillations in γ is possible. We also note that the results also depend strongly on γ̇(0), which we will show in <ref> could result in more dramatic effects, even when farther from resonance. Another potential application of this model is to the recently announced TRAPPIST-1 planetary system <cit.>. In this case, there are several planets arranged in a chain, with several near commensurabilities. If we focus on the planet TRAPPIST-1d, which has the most Earth-like irradiation, and is found in a mean-motion resonance with planet TRAPPIST-1e, we find that β∼ 1 for the nominal choices of e∼ 0.05, (B-A)/C ∼ 2 × 10^-5 and the estimated mass for TRAPPIST-1e (∼ 0.62 M_⊕). Other contributing factors to the larger β are the fact that this is a 3:2 resonance, which increases j and C_r, and that the host star is of very low mass (∼ 0.08 M_⊙). We infer ϕ̇(0) = 3.0 yr^-1 from the observed period ratio of the two planets if we assume ϕ̇ is currently at the maximum value in its oscillation. In Figure <ref>, we demonstrate our results for the evolution for γ in this system if we choose γ̇(0) = 0.3 yr^-1 (small-amplitude in natural libration). The red dots in the left-hand panel depict the Poincare points, recorded after each driving cycle, with their random placement illustrating the chaotic nature of that we happen to observe with these parameters. the right-hand panel shows a snapshot in a small time range, showing how γ chaotically switches between libration about 0 or π and circulation with either prograde or retrograde motion. For a broader picture, Figure <ref> shows the effect of the driving for different ϕ̇(0) for this system, starting with γ̇(0)=0.3 yr^-1. We see again a substantial effect around the separatrix (ϕ̇ (0) = 4.1 yr^-1). It can be shown that the separatrix in ϕ occurs at ϕ̇ = ω_S √(j β) and ϕ = 0. With the knowledge that the largest effects occur near the separatrix of ϕ, or around twice the separatrix in the 2:1 mean-motion resonance case, we can summarise the applicable range of parameters in Figure <ref>. Here we show how the separatrix of ϕ shifts as we vary β. For maximum effect, we would like to a find a system which lies roughly around the separatrix. We find that the K00255 system is likely too far from this range to yield interesting behavior, but the TRAPPIST-1d system lies squarely in this range, and indeed many of our simulations yield a circularised γ near the implied ϕ separatrix based on observation. The Planet TRAPPIST-1d is the one with the most earth-like insolation, but other planets in the system also reside close to mean motion resonances. Both TRAPPIST-1e and TRAPPIST-1f are also close to 3:2 and 4:3 commensurabilities respectively, and we estimate β = 0.39 and β=2.65 for these planets under the same assumptions as above. The large value of the latter is the result of both larger j and C_r and also a larger perturber mass. Our model can also potentially apply to TRAPPIST-1c, which lies near a second order (5:3) resonance. However, in this case ω_M^2 (and thus β) acquires an extra factor of eccentricity, making the estimated value smaller (β=0.04). A further point to note for this system is that its extreme compactness also implies short tidal synchronisation and circularisation times. If we repeat the analysis above, we estimate synchronisation and circularisation times ∼ 3 × 10^4 years and ∼ 8 × 10^6 years, respectively, for TRAPPIST-1d. This would suggest that perhaps we should set eccentrity equal to zero for these systems. However, resonant chains with more than two planets can excite eccentricities through zeroth order resonances not available to the two planet case, as is known to occur in the Laplace resonance of the Galilean moons <cit.>. This combination can potentially lead to a non-negligible contribution to the planetary heat budget due to tidal dissipation <cit.>, which may shift the optimal conditions of habitability in this system from planet d to e or f. We review the influence of tidal dissipation from our model in appendix <ref>. This system clearly warrants a more detailed investigation, but, for now, we simply assume that the more extended interactions generate eccentricity but that the spins can be described in the two planet formalism. §.§ Limit Cycles including Dissipation The preceding discussion focused on the interaction between the spin state and the external driving, in order to illustrate the kind of spin behaviours that are possible. In the true physical situation, we must also account for the damping effects of tides, which will drive the planet to synchronous rotation in the absence of an external forcing. When a forcing is present, we then expect the system to evolve to a finite amplitude limit cycle, or a stable circulation. In studies of harmonically driven damped pendula, the properties of this limit cycle can often be a sensitive function of the system parameters <cit.>. To examine the consequences of this, we now evolve equation (<ref>) for our nominal K00255 parameters (yielding β=0.11 and ϵ = 6 × 10^-8), assuming initially a large γ̇, for a range of initial ϕ̇(0), to examine the properties of the family of resulting limit cycles. Figure <ref> shows our results with the points plotted depicting the full range (i.e. not just the Poincare section) of γ probed by the solution at late times only (τ> 10^8 yrs). Thus, these represent the final limit cycles and indeed show a great deal of sensitivity to the nature of the driving via ϕ̇(0). The system can librate about either γ=0 or γ=π, and it can also circulate.The final pattern of the limit cycle seems quite sensitive to the imposed conditions. As mentioned previously, the observed period ratio for the K00255 system suggests ϕ̇(0)=2.0 yr^-1. The solutions in this part of the diagram can exhibit limit cycle behaviour anywhere from complete circulation to low amplitude librations. We note that this is different than our solution for an initially librating γ, as shown in Figure <ref>, without damping. This suggests that the initial conditions of the spin state, along with the strength of the forcing term, can dramatically alter the resulting limit cycle, even relatively far from the ϕ – ϕ̇ separatrix and with subtle variations of these initial conditions, with higher γ̇(0) giving a larger range of interesting behaviors. Figure <ref> shows the same thing for the TRAPPIST-1d system. In this case, both the forcing and the damping are stronger, yielding β=1.0 and ϵ=8 × 10^-3 for our nominal system parameters, and we set γ̇(0) = 5.0 yr^-1 (natural moderate-amplitude libration in γ). Once again we see similar behaviour, with more extensive regions where the limit cycle yields circulation. It is also interesting to note that ϕ̇(0)=3.0 yr^-1 is implied from observation, which yields a libration with semi-amplitude ∼ 42^∘, meaning that approximately 73% of the planet gets at least some level of illumination. Furthermore, this is right on the edge of the regime where the limit cycle circulates. If ϕ̇(0) were only slightly larger, or (B-A)/C slightly smaller, then the entire planet could potentially also receive substantial illumination, and slight changes in parameters can yield wildly different results. Indeed, the evolution of ϕ is likely to be more complicated in this resonant chain system, and the planet may move between spin regimes. We intend to investigate this further in a forthcoming paper. Figure <ref> shows two examples of the evolution to the limit cycle for the TRAPPIST-1d system near the separatrix of the ϕ – ϕ̇ system. The top panels depict results with ϕ̇(0)=3.4 yr^-1 and γ̇(0)=5 yr^-1, where we see that γ damps down into a limit cycle librating about γ=π with a semi-amplitude of 60^∘. The bottom panels of Figure <ref> shows the limit cycle with the same parameters except that ϕ̇(0)=3.8 yr^-1. In this case the limit cycle exhibits features of both circulation and libration – in essence the circulation is interrupted by a two beat libration at both γ = 0 and γ = π, before resuming. §.§ Timescales In order to determine the effects on climate, we must also determine typical timescales on which thespin states drift. The definition of γ̇ implies the degree of mismatch between the spin of the planet and the orbital frequency, so the planet still rotates almost synchronously unless γ̇ is quite large. However, a small finite mismatch means that the pattern of illumination will slowly drift over the surface of the planet, and the timescale over which it does this will determine the nature of the atmospheric response. If it occurs very slowly, the atmosphere should adjust to the change in illumination at each location, and should therefore present a similar atmospheric state to that of a synchronous planet, with the only likely variations due to whether the current dayside has more or less water present. On the other hand, if the drift is sufficiently rapid, the effect of the insolation is averaged over the surface of the planet. The most interesting cases are when the drift occurs on a timescale comparable to the atmospheric response time, as this can lead to nonlinear feedback. Specifically, we are interested in the time in which γ spans the entire range within its limit cycle (i.e. the time for the substellar point to return to the same physical location on the planet and begin a new cycle). For circulating cases, this corresponds to the time in which γ goes from -π to π, after spanning all γ in between. So we can say that this yields an effective length of a stellar day on the planet in these circulating cases. Meanwhile, for librating cases, it yields the time in which γ makes a full oscillation with its maximum amplitude. We will define this as our relevant “spin period.” In the case of our K00255 analogue, the large scale librations and circulations shown in Figure <ref> have characteristic timescales ∼10–25 years. We summarise this in Figure <ref>, where we show what our relevant spin period is for each simulation, where red dots depict drift periods for librating cases, and blue depicts circulating cases. If we consider the two limit cycles for TRAPPIST-1d shown in Figure <ref>, the period for the libration for the case in the top panel is 1.5 years. This suggests that portions of a planet in such a configuration could experience variations in levels of illumination not unlike the seasonal variations experienced on Earth because of the obliquity of Earth's spin. In the case of the hybrid libration-circulation limit cycle shown on the bottom panels of Figure <ref>, the period of the libration episodes is similar, and the period of a full cycle, including the circulation, is 6 years. We can again summarise this in Figure <ref>, with red dots depicting librating cases, and blue dots circulating cases. This shows that the TRAPPIST1-d system has much shorter spin periods than our K00255 system analogue, with many periods on the order of just a year, as well as having far more circulating cases. Overall, the large scale librations and circulations in the limit cycles of equation (<ref>) have characteristic timescales in the range Δτ∼ 1–25 years, which can imply different real-world timescales depending on the triaxiality and orbital period, but which are approximately decades for Earth-like planets in the habitable zones of M0 stars, and of order years for those in the habitable zones of very low mass stars, like TRAPPIST-1. § DISCUSSION Our model demonstrates that the presence of a companion in, or near, a mean motion resonance can perturb a planets spin sufficiently, even in the presence of tidal dissipation, to place it in a non-synchronous limit cycle. This can result in substantial drift, and even circulation, of the planetary spin. A consequence of this is thata greater fraction of the planetary surface can receive some level of illumination, relative to that expected from a planet tidally locked to a synchronous spin. Furthermore, the timescales for this drift are approximately years to decades, depending on the particular configuration. This could have profound implications for the climate, and thus the habitability, of such a planet. The response of a planetary atmosphere and surface conditions to variations in level of illumination is very complex and an entire subject of study in itself. Nevertheless, we can anticipate the broadest atmospheric consequences of our new spin dynamics by studying the one-dimensional energy balance models used to estimate the breadth of stellar habitable zones (e.g. <cit.>). In particular, if we consider the models of <cit.>, the atmospheric thermal response times for Earth-like planets range from years to decades, depending on the level of surface water (more water implies more thermal inertia and longer response times). This is comparable to the characteristic timescales estimated above for variations in the level of illumination, which implies that the atmospheric dynamics could be quite complex, as the conditions change on similar timescales to the response. In the context of exoplanet habitability then, our results suggest one more avenue that may allow a planet to avoid the consequences of tidal synchronism close to low mass stars <cit.>. Indeed, our model need not be exclusive of other proposals to avoid this fate. The formation of thick clouds near the substellar point <cit.> will presumably still be a part of the evolving climate dynamics, and the operation of strong atmospheric tides <cit.> may also operate, introducing an additional torque component into the spin dynamics that might also be worthy of consideration in the future. We began our consideration of this problem motivated by the anticipation of future discoveries of planetary systems around M dwarfs, and use the K00255 system as a prototype for the kinds of host stars likely to dominate magnitude limited samples of this type. However, the discovery of the planetary system around the very low mass (∼ 0.08 M_⊙) star TRAPPIST-1 illustrates that such models have a potentially very broad scope, and the presence of several near-commensurabilities in that system suggest that the configurations considered here are notrare. The fact that the drift timescales discussed above scale relative to the orbital period also suggests that the particular atmospheric responses may vary with stellar host mass too, with the habitable planets around the lowest mass stars like TRAPPIST-1 experiencing variations quite similar to the seasonal variations on Earth. In near resonant chains like the TRAPPIST-1 system, the variation in the spin dynamics may have an influence on the relative habitability of each of the systems as well. Figure <ref> shows the limit cycle for the three planets TRAPPIST-1d, TRAPPIST-1e and TRAPPIST-1f, that lie near first order resonances. In each case we have considered only the effects from the external neighbour, even though the inner companion may also provide non-negligible perturbations. In each case, there is a reason to suggest that the external perturber is more significant (the coupling between c-d is second order, and the external perturber is more massive in the case of planets e and f). We have performed this calculation assuming the nominal triaxiality and Q_⊕ for each planet. We see that TRAPPIST-1e executes moderate libration under these conditions, and TRAPPIST-1f a much smaller libration, both about γ = π. On the other hand, TRAPPIST-1d has a limit cycle that circulates and features an intermittent, wide-angle libration as well. The characteristic timescale for the full cycle is ∼ 1.1 years in this case. This demonstrates that planets in the same system can occupy different spin states, with some being essentially synchronised, while others experience a more distributed irradiation. Although the arrangement in Figure <ref> counter-intuitively suggests increased synchronism away from the star, this more has to do with our exact choices in initial conditions, with strong sensitivity to γ̇(0) and especially ϕ̇(0) within these regimes that could result in significantly different limit cycles. The general effect is typically regulated by the value of β, but the solutions could still change dramatically within certain regimes if we adjust ϕ̇(0) or γ̇(0), as can be seen in Figures <ref> and <ref>. Such spin states have potentially both direct and indirect observational consequences. The most obvious indirect consequence is simply the fact that spreading the stellar illumination over more of the surface area will help to make a planet more temperate and increase the fraction of the planet that is habitable. Figure <ref> demonstrates this by comparing the average flux received at each point on a planet for the limit cycles shown in Figure <ref>. In the case of libration for TRAPPIST-1d, we see a moderate increase in the average value of the flux and in the surface area irradiated. For the circulating case, the differences are much more dramatic, with a substantial averaging observed. More direct observational consequences may emerge if the observing technology reaches the point of identifying signals from surface features (e.g. <cit.>), because then one could potentially identify the years-to-decades drift on surface features over the course of many orbits. § CONCLUDING REMARKS As observations uncover more planets around other stars, it is becoming clear that many such systems contain multiple planets, orbiting in compact configurations with substantial mutual gravitational interaction. Such interactions are particularly strong when the mean motions of neighbouring planets are approximately commensurate with low order rational ratios. In that event, the induced periodic variation in mean motion is often enough to prevent the planetary spin from achieving a stable equilibrium, even under the influence of strong tidal damping. Contemporaneously with our work, <cit.> have presented an analysis of a similar problem, focussed on near-resonant systems discovered by Kepler, for which significant transit timing variations are observed. They identify the same qualitative behaviour as we do, namely the existence of asynchronous and possibly chaotic spin equilibria. As shown by our work above, the application of such considerations to planets in the habitable zones of low mass stars offers a mechanism to alleviate the long-standing problem of asymmetric illumination of such planets. We have examined the dynamics of the planetary spin and identified a variety of limit cycle behaviour, including libration, circulation and combinations of both. Furthermore, the characteristic timescales for the stellar illumination to drift over the planet appear to be from years to decades, which is comparable to the thermal inertia of terrestrial planet atmospheres. This promises a variety of interesting feedback behaviour and suggests that the climates of habitable planets around M dwarfs may be far more dynamically rich than the traditional synchronous rotation often assumed. Unfortunately the nature of the limit cycle solutions are quite sensitive to the parameters of the problem, some of which are difficult to constrain directly. Nevertheless, our results suggest that such systems may rewardefforts to characterise their atmospheric properties directly. Such considerations are likely to become ever more important in the forthcoming years, as the observational sample continues to grow through both ground-based and space based efforts such as TRAPPIST and TESS. The recent announcement of the discovery of the TRAPPIST-1 planetary system is particularly encouraging in this regard, since it demonstrates the reality of the mean motion resonant configurations discussed in this paper. mnras § THE SHIFT OF THE HABITABLE ZONE DUE TO TIDAL HEATING The traditional definition of a habitable planet is one that can maintain a surface temperature sufficient to allow for the long-term existence of liquid water at the surface. The particular conditions that allow this to occur are the subject of detailed study (see <cit.> for a discussion of the standard picture), but a simple estimate is often made by scaling relative to the amount of insolation that the Earth receives from the Sun. We shall adopt heating rates with a factor of two of Terrestrial Insolation as our simple definition of Habitability. For most stars, heating is equivalent to instellation – the amount of irradiation received from the central star. However, for very low mass stars, habitable planets may be quite close to the star, and tidal forces may generate significant heat if there is a non-zero eccentricity <cit.>. Since our model invokes a non-zero eccentricity to provide a resonant perturbation, we estimate here the contribution to total heating that arises from out tidal model. If we calculate tidal heating due to circularisation from equation (<ref>), within the formalism of <cit.>, and add this to the heating received from instellation (assuming the energy is redistributed evenly across the surface of the planet), we obtain an expression for the surface temperature of a planet orbiting a 0.08 M_⊙ star T^4 = 245^4 ( a/0.03AU)^-2 + 223^4 ( e/0.05)^2 ( R_p/R_⊕)^8 ( a/0.03AU)^-9. The second term represents the tidal heating, and depends strongly on the planetary radius andalso on the eccentricity. The consequences for planetary temperature are shown graphically in Figure <ref>, which shows curves for the radii of each of the three planets TRAPPIST-1d, TRAPPIST-1e and TRAPPIST-1f. These are the most likely to be habitable, and the tidal heating doesn't change this conclusion significantly, and actually improves the case for TRAPPIST-1f. As these are the three planets most likely to exhibit interesting spin dynamics, the contribution of tidal heating will add another dimension to the atmospheric dynamics. Equation (<ref>) and Figure <ref> are based on the orbital energy dissipated due to tidal circularisation. However, in this paper, we have focussed more on the dissipation of spin. Is dissipation of spin energy not of interest? In the same formalism, we can calculate the energy release due to spin dissipation relative to orbital circularisation, as Ė_spin/Ė_circ∼2/19θ̇/n( 1 - θ̇/n) ∼ 0.23 ( e/0.05)^-2γ̇ where we have normalised γ̇ by ω_s/√(2) as in the rest of the paper, assuming (B-A)/C = 2 × 10^-5. Therefore, we conclude spin energy is a minority contributor when considering the slow drifts relative to synchronism of interest here. | http://arxiv.org/abs/1705.09685v3 | {
"authors": [
"Alec M. Vinson",
"Brad M. S. Hansen"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170526191618",
"title": "On the Spin States of Habitable Zone Exoplanets Around M Dwarfs: The Effect of a Near-Resonant Companion"
} |
Reliability of Broadcast Communications UnderSparse Random Linear Network Coding Suzie Brown, Oliver Johnson and Andrea TassiThis work is partially supported by the University of Bristol Faculty of Science, the School of Mathematics and theProject, which is supported by Innovate UK under Grant Number 102202.S. Brown and O. Johnson are with the School of Mathematics, University of Bristol, UK (e-mail: [email protected], [email protected]).A. Tassi is with the Department of Electrical and Electronic Engineering, University of Bristol, UK (e-mail: [email protected]).December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Ultra-reliable Point-to-Multipoint (PtM) communications are expected to become pivotal in networks offering future dependable services for smart cities. In this regard, sparse Random Linear Network Coding (RLNC) techniques have been widely employed to provide an efficient way to improve the reliability of broadcast and multicast data streams. This paper addresses the pressing concern of providing a tight approximation to the probability of a user recovering a data stream protected by this kind of coding technique. In particular, by exploiting the Stein–Chen method, we provide a novel and general performance framework applicable to any combination of system and service parameters, such as finite field sizes, lengths of the data stream and level of sparsity. The deviation of the proposed approximation from Monte Carlo simulations is negligible, improving significantly on the state of the art performance bounds.Sparse random network coding, broadcast communication, multicast communications, Stein–Chen method.§ INTRODUCTIONIn next-generation networks, reliable broadcast communication is expected to be critical. In particular, this holds true in future networks of self-driving vehicles where road-side base stations (BSs) will broadcast live sensor data <cit.>. For example in the 5G-PPP's “bird's eye” use case, live 3D Light Detection and Ranging (LiDAR) scans of vehicles engaging a traffic junction are broadcast to the incoming vehicles –enabling them to take an informed decision on how to safely drive through the junction <cit.>. In this kind of network, a key performance indicator isthe user delivery probability, defined as the probability of a user successfully recovering the transmitted data stream.Generally, modern communication systems enhance the reliability of Point-to-Multipoint (PtM) data streams by employing Application Level-Forward Error Correction (AL-FEC) techniques, which are usually based on Luby Transform (LT) or Raptor codes <cit.>. These kinds of codes only operate to their capacity if large block lengths are employed, which could be a problem in the presence of delay-sensitive services <cit.>. For this reason, in our system model reliability of PtM data streams is ensured via the Random Linear Network Coding (RLNC) approach <cit.>.The RLNC approach requires the BS to split each PtM data stream into K source packets, which form a source message. A sequence of coded packets is obtained in a rateless fashion by linearly combining the source packets. A user recovers the PtM data stream as soon as it collects K linearly independent coded packets <cit.>. A drawback of the RLNC approach is the computational complexity of the decoding phase, which is a function of K and the finite field size q considered during the encoding phase <cit.>. Tassi et al. <cit.> observed that this complexity can be significantly reduced by adopting a sparse implementation of the RLNC approach, where the number of non-zero elements in the encoding matrix is smaller. However, as the encoding matrix becomes sparser, the number of coded packet transmissions needed by a user to recover the source message is likely to increase. To date, an exact expression for the user delivery probability as a function of the sparsity and number of coded packet transmissions is still unknown.The key contribution of this paper is a tight approximation to the user delivery probability in a system where broadcast source messages are protected by sparse RLNC (see Section <ref>). Our approximation is valid for any finite field size, sparsity level, and data stream length. As shown in Section <ref>, the deviation of the proposedmodel from simulation results is negligible. Our approximation enables service providers to increase the sparsity level (reducing the complexity of the decoding phase) while ensuring a target user delivery probability. The lack of an exact performance model for sparse RLNC implementations is caused by the lack of an accurate expression for the probability of a sparse random matrix, generated over a finite field, being full rank <cit.>. Garrido et al. <cit.> proposed models based on absorbing Markov chains to characterize the user performance, in a particular implementation of sparse RLNC where the number of source packets employed to generate each coded packet is fixed. This assumption significantly simplifies the performance modeling issue, yet <cit.> mostly relies onMonte Carlo simulation to estimate, via a regression technique, the statistical correlation between the rows of full rank sparse matrices.Unlike <cit.>, this paper refers to a more general sparse RLNC scheme where a source packet participates in the generation of a coded packet with probability 1-p, for 0 ≤ p ≤ 1. With regards to this general sparse RLNC formulation, Tassi et al. <cit.> proposed the first performance model valid for any finite field size, data stream length and probability p. In particular, the probability bound proved in <cit.> allowed the authors <cit.> to derive a tractable but not tight performance bound. More recently, the theoretical framework proposed by A. Khan et al. <cit.> extended <cit.> and can be directly used to upper- and lower-bound the user delivery probability. However, again often these bounds are not tight.In this paper, we address the limitations of the previous studies and provide the following contributions: * We propose an accurate expression for the user delivery probability suitable for general sparse RLNC formulations and applicable to any combination of system parameters, which overcomes the lack of generality of the model proposed in <cit.> and <cit.>. In particular, with regards to <cit.>, a new set of Monte Carlo simulations are required to re-derive a performance model as the field size, the number of source packets defining a source message or the number of source packets involved in the generation of each coded packet changes.* Unlike most recent works <cit.> whichbuild upon <cit.>, we approximate the user delivery probability by employing a novel mathematical framework based on the Stein-Chein method. In fact, for K > 10 and p ≥ 0.7, <cit.> notoriously is not a tight lower-bound to the probability of a sparse random matrix being full rank <cit.>, with a subsequent impact on the estimation of the user delivery probability.* Regardless of the field size and level of sparsity of the encoding matrix,our approximation of the user delivery probability is very close to simulated values. On the other hand, the state of the art upper- and lower-bound to the user delivery probability proposed in <cit.> significantly deviate from Monte Carlo simulations for a binary field, with the lower-bound performing better than the upper-bound. For larger field sizes, both our approximation and the upper-bound as per <cit.> to the user delivery probability tightly follow simulation results but, in this case, the lower-bound as per <cit.> significantly deviates from our Monte Carlo simulations. As such, unlike our approximation, neither the upper-bound nor lower-bound consistently give a tight approximation of the user delivery probability, regardless of the field size. The rest of the paper is organized as follows. Section <ref> presents the considered system model. Section <ref> discusses the proposed performance characterization model for sparse RLNC implementations and states our novel approximate result in Theorem <ref>. The accuracy of the proposed performance model is considered using Monte Carlo simulation in Section <ref>. Finally, in Section <ref>, we draw our conclusions.§ SYSTEM MODELWe consider a system model where one transmitter broadcasts a stream of coded packets to multiple receiving nodes, over a channel with packet error probability equal to ϵ. We assume that the transmission time of a coded packet is equal to one time step, and that the time needed to transmit N coded packets is equal to N time steps.We say that a source message consists of K source packets {𝐬_i}_i = 1^K where 𝐬_i consists of L elements of a finite field 𝔽_q of size q. A coded packet 𝐜_j is also formed by L elements from 𝔽_q and is defined as 𝐜_j = ∑_i = 1^K g_i,j·𝐬_i where g_i,j∈𝔽_q is referred to as a coding coefficient. Provided that N coded packets have been broadcast by the transmitter, the input to the broadcast channel can be expressed in matrix notation as [ 𝐜_1, …, 𝐜_N ] = [ 𝐬_1, …, 𝐬_K ] ·𝐆. The K × N matrix 𝐆 is defined by elements g_i,j, i.e., 𝐆∈𝔽_q^K × N. Coding coefficients are chosen at random over 𝔽_q, in an identical and independent fashion according to the following probability law <cit.>:ℙ(g_i,j = v) = {[p if v = 0;1-p/q-1 otherwise,;].where 0 ≤ p ≤ 1.The greater the value of p, the more likelythat a coding coefficient is equal to 0, so we observe that the average number of source packets actively participating in the generation of a coded packet is a function of p. The `classic' RLNC scheme refers to p equal to 1/q <cit.> (so the coding coefficients are uniform on 𝔽_q), `sparse' RLNC schemes are characterized by p > 1/q.Let {𝐜_j}_j = 1^n be the set of coded packets that have been successfully received by a user, for 0 ≤ n ≤ N. At the receiving end, each user populates a K × n decoding matrix 𝐌 with the n columns of 𝐆 associated with the n coded packets that have been successfully received. Finally, relation [ 𝐜_1, …, 𝐜_n ] = [ 𝐬_1, …, 𝐬_K ] ·𝐌 holds. The source message is recovered as soon as 𝐌 becomes full rank and hence, 𝐌 contains a K × K invertible matrix.§ PERFORMANCE ANALYSISBased on <cit.>, we observe that the probability of a user to recover a source message, i.e., the user delivery probability, as a function of ϵ can be expressed as follows:R(ϵ) = ∑_n = K^N Nn (1-ϵ)^n ϵ^N-nR_K,n(p),where R_K,n(p) is the probability of a K × n decoding matrix being full rank,as a function of p. In the case of classic RLNC, it is known thatR_K,n(p)_|_p = 1/q = ∏_t = 0^K - 1[1 - 1/q^n-t] exactly <cit.>. For sparse RLNC schemes, an exact expression for R_K,n(p) is still unknown but, as proposed in <cit.>, it can be approximated by means of the following lower-boundR_K,n(p) ≥ 1 -min{η_max(n); ∑_t = 1^KKt (q-1)^t-1ρ_t }and upper-boundR_K,n(p) ≤ 1 -max{η_min(n); ∑_t = 1^KKt p^nt(1-p^n)^K-t}where η_max(t) = 1-∏_w = 0^K - 1[1 - ( max{p, 1-p/q-1})^t-w] and η_min(t) = 1-∏_w = 0^K - 1[1 - ( min{p, 1-p/q-1})^t-w]. From <cit.>, it follows that η_max(t) and η_min(t) are the lower- and upper-bound to the probability, for t ≤ K, of a K × t being non-full rank, respectively. Finally, we write ρ_ℓ for the probability that any set of ℓ rows of a K × n matrix sums to the zero vector in 𝔽_q, which can be expressed (directly following from <cit.>), for ℓ = 1, …, K and n ≥ K, as:ρ_ℓ≐[ 1/q( 1 + (q-1) ( 1 - q(1-p)/q-1)^ℓ) ]^n. Both (<ref>) and (<ref>) are based on η_max(t) and η_max(t), which essentially account for the event that some sets of rowsform a non-full rank matrix. Overall, 1-η_max(t) and 1-η_min(t) give a notoriously not tight approximation of R_K,t(p). This holds true especially for K > 10 and p ≥ 0.7 <cit.>. In addition, the right-hand terms in the minimization of (<ref>) and in the maximization of (<ref>) represent the probability of having any set of rows that linearly combined sums to the zero vector and the probability of having any groups of rows that are equal to the zero vector, respectively. In both cases, these events are significantly different to the event that some submatrix in 𝐌is not full rank – thus impacting on the tightness of (<ref>) and (<ref>). In the remainder of this section, weaddress this issue by providing a novel expression for R_K,n(p), which tightly approximates the user delivery probability across a large range of system parameters.§.§ Proposed Performance Model for Sparse RLNCTherefore, we consider the key research question: Given a K × n decoding matrix 𝐌, formed according to the probability model (<ref>), what is the probability that 𝐌 has rank K? We remark that for n < K, the source message cannot be recovered, i.e., R_K,n(p) is equal to 0. In the remainder of this section, we focus on the case where n ≥ Kand we wish to know whether the K rows of 𝐌 form a linearly independent set, i.e., the rank of 𝐌 is K. We give the following definition. Write ≐{ 1, 2, …, ∑_t=1^K Kt} = {1, 2, …,2^K - 1 } for a set of labels. For each r ∈,we regard S_r as a subset of the set of indices { 1, …, K } composed of |S_r| items. It is immediate to prove that the following remark holds.Matrix 𝐌 is full rank if and only if no linear combinations of any sets of rows indexed by a S_r sums to the zero vector over the field 𝔽_q. For q = 2, we can consider the collection of eventsU_S_r≐{∑_i ∈S_rm_i = 0}, for r ∈,where we write m_i for the i-th row of 𝐌, and where addition is understood to be over 𝔽_2. We know that 𝐌 is full rank if and only if none of the events U_S_r occur for any r ∈. Consider the following matrix when q=2:𝐌 = ( [ 1 0 0 1 1 0 1; 0 1 1 0 0 0 0; 1 0 1 0 0 1 1; 1 0 0 1 1 0 1; 1 1 0 0 0 1 1 ]).In this case, rows 1 and 4 are identical, so U_{1,4 } occurs. Further, rows 2,3 and 5 sum to zero over 𝔽_2, so U_{ 2,3,5 } also occurs. In addition, since both these sets of rows sum to zero, their union must also sum to zero, so U_{ 1,2,3,4,5 } also occurs. Example <ref> illustrates why it is not sufficiently accurate to estimate the full-rank probability of 𝐌 byconsidering the expected number of events U_S_r which occur, using the expression for the probability of each individual U_S. This approach ignores the fact that such events are positively correlated.In general, given disjoint sets S_1, …, S_t such that U_S_1, …, U_S_t occur, then U_S will occur for each of the 2^t-1 sets S formed as unions of the S_iOur proposed performance framework builds upon a different set of statistical events, defined as follows.Let V_S_r be defined as followsV_S_r≐ U_S_r⋂( ⋂_T⊂S_r U_T^C ), for r ∈,which is the event that the rows indexed by S_r sum to the zero vector in 𝔽_2 but that no collection of rows indexed by a proper subset of S_r sums to the zero vector. In general, from Definition <ref>, we have the following remark. Matrix 𝐌 is full rank if and only if none of the events V_S_r occurs, for r ∈. This choice of events significantly mitigates the impact of thecorrelation among events observed in Example <ref>. There, V_{1,4 } and V_{ 2,3,5 } both occur (since no subset of them sums to zero), however V_{ 1,2,3,4,5 } does not. This enables us to derive a tighter approximation of R_K,n(p). The proposed derivation of R_K,n(p) involves two approximation steps: (i) We approximate the probabilityof event V_S_r happening for any set S_r consisting of a given number of items, and (ii) Since results based on the Stein-Chen method <cit.> show the sum of approximately independent zero–one variables with small probability of being one is close to Poisson, we approximate R_K,n(p) with a negative exponential function. Firstly, we consider the following quantities: For each ℓ = 1, …, K:* For each r ∈ such that the set S_r has cardinality ℓ,the event V_S_r has the same probability π_ℓ of happening, defined asπ_ℓ≐ℙ[V_S_r], ℓ = 1, … K. * We define a further quantity π̃_ℓ recursively as follows:π̃_ℓ≐ρ_ℓ - ∑_s=1^ℓ-1ℓ-1sρ_s π̃_ℓ-s,where (since taking ℓ = 1 gives an empty sum) π̃_1 ≐ρ_1.Term π_ℓ (defined in (<ref>)) can be approximated as π̃_ℓ (defined in (<ref>)). See Appendix <ref>. We observe that an obvious way in which 𝐌 can fail to have full rank is that a particular row is identically zero. Indeed, considering this event gives an upper bound on R_K,n(p), which for certain parameter values can be reasonably tight. For this reason,we condition out these events as follows: R_K,n(p)=({}) ·( {} | {}),where we can write the first term of (<ref>) directly as (1-p^n)^K.From Lemma <ref>, we prove the following result. Weapproximate the second term of (<ref>) as( | ) ≃exp(-λ),whereλ≐∑_ℓ=2^K λ_ℓ, for λ_ℓ≐Kℓπ̃_ℓ/(1 - p^n)^ℓ,so we approximate R_K,n(p) as follows:R_K,n(p) ≅ (1 - p^n)^K exp(-λ).See Appendix <ref>. The most computationally intensive part of calculating (<ref>) is the derivation of π̃_ℓ, which requires O(K^2) operations. However, since the expression of π̃_ℓ is independent of K, it has to be computed only once to approximate R_K,n(p). In addition, for K = 10, 20, 50 and 100, the average time needed to compute[Tests performed by running our benchmark code on one core of an Intel Xeon CPU E5-2650v4 operated at 2.20.] π̃_1, …, π̃_K (normalized by K) is equal to 2.7 · 10^-3, 7.7 · 10^-3, 7.3 · 10^-2 and 4.3 · 10^-1, respectively. It is also key to note that Theorem <ref> allows us to decouple the impact that any ℓ× n submatrix of 𝐌 has on the approximation of R_K,n(p) as in (<ref>), for ℓ = 2, …, K. Asthe following remark explains, thisallows us to further approximate (<ref>) by reducing the number of summation terms defining λ and hence, reducing the computational complexity of the approximation (<ref>).Consider the set of all the ℓ× n submatrices of 𝐌, then λ_ℓ approximates the probability that at least one of these submatrices is not full rank, assuming 𝐌 has no zero rows. For this reason, the approximation of R_K,n(p) given in (<ref>) can be further approximated by referring to those submatrices of 𝐌 composed by up to m rows, for m = 2, …, K. As such, with define the m-th approximation order of (<ref>) as follows:R_K,n^(m)(p) ≐ (1 - p^n)^K exp(-∑_ℓ = 2^m λ_ℓ).Let us consider the following approximation order optimization (AOO) problem[For the sake of compactness, with a slight abuse of notation, we say that R_K,n^(m)(p) is always equal to R_K,n^(K)(p), for any m > K.]:AOO min_m ∈{2, …, K}ms.t.e(m) ≤τ ⋁m ≤mwhere function e(m) is defined as R_K,n^(m)(p) - R_K,n^(m+1)(p). The solution m^* to the AOO problem represents the smallest-order approximation of (<ref>) associated with a target error value τ∈ [0, 1] or such that m^* is smaller than m, for 2 ≤m≤ K.From (<ref>), it follows that term ∑_ℓ = 2^m λ_ℓ is a non-decreasing function of m, i.e., relation R_K,n^(m)(p) ≥R_K,n^(m+1)(p) ≥R_K,n^(K)(p) holds. As such, for any given K, n and p, the error function e(m) attains only one maximum, for m ∈{2, …, K}. For this reason, the AOO problem can be solved iteratively evaluating R_K,n^(m^*)(p), for m^* = 2, …, K, until e(m^*) ≥ e(m^* + 1) and e(m^* + 1) ≤τ or m^* is smaller than or equal to m̂.From Remark <ref>, 𝐌 is full rank if and only if none of the events V_S_r occurs, for r ∈, and q = 2. However, for non-binary fields, the aforementioned statement captures a subsets of events when a random matrix is full rank. As such, we propose to use (<ref>) to approximate R_K,n(p), for q > 2.§ ANALYTICAL RESULTSThis section compares the approximation we proposed in Theorem <ref> against the approximation (<ref>) and (<ref>). Both our simulator and the implementation of the proposed theoretical framework are available online <cit.>.Fig. <ref> shows the relationship that exists between the order m of the approximation as in (<ref>) and the number n of received coded packets in R_K,n^(m), for q = 2, a source message composed by K = 20 packets and p = 0.8. From (<ref>), we remark that, for a given value of n, R_K,n^(m) is a non-increasing function of m. This is directly related to the fact that small approximation orders account for submatrices of 𝐌 composed of a reduced number of rows. This can be intuitively explained by considering the extreme case where n is large compared to K. In this case, the probability of 𝐌 being full rank can be approximated by the probability of havinga set of K (non-zero) rows of 𝐌 where no rows are identical – this corresponds to the case where m is set equal to 2.The aforementioned facts are confirmed by Fig. <ref>. For instance, for n = 20, the value of R_K,n^(m) drops from 0.72 (m = 2) to 0.21 (m = 14) to remain almost unchanged for 14 ≤ m ≤ 20. In particular, by solving the AOO problem for τ = 10^-4, m̂ = K and n = 20, we obtain an optimal value of m^* equal to 18 as per Remark <ref>. We also observe that the value of m^* appears to sharply decrease as n increases, which makes computationally convenient to approximate R_K,n with R_K,n^(m^*). For instance, Fig. <ref> shows that the error function e(m^*) takes values smaller than or equal to τ = 10^-4 for n = 31 and m^* = 4 – thus making it pointless to approximate R_K,n withan heuristic order equal to or greater than 5. In the remainder of this section, to highlight the accuracy of our approximation, we will refer to a value of τ = 10^-10 or m̂≤⌈ 3K/4 ⌉.Fig. <ref> compares the user delivery probability R(ϵ) for K = 10, 20 and 50 and q=2. We compare the value for R(ϵ) implied by (<ref>), substituting the approximations to R_K,n given by (<ref>), (<ref>) and our proposal in (<ref>) to the probability R(ϵ) estimated byMonte Carlo simulations. Results are given as a function of N-K, which represents the transmission overhead, i.e., the number of coded packets in excess of K that are transmitted. Assuming that the time needed to transmit each coded packet is fixed and equal to one time slot, the goodput of the system can be immediately expressed as the bit length of the source message divided by the time duration of N time slots. For concreteness, we considered a value of packet error probability ϵ = 0.1, which is the maximum transport block error probability regarded as acceptable in a Long Term Evolution-Advanced (LTE-A) system <cit.>. In particular, in the case where p = 0.7, Fig. <ref> shows that the maximum gap between our proposed approximation (<ref>) and simulation results is equal to 3.1 · 10^-2, which occurs for K = 20. Fig. <ref> refers to the case when p = 0.9 and shows that the gap between (<ref>) and simulation results is negligible.In contrast, both Fig. <ref> and <ref> show that approximating R_K,n using the state of the art (<ref>) and (<ref>) leads R(ϵ) to significantly deviate from the simulation results. For instance, for p = 0.9, K = 50 and N-K = 7, the absolute deviation can be up to 0.14 and 0.22, in the case of (<ref>) and (<ref>), respectively. In general, the maximum Mean Squared Error (MSE) between simulations and our proposed approximation (<ref>) is experienced for K = 50 and p = 0.7 and it is equal to 7 · 10^-4. That is smaller than the corresponding MSEs between simulation and approximations (<ref>) and (<ref>), which are equal to 1.1 · 10^-3 and 7.7 · 10^-3, respectively (between 1.6 and 11 times smaller). In addition, Fig. <ref> shows that, for p = 0.9, our proposal overlaps simulation results while the considered alternatives significantly deviate. In this case, the MSE of our approximation is between 237 times (for K = 50) and 1063 times (K = 20) smaller than in the case of (<ref>) and, between 722 times (K = 50) and 857 times (for K = 10) smaller than in the case of (<ref>).Fig. <ref> shows that for q = 2^4, our approximation either overlaps (for p = 0.7) or marginally diverges from simulation results (p = 0.9). In the latter case, we observe that the (absolute gap) never exceeds 0.11 and the maximum MSE is equal to 3.8 · 10^-3. Similar behavior is also exhibited by the case where R_K,n is approximated as in (<ref>) or (<ref>) and p = 0.7. However, as p increases to 0.9, the approximation based on (<ref>) significantly deviates from simulation results even of a quantity larger than 0.51 (K = 20 and N-K = 9).Generally, from Figs. <ref> and <ref> we observe that, for binary fields (with the only exception of K = 50 and p = 0.9), the approximation based on (<ref>) is tighter than that based on (<ref>). However, the exact opposite holds as both q and p increase. Our proposed approximation avoids this issue. In fact, our solution tightly approximates simulation results, for all the cases considered. These conclusions are also confirmed by Fig. <ref>, which shows the probability R(ϵ) as a function of p, for q = {2,2^4}, K = {20, 50}.As an immediate application of a tighter approximation of the user delivery probability, we can accurately estimate the average transmission overhead needed for a user to recover a source message (that is ∑_t = K^∞ t ·R_|_N = t(ϵ) - K). In particular, Fig. <ref> shows the average transmission overhead as a function of ϵ, for K = 20, 50 and 100 and, p = 0.7 and 0.9. Direct proof of the quality of the proposed approximation is given by the fact that the deviation between theoretical and simulation results is negligible across the whole range of parameters – the maximum gap between theory and simulations is equal to 2 and occurs for K = 100 and p = 0.9.§ CONCLUSIONThis paper presented a novel approximated performance model for a sparse RLNC implementation. The proposed model exploits the Stein-Chen method to derive a tight approximation to the probability of a user recovering a source message. Analytical results show that the Mean Squared Error (MSE) between our approximation and simulation results, for q = 2 and 2^4, never exceeds 7 · 10^-4 and 3.8 · 10^-3, respectively. On the other hand, the state of the art bounds are not always tight. For instance, when q = 2 our proposal is between 1.5 and 1063 times closer in MSE to simulations.§[Proof of Lemma <ref>] By symmetry it is enough to consider a subset of rows of the form S = { 1, 2, …, ℓ}, where ℓ≤ K. The key is to fix one row (say row 1) and to consider the smallest set of rows containing row 1 which sums to zero. Consider W, a subset of S with 1 ∈W, and say that event T_W,S occurs when both:(i) the rows of 𝐌 with indices in S add to zero and, (ii)rows with indices in W add to zero, but no subset of these rows add to zero, i.e. event V_W occurs see (<ref>). By considering (i) and (ii) together, T_W, S occurs when the rows in both the setsS∖W and W (but no subset of W) add to zero. In other words T_W,Sequals U_S∖W⋂ V_W. Since rows in 𝐌 are statistically independent, for each W of size (ℓ-s), the event T_W,S occurs with probability( T_W,S)= ( U_S∖W⋂ V_W)= ( U_S∖W) ( V_W) = ρ_s π_ℓ-s.Furthermore,since U_S = ⋃_W T_W,S for any S we observe:ρ_l = ( U_S )=( ⋃_W T_W,S)≃∑_W( T_W,S )= ∑_s=0^ℓ-1ℓ-1sρ_s π̃_ℓ-s= ∑_s=1^ℓ-1ℓ-1sρ_s π_ℓ-s + π_ℓ.This relation holds because (i) we assume events T_W,S are approximately disjoint (ii) there are ℓ-1s possible sets W of size (ℓ-s) containing 1. In Example <ref>, if S = { 1,2,3,4,5 } then U_S occurs as previously discussed, indeed so does T_W,S with W = {1,4 }. This concludes the proof. [Proof of Theorem <ref>] We write ^* for the collection of indices r such that | S_r |≥ 2 and define the random variableW ≐∑_r ∈^*V_S_r |where indicator function · equals1 if a particular event has occurred, or 0 otherwise.Observe that ( | {}) = (W = 0).By (<ref>),if S_r has ℓ≥ 2 elements, independence of the rows means that the probability( V_S_r | {})=( V_S_r⋂{})/(1-p^n)^K = ( V_S_r) ( {})/(1-p^n)^K≅π̃_ℓ (1-p^n )^K-ℓ/(1-p^n)^K= π̃_ℓ/(1 - p^n)^ℓ.Hence, by counting sets of different sizes in ^*, we it follows that [W] = λ. Further, W is the sum of a large number of zero–one variables, each of which equals one with small probability, and where each random variable in the sum is independent of a large proportion of the other terms. These are the conditions under which W is close to Poisson, as shown by the Stein–Chen method <cit.>. Approximation (<ref>) follows because <cit.> means that (W = 0) ≃exp(-λ). IEEEtran | http://arxiv.org/abs/1705.09473v4 | {
"authors": [
"Suzie Brown",
"Oliver Johnson",
"Andrea Tassi"
],
"categories": [
"cs.IT",
"cs.PF",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170526082100",
"title": "Reliability of Broadcast Communications Under Sparse Random Linear Network Coding"
} |
ℒ Ãρ̃ θ̃ | http://arxiv.org/abs/1705.09716v1 | {
"authors": [
"Christopher D. Carone",
"Shikha Chaurasia",
"John C. Donahue"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526205625",
"title": "Universal Landau Pole and Physics below the 100 TeV Scale"
} |
Rejection-Cascade of Gaussians: Real-time adaptive background subtraction framework B Ravi Kiran^1 Arindam Das^2 Senthil Yogamani^3 December 30, 2023 ===================================================================================§ INTRODUCTIONThere have been fascinating new developments in the understanding of perturbative scattering amplitudes which have at their heart a map from the space of kinematics of n massless particles to the moduli space of n-punctured Riemann surfaces. This map is provided by the scattering equations, and it allows the reformulation of tree and loop level scattering amplitudes of several quantum field theories in terms of integrals over said moduli space. These new representations come in two very different flavours: On the one hand there are intrinsically 4d representations like the Roiban-Spradlin-Volovich-Witten formula <cit.> for super-Yang-Mills and the Cachazo-Skinner formula for supergravity <cit.>. They have their origin in twistor theory, use spinor-helicity variables and accommodate supersymmetry rather naturally via the use of on-shell supersymmetry. On the other hand is the dimension agnostic Cachazo-He-Yuan framework, which has its roots in ambi-twistor space and can describe a plethora of scattering amplitudes in any number of space-time dimensions, but so far is largely limited to bosonic states. It is a rather non-trivial fact that the CHY formulae reduce to the corresponding twistor formulae once the external kinematics is four dimensional. Both are underpinned by the same set of equations, albeit in very different representations, which is widely understood <cit.> and we review briefly in <ref>. However the functions on the moduli space which determine the states and interactions look very different in the CHY and twistor representations. The two main contributions of this paper are that we * advance the understanding of the translation between the CHY and (ambi-)twistor representations by demonstrating and proving the splitting of general holomorphic correlators on the Riemann sphere of spinors valued in the tangent bundle of Minkowski space T 𝕄, (which appear e.g. in the RNS string or ambi-twistor string,) into correlators of spinors valued in the spin bundles 𝕊^+ and 𝕊^-. All kinematical CHY integrands (for gravity, EYM etc.) are special cases or limits of this general type of correlator. * find a formula for all tree level scattering amplitudes, in all trace sectors, in4d supersymmetric Einstein-Yang-Mills theory.This paper extends the known formulas for purely bosonic states <cit.>, previous work on 4d formulas for EYM of <cit.>, which gave a formula for all single trace amplitudes, and the translation of CHY Pfaffians <cit.>. Both results are each interesting in their own right. Chiral splitting can be stated as the splitting of fermion correlators of the form ⟨∏_i (λ_i _i ) ·ψ(z_i) ⟩ ,S= ∫ _^1η_μν ψ^μ(z) ψ^ν(z) where (λ_i _i)_μ is a 4d null vector in spinor helicity notation and ψ^μ(z) is a left-moving fermionic spinor on the Riemann sphere, into two factors, each only involving the left handed λ_i and right handed _i respectively. While chiral splitting is a general property of the correlators (<ref>), rooted in the fact that the tangent bundle of Minkowski space splits asT 𝕄≃𝕊^+ ⊗𝕊^-into the left and right handed spin bundles, the present paper is interested in this because of their role in the CHY formulae. Indeed, all kinematic Pfaffians appearing in the CHY arise as correlators of this type. This means that the chiral splitting of the worldsheet correlators <ref> on the sphere lifts, via the scattering equations, to a chiral splitting of 4d quantum field theory amplitudes. In other words, all QFT amplitudes that can be described by a CHY formula will exhibit this chiral splitting in 4d.Following the early work <cit.> there has been renewed interest recently in the study of Einstein-Yang-Mills amplitudes and their relation to pure Yang-Mills from the perspective of the double copy construction <cit.>, string theory <cit.> and the CHY formulae <cit.>. We believe that the new formulas for EYM scattering amplitudes (<ref>) can provide a new tool to study these relations, particularly in light of the 4d KLT and BCJ relations <cit.>.We begin by very briefly reviewing the 4d scattering equations in <ref>. In <ref> we discuss the technicalities and give examples of the splitting of CHY type Pfaffians, which we prove in <ref>, and in <ref> we present the 4d scattering amplitude for EYM on (ambit)twistor space. Both <ref> are largely self-contained, so the reader may skip directly to her/his point of interest. §.§ Representations of Scattering Equations It is well known that in four dimensions the scattering equations split into R-charge sectors, also known as N^k-2MHV sectors. These sectors are labelled by an integer k, or d ≡ k-1, or ≡ n - k-1, where n is the total number of particles. There are many representations of these refined scattering equations, and we will now briefly recall the three we use for this paper. The idea is to write the particles' momenta as matrices in spinor-helicity notation, and then solve the scattering equations[Since P is a meromorphic (1,0)-form, P =0 contains n-3 independent equations] P = 0 , where P^α (z) := ∑_i∈pλ_i^α_i^ z/z-z_i , by factorizing <cit.> it as P^α(z) = λ ^α (z )^ (z) , globally on the sphere. The factorization involves a choice of how to distribute the zeros of P among the two factors, and this choice labels the different refinement sectors. It also requires a choice of how to distribute the poles of P among the two factors and this choice labels the various equivalent representations of the scattering equations. The first representation of the refined scattering equation is given by the splitting P(z)= λ_T(z)_T(z) withλ_T ^α∈ H^0 ( 𝒪(d) ) , _T^∈ H^0 ( 𝒪(-d) ⊗ K[∑ _i ∈p z_i ] ) ,where the subscript stands for twistor. The notation here means that for α = 0,1, λ_T^α is a holomorphic polynomial of degree d while _T^ is a meromorphic (1,0)-form of homogeneity -d with simple poles at all marked points. In these variables the scattering equations read_z_i_T = t_i_i ,t_iλ_T(z_i) = λ_i ∀ i ∈p= {1, ⋯ ,n } . They fix the sections λ_T, _T, the scaling parameters[Both λ_T(z) and λ_i are only defined up to rescaling by a non-zero complex number. Hence the scattering equations can only require them to be proportional, and the scaling parameters t_i are introduced to account for the rescaling covariance.] t_i and locations z_i (up to Möbius invariance), and also enforce momentum conservation. The distinct refinement sectors are labelled by the integer k = d+1, and the original scattering equations P^2=0 are equivalent to the union of the refined scattering equations for k = 1 , ⋯ , n-1.The second representation is the parity conjugate of the previous one and is given by the splitting P(z)= λ_T̃(z)_T̃(z) withλ_T̃ ^α∈ H^0 ( 𝒪(-) ⊗ K[∑ _i ∈p z_i ] ) , _T̃^∈ H^0 ( 𝒪() ) ,where the subscript stands for dual twistor. Here λ_T̃ is a meromorphic (1,0)-form of homogeneity - with simple poles at all marked points, while _T̃ is a holomorphic polynomial of degree . In these variables the scattering equations read_z_iλ_T̃ = t̃_iλ_i , _i_T̃(z_i) = _i ∀ i ∈pand they again fix the sections λ_T̃, _T̃, the scaling parameters _i and locations z_i (up to Möbius invariance) and enforce momentum conservation. The third representation is useful if there is a natural splitting of the set of external particles p into two subsets p^+ ∪p^- = p. Then we can require that P(z)= λ_A(z)_A(z) withλ_A^α∈ H^0 ( K^1/2[∑ _i ∈p^- z_i ] ) , _A^∈ H^0 ( K^1/2[∑ _i ∈p^+ z_i ] ) , where the subscript stands for ambi–twistor. Here λ_A ,_A are both meromorphic (12,0)-forms of homogeneity 0 and have simple poles at the marked points in p^- , p^+ respectively.In these new variables the scattering equations read_z_iλ_A = ũ_iλ_i , ũ_i_A(z_i) = _i ∀ i ∈p^- _z_i_A = u_i_i ,u_iλ_A(z_i) = λ_i ∀ i ∈p^+and they fix the sections λ_A, _A, the scaling parameters u_i , ũ_i and locations z_i (up to Möbius invariance) and enforce momentum conservation. We can easily switch between these three representation via the relations λ_T (z) ∝ ∏_i∈p^- (z-z_i)/√( z) λ_A(z) ∝ ∏_i∈p (z-z_i)/ z λ_T̃(z)_T (z) ∝ √( z)/∏_i∈p^- (z-z_i) _A(z) ∝ z/∏_i∈p (z-z_i) _T̃(z) for the sections, where the factor of proportionality is independent of z, and also for the scaling parameters t_j/t_i ∏_k ∈p^- \{ i}z_j - z_k/z_i - z_k = ũ_i u_j √( z_i z_j)/z_i-z_j= _i/_j ∏_k ∈p^+ \{ j}z_i - z_k/z_j - z_k for any choice of i ∈p^- , j ∈p^+. The locations z_i are identical among the three representations.Notice that among the three representations the number of zeros in λ_A, λ_T , λ_T̃ and _A, _T , _T̃ is always d andrespectively, and only the poles are redistributed. Of course one may define many more representations of the same equations by choosing different ways of distributing the poles among the two factors, but for the present paper we will only need these three.§ CHIRAL SPLITTING OF FERMION CORRELATORS & CHY PFAFFIANS The key step in the translation of the CHY integrands into spinor–helicity language is the factorization of the kinematic Pfaffians into Hodges matrices <cit.>. Take 2n points on the sphere z_i and to each point associate one un-dotted (left-handed) spinor λ_i and one dotted (right-handed) spinor _i. The basic identity which we found and use throughout the rest of the paper is the factorization of the Pfaffian ( λ_iλ_j_i_j/z_i - z_j)^i,j=1,⋯ , 2n = ( λ_iλ_j/z_i - z_j)^i∈b_j∈b^c/V( b) V( b ^c) ( _i_j/z_i - z_j)^i∈b̃_j∈b̃^c/V( b̃) V( b̃ ^c) V( { 1 , ⋯ , 2n } ) where b, b̃ are arbitrary ordered[The expression <ref> is easily seen to be independent of the ordering of b, b̃, but the Hodges determinant and the Vandermonde determinant separately are not, so we keep track of the ordering.] subsets of {1, ⋯ , 2n } of size n and b^c , b̃ ^c are their complements. We use the notation that (M_ij)^i∈a_j∈b denotes the determinant of the matrix M, with rows indexed by the set a and columns by the set b. Since the Pfaffian is only defined for antisymmetric matrices, it's rows and columns are necessarily indexed by the same set. We also use the Vandermonde determinant, defined as usual V( b ) = ∏_i<j ∈b (z_i - z_j)for an ordered set of points on the sphere. It is worth emphasizing that this factorization does not require the scattering equations to hold (and that the spinors λ _i , _i need not have any interpretation in terms of null momenta or polarization vectors, though of course that is how we will employ this formula below). The kinematic Pfaffians for gravity, EYM, etc. may all be realized as appropriate limits or special cases of this Pfaffian.To prove <ref> we simply compute the residues as any z_i - z_j → 0 on both sides and invoke induction. We outline the idea of the proof here and point the interested reader to <ref> for details: At first glance it seems as though the right hand side depends on the splitting of the 2n points into the two halves b , b ^c and b̃ , b̃ ^c respectively, which would be at odds with the manifest S_2n antisymmetry of the Pfaffian on the left. This tension is resolved by the surprising fact that the combination (ij/z_i - z_j)^i∈b_j∈b^c/V( b) V( b ^c) is totally S_2n permutation symmetric, despite making only the permutation invariance under a S_n × S_n ×ℤ_2 subgroup manifest. To exhibit full permutation invariance we may go to an alternative representation (ij/z_i - z_j)^i∈b_j∈b^c /V( b ) V( b ^c ) = (-1)^n(n-1)/2∑_p⊂{ 1 , ⋯ , 2n }|p| =n∏_i ∈p(λ_i) ^0 ∏_j ∈p^c (λ_j) ^1 /∏_i ∈pj ∈p^c(z_i - z_j) where the sum runs over all unordered subsets p⊂{ 1 , ⋯ , 2n } of size n. The right hand side is now manifestly S_2n permutation invariant (though it has lost its manifest SL(2) Lorentz invariance). This last equality is interesting in its own right, but will not be used in the present paper other than to prove the S_2n symmetry of <ref>. Below we will use the S_2n symmetry of <ref> repeatedly in order to streamline the calculations.We now demonstrate the chiral splitting formula <ref> by translating various CHY formulae into 4d spinor helicity variables. §.§ Refinement of the Scalar Mode Pfaffian As a warm-up we demonstrate how to use <ref> to factorize the scalar mode CHY Pfaffian (A) = ( p_i · p_i / z_i-z_j )^i,j=1, ⋯ , n = ( λ_iλ_j_i_j/z_i - z_j)^i,j=1,⋯ ,n with n even. Now we have to choose two ways of splitting of the n labels into two halves, one for the rows and one for the columns respectively. Choosing, for example, the splitting 1 ,⋯ ,n/2 and n/2+1 , ⋯ , n for both for both angle and square brackets, by <ref>we find (A) = ( λ_iλ_j/z_i - z_j)^i = 1 , ⋯ , n/2_j = n/2+ 1 , ⋯ , n · ( _i_j/z_i - z_j)^i = 1 , ⋯ , n/2_j = n/2+ 1 , ⋯ , n ·∏_i = 1 ^n/2∏_j = n/2+1 ^n (z_i-z_j) /∏_i,j = 1 ^n/2 (z_i-z_j) ∏_i,j = n/2+1 ^n (z_i-z_j) Of course this is just one way to chose the distribution of the n row/column labels of the Pfaffian onto the two determinants, and they're all equivalent (after taking into account the appropriate Vandermonde ratios).Since (A) has corank 2 when evaluated on solutions to the scattering equations <cit.>, the above Pfaffian actually vanishes and we instead consider the reduced Pfaffian, defined to be the Pfaffian of any n-2 × n-2 minor of A, together with a Jacobian factor which preserves the S_n permutation invariance of the construction. We can adapt the above easily ^' (A)= 1/z_1 - z_n ( λ_iλ_j_i_j/z_i - z_j)^i,j=2,⋯ ,n-1 = ( λ_iλ_j/z_i - z_j)^i = 2 , ⋯ , n/2_j = n/2+ 1 , ⋯ , n-1 · ( _i_j/z_i - z_j)^i = 2 , ⋯ , n/2_j = n/2+ 1 , ⋯ , n-1 ·1/z_1 - z_n ∏_i = 2 ^n/2∏_j = n/2+1 ^n-1 (z_i-z_j) /∏_i,j = 2 ^n/2 (z_i-z_j) ∏_i,j = n/2+1 ^n-1 (z_i-z_j) . Again, for concreteness, we display just one of many equivalent ways of splitting the reduced Pfaffian into two determinants. Notice that the splitting does not require the scattering equations to hold, but on the support of the scattering equations the above expression becomes S_n symmetric. §.§ Refinement of the Vector–Mode Pfaffian As next example we shall translate the probably best known CHY integrand, the kinematic Pfaffian for massless vector modes <cit.> ( [A -C^T;CB ]) with the entries of each block-matrix given as A_ij = p_i · p_jS( z_i , z_j ) , B_ij = ε_i ·ε_jS( z_i , z_j ) , C_ij = ε_i · p_jS( z_i , z_j ) andA_ii =0 , B_ii = 0 ,C_ii = ε_i ·P(z_i) , where S(z , w) := √( z w)/z-w is the free fermion propagator (Szegó kernel) on the Riemann sphere[The factors of √( z) can be removed using the multilinearity of the Pfaffian, but we keep them in place to highlight its CFT origin.]. It is convenient to use the following parametrization for polarization vectorsε^- _i = λ_i ξ_i/ξ_i _i , ε^+_i = ξ_i_i /ξ_i λ_i , for states of negative and positive helicity respectively. For the following discussion we fix the degree of the refined scattering equations to be d = |p^-| -1, for which we give a justification below. In order to use the factorization formula <ref> on this Pfaffian we first have to cast it into the form ( q_i · q_j S(z_i , z_j) ) ^i,j = 1 , ⋯ ,2n . We would like to identify q_i = p_i and q_i+n = ε_i as well as z_i = z_i+n for i = 1 , ⋯ , n, but the diagonal terms in the block C present an obstruction to doing so. We can resolve this obstruction by employing a point splitting procedure and using the scattering equations. The idea is to introduce n new marked points on the sphere, one w_i associated to each z_i, and write the momenta as p_i = lim _w_i → z_ip_i(w_i)≡lim _w_i → z_i {t_i_i λ_T(w_i) _T̃ (w_i) } and also writeε^-_i =t_i λ_T (z_i) ξ_i/ξ_i _iandε^+_i =_i ξ_i _T̃(z_i) /ξ_i λ_i .Here we use the functions λ_T and _T̃ from the twistor and dual twistor representation of the refined scattering equations, respectively. We shall work in this enlarged description[In the language of the Ambitwistor String model of <cit.> this means that we write the descended vertex operators as a product ε_i · P(z_i) + : ε_i ·ψ (z_i) p_i ·ψ (z_i) : = lim_w_i → z_i p_i(w_i) ·ψ(w_i) ε_i ·ψ(z_i) . The correlator of these point–split vertex operators gives rise to the Pfaffian in <ref>.] to facilitate the factorization of the Pfaffian, and take the limit w_i → z_i only at the very end, where we recover the original momentum vectors. The upshot is that the diagonal terms of C may now be written as ε^±_i · P(z_i)= lim_w_i → z_i( ε_i ^±· p_i(w_i) S(z_i , w_i)) and hence we have succeeded in bringing the Pfaffian into the desired form ( [A -C^T;CB ]) = lim_w_i → z_i( [ p_i(w_i) · p_j(w_j) S(w_i ,w_j) p_i(w_i) ·ε_jS(w_i,z_j); ε_i· p_j(w_j) S(z_i ,w_j) ε_i·ε_jS(z_i,z_j) ]) . Clearly, the only non-trivial part of this statement is that the diagonal terms in C indeed have the correct limit, which we demonstrate below.Having brought the Pfaffian into the canonical form <ref> we may now use <ref> to factorize it and find ( [ p_i(w_i) · p_j(w_j) S(w_i ,w_j) p_i(w_i) ·ε_jS(w_i,z_j); ε_i· p_j(w_j) S(z_i ,w_j) ε_i·ε_jS(z_i,z_j) ]) = ( [ λ(z_i)λ(w_j) S(z_i,w_j);ξ_kλ(w_j) S(z_k,w_j) ])^i ∈p^- , k ∈p^+_j ∈p^- ∪p^+ ·∏_i∈p^+1/S(w_i,z_i)_i^2/λ_iξ_i·( [ ξ_i(w_j) S(z_i,w_j); (z_k)(w_j) S(z_k,w_j) ])^i ∈p^- , k ∈p^+_j ∈p^- ∪p^+ ·∏_i∈p^-1/S(w_i,z_i)t_i ^2/_iξ_i·∏_i≠ j =1^n (z_i - w_j)/∏_i< j =1^n (z_i - z_j)(w_i - w_j) where we have chosen the splitting such that the rows are labelled by the punctures z_i while the columns are labelled by the w_i. The sub–blocks of both determinants are of size |p^-| × n and |p^+| × n respectively.The last step is to take the limit w_i → z_i. Notice that each line is finite in the limit, as the potential singularities on the diagonal of the determinants are cancelled by the inverse Szego kernels multiplying them. Explicitly, we see for example that the first line becomes lim _w_i → z_i( [ λ(z_i)λ(w_j) S(z_i,w_j);ξ_kλ(w_j) S(z_k,w_j) ])^i ∈p^- , k ∈p^+_j ∈p^- ∪p^+ ·∏_i∈p^+1/S(w_i,z_i)1/λ_iξ_i=(Φ)^i∈p^-_j∈p^- where we recover the dual Hodges matrix with entries Φ_ij = λ(z _i) λ(z_j)S(z_i ,z_j) ,andΦ_ii =λ (z_i) λ(z_i) . To take the limit w_i → z_i in (<ref>) we used the Leibniz formula for the determinant and then noticed that the terms surviving in the limit reassemble into ( Φ ). The second line similarly gives rise to the determinant of the Hodges matrix with entries _ij = (z _i) (z_j) S(z_i ,z_j) ,and_ii = (z _i) (z_j) . Recall that the polynomials λ(z), (z) belong to the twistor/dual twistor representation of the refined scattering equations, respectively, so these definitions of the Hodges matrices agree with the usual ones on the support of the scattering equations (of the correct degree).To summarize, we have shown that the vector–mode CHY Pfaffian factorizes into a product of a Hodges determinant times a dual Hodges determinant( [A -C^T;CB ]) = (Φ) ^i ∈p^-_j ∈p^- · () ^i ∈p^+_j ∈p^+·∏_i∈p^- t_i^2·∏_j∈p^+_j^2 on the support of the scattering equations of degree d = |p^-| -1. §.§.§ Degree, Kernel and C_ii Diagonal Elements There are several loose ends to tie up in the above discussion. Firstly, we have used the refined scattering equations of degree d = |p^-| -1 without justification for fixing the degree of the scattering equations in terms of the number of particles with negative helicity. Indeed, the CHY scattering equations are equivalent to the union of the refined scattering equations of all possible degrees, so a priori there is no reason to restrict our attention to the refined scattering equations in the sector k = |p^-| only. It is however known <cit.> that the Pfaffian <ref> actually vanishes when evaluated on solutions to the refined scattering equations of the wrong degree. One way to show this is to perform the same steps as above[In the twistor and dual twistor representation the diagonal terms in the C matrix block remain of the same form even when d ≠ |p^- | - 1. See below for further comments.] and then discover that one of the Hodges matrices has a larger than expected kernel. Though this is straightforward, we want to take an alternative route here.We can actually construct the kernel of the CHY–matrix evaluated on the wrong degree explicitly. In fact, if 0 < Δ := |p^- | -1 -d, then define v_ i = γ(z_i ) t_i^-1 [ξ _i| _z_iζ̃]/ [ ξ_i_i ] ,w_ i = -γ(z_i ) t_i^-1 [_i | _z_iζ̃]fori = 1 , ⋯ ,n where γ∈ H^0( T^1/2) is any holomorphic section of T^1/2 and ζ̃^α̇∈ H^0 ( 𝒪(-d) ⊗ K [∑_i=1^n z_i ] ) with the requirement that _z_iζ= t_i_i ∀ i ∈p^+ . Here ξ_i are the auxiliary spinors that enter the definition of the polarization vectors when i ∈p^-, and arbitrary spinors when i ∈p^+. (Note that this requirement implies some simplifications of the kernel, e.g. w_i = 0 for i ∈p^+. Also we find that under a gauge transformation ε_i →ε_i + p_i the kernel transforms as v_i → v_i - w_i, which is necessary for the following equation to be gauge covariant.) With these definitions a straightforward, though somewhat tedious, calculation shows that the scattering equations imply ( [A -C^T;CB ]) ( [ v; w ]) = 0 Counting the free parameters in ζ and γ we find that the kernel is of dimension 2Δ +2.If Δ <0 we have to take the parity conjugate of the above construction and find that the kernel is of dimension 2 |Δ| +2.The second loose end to tie up is that even when evaluated on solutions of the correct degree, the Pfaffian and the Hodges matrices have a non–empty kernel. Hence, the above discussion needs to be adapted to the reduced Pfaffian ^'( [A -C^T;CB ]) = S(z_1 , z_2) ( [A -C^T;CB ]) ^1̌, 2̌ where the superscript 1̌, 2̌ is the instruction to remove the first two rows and columns from the matrix before taking its Pfaffian. The scattering equations ensure that the reduced Pfaffian is still fully permutation symmetric, albeit not manifestly so. We may assume without loss of generality that 1 ∈p^- and 2 ∈p^+. Then, retracing the steps from above with one fewer row/column in each matrix, we find^'( [A -C^T;CB ])= ( [ λ(z_i)λ(z_j) S(z_i,z_j);ξ_2λ(z_j) S(z_2,z_j) ])^i ∈p^- \{1}_j ∈p^- · 1/λ_2ξ_2 _2·∏_i∈p^+_i^2·( [ ξ_1(z_j) S(z_1,z_j); (z_k)(z_j) S(z_k,z_j) ])^k ∈p^+ \{2}_j∈p^+ · 1/_1ξ_1t_1·∏_i∈p^- t_i^2where the two matrices each consist of two sub-blocks, with dimensions 1 × d+1 and d × d+1 and 1 ×+1 and × +1 respectively. To bring this into the desired form of two reduced Hodges determinants we use that one can add linear combinations of the columns in a matrix onto each each other without changing the value of the determinant. Thus we can show that for instance( [ λ(z_i)λ(z_j) S(z_i,z_j);ξ_2λ(z_j) S(z_2,z_j) ])^i ∈p^- \{1}_j ∈p^-= ( [ λ(z_i)λ(z_j) S(z_i,z_j);ξ_2λ(z_2)δ_1,j ])^i ∈p^- \{1}_j ∈p^- · S(z_1,z_2) ∏_k ∈p^- \{1}z_1 - z_k/z_2-z_k = ( Φ)^i ∈p^- \{1}_j ∈p^- \{1} · ξ_2 λ (z_2)· S(z_1,z_2) ∏_k ∈p^- \{1}z_1 - z_k/z_2-z_kwhere we added the columns for j ∈p^- \{1} with coefficients√( z_1)/√( z_j )∏_k ∈p^- \{1, j}z_1 - z_k/z_j-z_k , onto the column j =1. This simplifies the determinant because the last row now has only a single non-vanishing entry. Indeed, Cauchy's theorem tells us that ξ _2 λ (z_1)S(z_2,z_1) + ∑_j ∈p^- \{1}ξ _2 λ (z_j)S(z_2,z_j) √( z_1)/√( z_j )∏_k ∈p^- \{1, j}z_1 - z_k/z_j-z_k =ξ _2 λ (z_2)S(z_1,z_2)∏_k ∈p^- \{1, j}z_1 - z_k/z_2-z_k for the last row, as well as Φ_i1 + ∑_j ∈p^- \{1}Φ_ij √( z_1)/√( z_j )∏_k ∈p^- \{1, j}z_1 - z_k/z_j-z_k =0 for all rows labelled by i ∈p^- \{1}. After the analogous argument for the Hodges matrix we find that the reduced Pfaffian factorizes as expected into two reduced Hodges determinants ^'( [A -C^T;CB ]) = ( Φ)^i ∈p^- \{1}_j ∈p^-\{1}·()^k ∈p^+ \{2}_j∈p^+ \{2}·1/ũ_1^2 u_2^2·∏_i∈p^- \{1} t_i^2·∏_i∈p^+ \{2}_i^2 Recall that the scaling parameters t_i, _i , u_i ,ũ_i come from the various representations of the refined scattering equations in the sector d= |p^-|-1. Finally we would like to spell out the details pertaining to the diagonal elements of the C block–matrix that were used in the point–splitting procedure above. It is known that on the support of the scattering equations the C_ii are gauge invariant, and in fact reduce to the diagonal elements of the Hodges matrices. Indeed for a negative helicity particle, i ∈p^-, we compute ϵ_i^- · P(z_i) = lim_z → z_iϵ_i^- · P(z) = lim_z → z_iλ_i λ_A(z)ξ_i _A(z)/ξ_i _i= λ_i λ_A(z_i)ũ_i ^-1 using the scattering equations in ambi–twistor form, and likewise ϵ_i^+ · P(z_i) = _i _A(z_i) u_i ^-1 for i ∈p^+. Notice that while, for instance, λ_A(z) has a pole at z_i for i ∈p^-, the combination λ_i λ_A(z) is regular at z → z_i. Analogous comments apply to _A(z) and of course P(z). Using the relations between the three representations of the refined scattering equations we may write these diagonal terms equivalently as ϵ_i^- · P(z_i) = λ_i λ_T(z_i) t_i = λ_i λ_T̃(z_i) _i^-1 ϵ_i^+ · P(z_i) = _i _ T(z_i) t_i^-1 = _i _T̃(z_i) _i for i ∈p^- ∪p^+. Note that the twistor and dual twistor representations of these terms do not rely on the fact that d= |p^-| -1, so they can be used straightforwardly even in solution sectors with d≠ |p^-| -1. §.§ Refinement of the `Squeezed' Vector–Mode Pfaffian Having discussed the factorization of the vector mode Pfaffian in great detail, we may now apply the same technique to the squeezed Pfaffian appearing in the CHY formula for Einstein-Yang-Mills. Recall from <cit.> the half-integrand for EYM tree amplitudes in the sector with τ colour traces reads^' ( Π (_1 , ⋯ , _τ : h) ) =∑_i_2 < j_2 ∈ _2⋯i_τ < j_τ∈ _τ∏_α =2^m (z_i_α -z_j_α ) (M(h∪I∪J : h))with the abbreviation for the gluon labelsI≡{ i_2 , ⋯ , i_τ}andJ≡{ j_2 , ⋯ , j_τ} . Note that ^'Π only makes explicit reference to τ-1 traces. On the support of the scattering equations this reduced Pfaffian does not depend on which trace is being removed from the expression.To factorize this we may use the splitting formula <ref> term by term in the sum. Since the formula for EYM scattering amplitudes contains also a vector mode Pfaffian for gravitons and gluons, we find using the Kernel argument from above that the amplitude localizes to solutions of degree d = n_gr^- +n_gl^- -1.For the gravitons we have to employ the point-splitting procedure as above, and the structure is identical to the pure vector Pfaffian. For the gluons we don't need to point-split and can just take the result for the scalar mode Pfaffian. Combining the two we find(M(h∪I∪J : h)) = ( Φ ^i ∈h^- ∪ I_j ∈h^- ∪ J) ( ^i ∈h^+ ∪ I_j ∈h^+ ∪ J) V( I ∪ J ) / V( I) ^2 V(J ) ^2 ,with Φ, the Hodges matrices as defined above. We have chosen to split the rows/columns in a symmetric way, but again, many others are possible using <ref>.§ EYM TREE SCATTERING AMPLITUDES Having factorized the CHY integrand for 4d EYM into two chiral halves, we can lift it to a formula for all tree–level scattering amplitudes in maximally supersymmetric Einstein-Yang-Mills.It is well known that 4d scattering amplitudes can be organized by MHV sector <cit.>, which counts the number of states of one helicity, and is independent of the number of states of the other helicity. The remarkable simplicity of the maximally helicity violating amplitudes can be traced back to the integrability properties of the underlying (anti-self-dual) field equations, and the higher N^k-2MHV amplitudes are an expansion around this integrable sector. While this perspective breaks manifest parity invariance, it retains a natural action of parity and the emergence of parity invariance is understood <cit.>. After incorporation of supersymmetry, the MHV sectors are generalized to R-charge super-selection sectors. This continues to be true in EYM, which is expected already from the CHY representation: Since the CHY integrand for EYM still contains one vector mode, alongside one squeezed vector mode Pfaffian, the specialization to a definite degree d = k-1 of the scattering equations still occurs, where k is the the R-charge sector of the amplitude. The spacetime Lagrangian dictates that a tree level scattering amplitude in Einstein-Yang-Mills in the τ trace sector comes with a factorκ^n_gr +2 τ -2of the gravitational coupling constant κ∼√(G_N), where n_gr denotes the number of external gravitons. In <cit.> it was explained that, when written in terms of a worldsheet model, these powers of κ must be accompanied by the same number of powers of ⟨, ⟩ or [, ] brackets. Indeed, from dimensional analysis we find that#⟨, ⟩ + # [,] = n_gr^+ + n_gr^- +2 τ -2 .Parity conjugation exchanges ⟨, ⟩ and [, ], which fixes#⟨, ⟩= n_gr^- + τ -1 ,# [ ,] = n_gr^++τ -1 . From the perspective of twistor theory, the appearance of the SL(2)_L,R invariants ⟨, ⟩ and [,] controls the breaking of conformal symmetry of a theory, and the very existence of a well defined counting is a hallmark of the natural action (and breaking) of this symmetry on twistor space. §.§ Einstein-Yang-Mills amplitudes in 4d spinor helicity variables There are as many representations of any 4d refined scattering amplitude as there are representations of the 4d refined scattering equations themselves, and they each make different properties manifest. We begin with the non-supersymmetric ambitwistor representation, which makes parity manifest. Now the R-charge sector k is simply given by the number of negative helicity particles, so d = |p^-|-1. Using the known behaviour of the Jacobian <cit.> which arises in going from the CHY to the refined scattering equations we find the Einstein-Yang-Mills scattering amplitudes in the τ-trace sector∫1/vol GL(2 , ℂ)∑_i_2 < j_2 ∈ _2⋯i_τ < j_τ∈ _τ( Φ ^i ∈h^- ∪I_j ∈h^- ∪J) ( ^i ∈h^+ ∪I_j ∈h^+ ∪J) V( I∪J ) / V(I) ^2 V(J) ^2∏_α = 2 ^τ( z_i_α - z_j_α) / z_i_αz_j_α∏_α = 1^τPT(_α) ∏_i ∈p^-ũ_i/ũ_i ^2( _i - ũ_i(z_i) )∏_i ∈p^+ u_i/u_i ^2( λ_i - u _iλ(z_i) )with the abbreviationsI≡{ i_2 , ⋯ , i_τ}andJ≡{ j_2 , ⋯ , j_τ} .We use the familiar Hodges matrices (in the ambitwistor representation)Φ_ij = λ_i λ_jS(z_i,z_j) ,Φ_ii = - ∑_j ∈p^- \{i}Φ_ij ũ _j/ũ _iand_ij = _i _jS(z_i,z_j) ,_ii = - ∑_j ∈p^+ \{i}_ij u _j/u _i ,for the integrand and the functions λ(z) , (z) are solutions to the ambitwistor scattering equations,λ(z) = ∑_i ∈p^-λ_iũ_i S(z,z_i) ,(z) = ∑_i ∈p^+_i u_i S(z,z_i) ,while u_i , ũ_i are the corresponding scaling parameters. Furthermore we used the world-sheet Parke-Taylor factor of a gluon trace, defined asPT() := ∑_σ∈ S_|| / ℤ_||[ 𝐓_σ(1)⋯𝐓_σ(||)] ∏_i ∈ S(z_σ(i) , z_σ( i+1)) ,with the gauge group generators 𝐓_i associated to each gluon in the trace. We emphasize again that while _1 appears to be singled out, the scattering equations guarantee that the formula is independent of this choice, so is actually S_τ permutation symmetric.It is well known that one of the major advantages of the refined formulas is that we can actually go beyond the bosonic amplitudes given by the CHY formula and find 𝒩≤ 3 super-symmetric amplitudes in a remarkably simple fashion. Given the Grassmann numbers η_i , η̃_i (transforming in the fundamental/anti-fundamental of the SU(𝒩) R-Symmetry, respectively) from the external supermomenta, we can promote <ref> to the full superamplitude by including the factorexp( ∑_ i ∈p^- i ∈p^+η_i ·η̃_j ũ_i u_jS(z_i,z_j) ) , whose behaviour under factorization is simple and well understood <cit.>. This is astonishing not just because of its simplicity, but also because it makes space-time supersymmetry manifest. It is a consequence of the natural incorporation of onshell SUSY on twistor space. §.§ sEYM on Twistor Space Now we shall give the twistor space representation of the scattering amplitude, which will break manifest parity invariance, but allow for manifest supersymmetry.In general, a scattering amplitude is a multi-linear functional of the external wave functions. Most commonly it is simply given in a basis of plane waves (as e.g. above), but on twistor space it is actually more natural to maintain the full structure. Using onhell SUSY we may write the wave function for a whole SUSY multiplet as a single function on onshell superspace. In sEYM there are two colour neutral multiplets, h_i, ϕ_i, which contain the graviton as their highest/lowest spin state, and one adjoint-valued multiplet A _i, containing the gluons.Via the Penrose transform the external wave functions of the super-multiplets are given by cohomology classes with a certain homogeneity on super twistor space := ^3|4\^1|4h_i ∈ H^1 (,𝒪(2) ) , A_i ∈ H^1(,𝒪(0) ) , ϕ_i ∈ H^1 (,𝒪(-2) ), of helicity -2,-1,0 respectively.As usual, the coefficients in the Taylor expansion w.r.t. the Grassmann coordinates ofcorrespond to the various components of the supermultiplet. With these definitions in place we now present the sEYM scattering amplitude in the τ colour trace sector on twistor space∑_d ∫ _ℳ_0,n(d)μ_d∑_i_2 < j_2 ∈ _2⋯i_m < j_m ∈ _m( Φ ^i ∈ϕ∪I_j ∈ϕ∪J) ( ^i ∈h∪I_j ∈h∪J) V( I∪J ) / V( I) ^2 V(J ) ^2∏_α = 2 ^τ( z_i_α - z_j_α) / z_i_αz_j_α∏_α = 1^τPT(_α) ∏_i ∈ hh_i(Z(z_i)) ∏_i ∈ gA_i(Z(z_i)) ∏_i ∈ϕϕ_i(Z(z_i))where we abbreviated the setsI≡{ i_2 , ⋯ , i_τ}andJ≡{ j_2 , ⋯ , j_τ} ,as well as the measure μ _d ≡^4(d+1)|4(d+1) Z/≡∏_a=0^d ^4|4 Z_a/ on ℳ_0,n(d), the moduli space of holomorphic maps of degree d from the n-punctured Riemann sphere to super twistor space. Here we have chosen to coordinatize this space as Z(z) = ∑_a=0^d Z_a s_a(z), for some fixed basis[Note that sinceis a Calabi-Yau supermanifold, the holomorphic measure μ_d is independent of the choice of basis {s_a}_a=0^d by itself. Indeed, when the target space is ^m|𝒩, a change of basis in H^0(ℙ^1 , 𝒪(d)) with JacobianJ({ s_a },{ s^'_a })induces the integration measure to transform as μ _d →μ_d J({ s_a },{ s^'_a })^m+1-𝒩.] of polynomials {s_a}_a=0^d spanning H^0(ℙ^1, 𝒪(d)). The Hodges matrices Φ , have been lifted to twistor space, with entries given by Φ_ij = S(z_i ,z_j) Z(z _i) Z(z_j) , Φ_ii =Z (z_i)Z(z_i)and _ij = S(z_i ,z_j) [ ∂/∂ Z(z _i) ∂/∂ Z(z _j)] , _ii = - ∑_j ≠ i_ij p(z_j)/p(z_i) for some arbitrary section p ∈ H^0(T^1/2⊗𝒪(d)). Here ⟨· , ·⟩ and [ · , · ] are the infinity twistor[These are fixed simple bitvectors (antisymmetric matrices of rank 2) on twistor space, which arise in the decompactification of 𝕄̅ to 𝕄.] and dual infinity twistorftn:infinittTwistor respectively, i.e. ⟨ Z Z^'⟩ = ℐ_IJ Z^I Z^' J for two twistors Z , Z^'∈ and [ W W^' ] = ℐ̃^IJ W_I W^'_J for two dual twistors W , W^'∈^∗. Generally, the appearance of the infinity twistor signals and controls the breaking of space-time conformal symmetry, which on twistor space is represented by general linear transformations. In the present case, they reduce as ⟨ Z Z^'⟩ = ⟨λ λ^'⟩ and [ W W^' ] = [^' ]to the Lorentz invariant pairings of left- and right-handed spinors, respectively. There are two remarkable features of this formula. Firstly, on twistor space the dependence on the infinity twistor and dual infinity twistor has separated. This is akin to the separation in supergravity amplitudes <cit.>, but the addition of Yang-Mills interactions leads to a sum of such products. In other words, the presence of gluon traces obstructs a complete separation of the infinity twistor and dual infinity twistor, albeit in a rather systematic way. Secondly, the amplitude is now manifestly space-time supersymmetric, so <ref> includes amplitudes for space-time fermions. Neither of these properties is obvious/accessible from simply using the substitution p_i · p_i →λ_iλ_jλ_iλ_j in the dimension agnostic CHY formulae. Another property of the amplitudes can be learned from <ref>: since the Hodges determinant is an antisymmetric polynomial of degree d-1 in the marked points, it will vanish identically if d < |h | + τ -1. Hence we find (for non-trivial amplitudes) the inequality d +1 ≥|h | + τ, and similarly the parity conjugate+1 ≥|ϕ| + τ. Moreover, since the R-charge selection rules follow from the fermionic part of the map and wave-functions, which completely separates from the rest of the formula, we manifestly have the usual selection rules for SUSY (in particular k = d+1). This completely fixes the degree d in terms of the external states, e.g. for external gravitons and gluons only, we recover d+1 = n_gr^- +n_gl^- as expected. A corollary of this is that n_gl^- ≥τ andn_gl^+ ≥τ, so any amplitude with less negative/positive gluons than traces will vanish.We may easily go from the twistor space representation back to the ambi-twistor representation (<ref>) <cit.> by specifying the external states to be plane wave states and use the explicit form of the Penrose representative ∫ t_i/t_i t_i ^2s_i +2 ^2(λ_i - t_iλ ) exp( t_i [_i μ ] +t_iη_i ·χ)for a multiplet of helicity s_i. Indeed, by judiciously choosing a coordinate basisftn:measure for the space of maps that is adapted to the external data, Z(z) = ∑_i∈p^-Z_i∏_j∈p^- \{i}z-z_j/z_i-z_j while keeping the punctures fixed, we may perform the integral over the moduli space of the map Z(z) trivially, upon which we recover the ambi-twistor representation. It is also worth pointing out that <ref> reduces correctly to previously known 4d expressions. Indeed, in the pure gluon (single trace) sector it agrees immediately with the RSVW formula super Yang-Mills. In the pure graviton sector it reduces to the restriction of the CS formula, while the the enhancement to follows from the special properties of the Hodges determinants and scattering equations. A special case of this is given by the Einstein-Maxwell sector, where each trace contains exactly two gluons. § CONCLUSION & OUTLOOKWe have presented and proven a formula for the splitting of certain fermion correlators into left and right handed Hodges type determinants. This factorization holds for general correlators of spinors with values in T 𝕄 on the Riemann sphere, and in particular does not require the scattering equations to hold. We have then applied this splitting formula to the translation of d-dimensional CHY fomulas into the 4d spinor helicity formalism, which crucially involves the refined scattering equations, both for known examples and to find a new formula for all tree scattering amplitudes in EYM.The rational functions given in terms of Pfaffians and determinants that enter <ref> have natural origins in 2d CFT on the punctured Riemann sphere. Even though we completely ignored this origin for the purpose of this paper, we believe that 2d CFT is the proper realm for understanding <ref> (and in fact the crucial parts of the proof were discovered using that CFT description). Here we briefly sketch this relation, but leave the details for an upcoming publication. The left hand side of <ref> is given by the correlator ⟨∏_i=1^2n (λ_i _i) ·ψ(z_i) ⟩ = (λ_iλ_j_i_jS(z_i , z_j) )^i,j=1,⋯ , 2n with the action S = ∫_ℙ^1ψ^μψ^ν η_μν , ψ∈ΠΩ^0(K^1/2⊗T 𝕄) , and no constraints on the locations z_i. On flat Minkowski space 𝕄 =ℝ^3,1 the tangent bundle splits into a product of the left-handed and right-handed spin bundlesT 𝕄 ≃ 𝕊^+ ⊗𝕊^- , where the isomorphism is provided by the van der Waerden symbols σ^μ_α. We can write the world-sheet current[The argument in <cit.> is based on this identity.] associated to Lorentz transformations on either side of the isomorphism as ψ^[μψ^ν] ≃ ρ^a_α ρ^b_β ε_ab ε_ + ε_αβ _a_b ε^ab with the new fieldsρ^a ∈ΠΩ^0(K^1/2⊗𝕊^- ) , _a ∈ΠΩ^0(K^1/2⊗𝕊^+ ) , where the Roman indices a,b=1,2 label the fundamental representation of a new SL(2) gauge symmetry. This can be seen to arise as redundancy in the change of variables from ψ to ρ , and is responsible for the permutation symmetry of (<ref>). It is hence natural to suspect that the corresponding 2d sigma sigma models are also related. Indeed, the right hand side of <ref> involves the correlators⟨∏_i ∈b⟨λ_iρ ^1 (z_i)⟩ ∏_j ∈b^c⟨λ_jρ ^2 (z_j)⟩⟩ = (λ_iλ_jS(z_i,z_j) )^i∈b_j∈b^cin the actionS =∫_ℙ^1ε_ab ⟨ρ^a ρ ^b ⟩ ,and likewise for the right-handed determinant. Taking into account the appropriate Vandermonde determinants, we have proven (<ref>) that this correlator is indeed permutation symmetric. This suggests that there is an equivalence of CFTs that underlies the splitting formula <ref>. Closely related to the previous comments is the fact that the twistor space connected formula (<ref>) for EYM amplitudes can naturally be incorporated into the twistor string of <cit.>, which we describe in an upcoming publication. Finally, it is very tempting to apply <ref> to scattering-equation based formulas for higher-loop amplitudes, which currently come in two flavours. On the one hand, the ambitwistor string model <cit.> gives rise to amplitudes for supergravity on the torus <cit.>. Indeed, <ref> has a natural generalization to higher genus surfaces, and we expect a generalization of the proof here to carry over. It is however believed that the ambitwistor string is only modular invariant in 10d, so even though the external states can easily be restricted to lie in a 4d subspace, the loop momentum would have to be integrated over a 10d space, which will make P(z) generically a 10d vector. This obstructs the use of <ref> as shown here, since the C_ii elements of the CHY type Pfaffian cannot be split straightforwardly. Further complications might arise from Ramond sector fields or the spin-structure dependence of the Szegó kernel.On the other hand there are formulas for loop amplitudes on the nodal sphere <cit.>. While these seem to be well defined (or at least come with a canonical regularization scheme) in any dimension, the above obstruction remains: on the nodal sphere the scattering equations imply generically P(z)^2 ≠ 0, so we again cannot split the C_ii elements of the CHY type Pfaffian straightforwardly. In this situation the resolution might be more apparent: we can write P(z) as a sum of null vectors, and, using the multi-linearity of the Pfaffian, apply the factorization <ref> to each summand separately. There have just been promising new results <cit.> for 4d loop amplitudes based on the 4d refinement of the scattering equations on the nodal sphere, which might be combined naturally with the present work to find n-point SUGRA integrands. We leave these exciting thoughts and questions for future work.§ ACKNOWLEDGEMENTS It is a privilege to thank David Skinner for many inspiring and insightful discussions as well as continuous encouragement. The author would also like to thank Tim Adamo, Piotr Tourkine and Lionel Mason for interesting discussions.The author is supported in part by a Marie Curie Career Integration Grant (FP/2007-2013/631289).§ PROOF OF CHIRAL SPLITTING IN 4D In this appendix we give the full details of the proof of <ref>, which we repeat here for the reader's convenience (ijij/z_i - z_j)^i,j=1,⋯ , 2n = (ij/z_i - z_j)^i∈b_j∈𝔟^c/V( b) V( b ^c) (ij/z_i - z_j)^i∈b̃_j∈b̃^c/V( b̃) V( b̃ ^c) V( { 1 , ⋯ , 2n } ) where b, b̃ are any ordered subsets of {1, ⋯ , 2n } of size n and b̃ ^c, b^c are their complements. First we recall the definitions. Take 2n points on the Riemann sphere, given in inhomogeneous coordinates by z_i, as well as one left- and right-handed spinor λ_i, _i associated to each puncture, with i = 1 ,⋯ , 2n. We also have the Vandermonde determinant of an ordered set, defined as usual V( b ) := ∏_i<j ∈b (z_i - z_j) We begin by proving that the building block(ij/z_i - z_j)^i∈b_j∈b^c/V( b) V( b ^c)is independent of the split of the labels {1 , ⋯ , 2n } into the two ordered subsets b , b^c – in other words, it is still S_2n permutation symmetric, albeit not manifestly so. §.§ Proof of S_2n Symmetry For definiteness, and without loss of generality, we assign the labels {1 , ⋯ , n } to the rows and {n+1 , ⋯ , 2 n } to the columns, so we prove the S_2n permutation symmetry of( Φ ^i = 1 , ⋯ , n _j = n+1 , ⋯ , 2n)/V({ 1, ⋯ ,n} ) V({n+1, ⋯ ,2n} ) which is used in the main text. Recall that the numerator is given by the determinant of the n× n matrix Φ with elements Φ^i_j = ⟨λ_i , λ_j ⟩/z_i-z_j ,fori = 1, ⋯ ,n and j = n+1 , ⋯ , 2n , where ⟨· , ·⟩ is the SL_2 invariant pairing of the left-handed spinors. We emphasize that while the set of punctures that label the rows is disjoint from the set of punctures that label the columns, we can easily achieve any overlap between the sets labelling rows and columns by taking an appropriate limit of the above matrix, with a “diagonal element” that we can specify freely. This is necessary e.g. for amplitudes involving gravitons.Notice that, due to the antisymmetry of the Vandermonde determinant as well as the numerator <ref> is manifestly S_n × S_n ×ℤ_2 symmetric, i.e. permutation symmetric in each of the two sets {1 , ⋯ , n } and {n+1 , ⋯ , 2n } separately, while the ℤ_2 factor swaps the two sets and transposes the matrix. In order to show that it has in fact S_2n symmetry we will show that as a rational function of the locations z_i and spinors λ_i it is equal to the expression (-1)^n(n-1)/2∑_b⊂{ 1, ⋯ , 2n}|b| =n ∏_i ∈b (λ_i)^0 ∏_j ∈b^c (λ_j)^1 ∏_ i ∈bj ∈b^c1/z_i - z_j Since this expression is manifestly S_2n invariant (even though it has lost its manifestLorentz SL_2 invariance), proving equality of <ref> and <ref> will establish the S_2n symmetry of <ref>.The plan is to examine the poles and residues of each expression as any of the two punctures coincide, and then use a recursion argument to show that the residues agree. First, we rewrite the claim as( Φ ^i = 1 , ⋯ , n _j = n+1 , ⋯ , 2n)= (-1)^n(n-1)/2∑_b⊂{ 1, ⋯ , 2n}|b| =n ∏_i ∈b (λ_i)^0 ∏_j ∈b^c (λ_j)^1 V(1, ⋯ ,n ) V(n +1, ⋯ ,2n )/∏_ i ∈bj ∈b^cf (z_i - z_j) , Each side is now a section of ⊗ _i 𝒪_i(-1) with at most simple poles as any two punctures coincide, so by Cauchy's theorem, comparing residues is sufficient to prove equality. Furthermore, given the already manifest S_n × S_n ×ℤ_2 symmetry, it is sufficient to check the residues at z_1= z_2 andz_1 = z_2n. It is actually immediately clear that both sides have vanishing residue at z_1 = z_2, so we only have to put some effort into checking the residue at z_1 =z_2n. On the left hand side we find lim _ z_1 → z_2n{(z_1 -z_2n) ( Φ ^i = 1 , ⋯ , n _j = n+1 , ⋯ , 2n) } = (-1)^n ⟨λ_1 , λ_2n⟩ ( Φ ^i = 2 , ⋯ , n _j = n+1 , ⋯ , 2n-1) ,while on the right hand side we find lim _ z_1 → z_2n {(z_1 -z_2n)∑_b⊂{ 1, ⋯ , 2n}|b| =n ∏_i ∈b (λ_i)^0 ∏_j ∈b^c (λ_j)^1 V(1, ⋯ ,n ) V(n+1, ⋯ ,2n )/∏_ i ∈bj ∈b^c (z_i - z_j) } =⟨λ_2n ,λ_1⟩ ∑_b⊂{ 2, ⋯ , 2n-1}|b| =n-1 ∏_i ∈b (λ_i)^0 ∏_j ∈b^c (λ_j)^1 V(2, ⋯ ,n ) V(n+1, ⋯ ,2n -1)/∏_ i ∈bj ∈b^c(z_i - z_j) where, going to the second line, we observed that only those terms in the sum where 1 and 2n are in different subsets contribute to the pole. We immediately recognize the condition for the residues to agree as the very same claim we're trying to prove but for n-1. Hence, we may conclude the proof by invoking a simple induction argument from n to n-1.§.§ Proof of Splitting Armed with the knowledge that the factor <ref> is secretly S_2n symmetric, we may now establish the factorization formula <ref> by comparing residues. Both sides are again sections of ⊗_i 𝒪_i(-1) so comparing residues as any pair of punctures collide is sufficient to prove equality.Using the S_2n symmetry of both sides we may simply look at the residue as z_1 → z_2, where we find for the left hand side lim _ z_1 → z_2 {(z_1 -z_2)(ijij/z_i - z_j)^i,j=1,⋯ , 2n} = 1212 (ijij/z_i - z_j)^i,j=3,⋯ , 2n . On the right hand side we first make a judicious choice for the splitting of labels into rows and columns such that the rows of the first matrix be labelled by the set {1}∪b^' and the columns by {2}∪b^'^c where b^'∪b^'^c= {3 , ⋯ , 2n } is a partition of the remaining labels and similarly for the second matrix. (For the sake of clarity we drop the primes below.) Hence we find the residue on the right hand side lim _ z_1 → z_2{(z_1 -z_2) (ij/z_i - z_j)^i∈{1}∪b_j∈{2}∪b^c /V( {1}∪b) V({2}∪b^c ) (ij/z_i - z_j)^i∈{1}∪b̃_j∈{2}∪b̃ ^c /V({1}∪b̃) V( {2}∪b̃ ^c) V( { 1 , ⋯ , 2n } )}= 1212 (ij/z_i - z_j)^i∈b_j∈b^c /V(b) V(b^c ) (ij/z_i - z_j)^i∈b̃_j∈b̃ ^c /V(b̃) V( b̃ ^c) V( { 3 , ⋯ , 2n } ) Notice that each determinant has a simple pole as z_1 → z_2, while the big Vandermonde factor has a simple zero, and on the location of the residue there are cancellations between the various Vandermonde factors.We again recognize the condition for the residues to agree as the very same claim we're trying to prove but for n-1 so, after invoking recursion, this concludes the proof of the splitting formula <ref>. JHEP | http://arxiv.org/abs/1705.09315v2 | {
"authors": [
"Kai A. Roehrig"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170525181958",
"title": "Chiral Splitting and $\\mathcal N = 4$ Einstein--Yang--Mills Tree Amplitudes in 4d"
} |
A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems Christopher Coley,[Ann and H.J. Smead Aerospace Engineering Sciences, 426 UCB, University of Colorado Boulder 80309, USA] [ E-mail: [email protected]]Joseph Benzaken[Department of Applied Mathematics, 526 UCB, University of Colorado, Boulder, CO 80309, USA],John A. Evans^*====================================================================================================================================================================================================================================================================================================== When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.*Equal contribution. § INTRODUCTION Learning in statistical models via gradient descent is straightforward when the objective function and its gradients are tractable. In the presence of latent variables, however, many objectives become intractable. For neural generative models with latent variables, there are currently a few dominant approaches: optimizing lower bounds on the marginal log-likelihood <cit.>, restricting to a class of invertible models <cit.>, or using likelihood-free methods <cit.>. While likelihood-free methods are popular and result in models that generate crisp samples, they do not compare as favourably in quantitative log-likelihood evaluations <cit.>. In this work, we focus on the first approach and introduce filtering variational objectives (FIVOs),a tractable family of objectives for maximum likelihood estimation (MLE) in latent variable models with sequential structure.Specifically, letdenote an observation of an -valued random variable. We assume that the process generatinginvolves an unobserved -valued random variablewith joint density p(, ) in some family 𝒫. The goal of MLE is to recover p ∈𝒫 that maximizes the marginal log-likelihood, log p() = log( ∫ p(, )d)^1^1We reuse p to denote the conditionals and marginals of the joint density. . The difficulty in carrying out this optimization is that the log-likelihood function is defined via a generally intractable integral. To circumvent marginalization, a common approach <cit.> is to optimize a variational lower bound on the marginal log-likelihood <cit.>. The evidence lower bound (, p, q) (ELBO) is the most common such bound and is defined by a variational posterior distribution q( | ) whose support includes p's,(, p, q)= _q(| )[logp( , )/q( | )] = log p() - q( | )p( | )≤log p() .(, p, q) lower-bounds the marginal log-likelihood for any choice of q, and the bound is tight when q is the true posterior p(|). Thus, the joint optimum of (, p, q) in p and q is the MLE. In practice, it is common to restrict q to a tractable family of distributions (e.g., a factored distribution) and to jointly optimize the ELBO over p and q with stochastic gradient ascent <cit.>. Because of the KL penalty from q to p, optimizing (<ref>) under these assumptions tends to force p's posterior to satisfy the factorizing assumptions of the variational family which reduces the capacity of the model p. One strategy for addressing this is to decouple the tightness of the bound from the quality of q. For example,observed that Eq. (<ref>) can be interpreted as the log of an unnormalized importance weight with the proposal given by q, and that using N samples from the same proposal produces a tighter bound, known as the importance weighted auto-encoder bound, or IWAE.Indeed, it follows from Jensen's inequality that the log of any unbiased positive Monte Carlo estimator of the marginal likelihood results in a lower bound that can be optimized for MLE. The filtering variational objectives (FIVOs) build on this idea by treating the log of a particle filter's likelihood estimator as an objective function. Following <cit.>, we call objectives defined as log-transformed likelihood estimators Monte Carlo objectives (MCOs). In this work, we show that the tightness of an MCO scales like the relative variance of the estimator from which it is constructed. It is well-known that the variance of a particle filter's likelihood estimator scales more favourably than simple importance sampling for models with sequential structure <cit.>. Thus, FIVO can potentially form a much tighter bound on the marginal log-likelihood than IWAE.The main contributions of this work are introducing filtering variational objectives and a more careful study of Monte Carlo objectives. In Section <ref>, we review maximum likelihood estimation via maximizing the ELBO. In Section <ref>, we study Monte Carlo objectives and provide some of their basic properties. We define filtering variational objectives in Section <ref>, discuss details of their optimization, and present a sharpness result. Finally, we cover related work and present experiments showing thatsequential models trained with FIVO outperform models trained with ELBO or IWAE in practice.§ BACKGROUND We briefly review techniques for optimizing the ELBO as a surrogate MLE objective. We restrict our focus to latent variable models in which the model p_θ(, ) factors into tractable conditionals p_θ() and p_θ( | ) that are parameterized differentiably by parameters θ. MLE in these models is then the problem of optimizing log p_θ() in θ. The expectation-maximization (EM) algorithm is an approach to this problem which can be seen as coordinate ascent, fully maximizing (x, p_θ, q) alternately in q and θ at each iteration <cit.>. Yet, EM rarely applies in general, because maximizing over q for a fixed θ corresponds to a generally intractable inference problem. Instead, an approach with mild assumptions on the model is to perform gradient ascent following a Monte Carlo estimator of the ELBO's gradient <cit.>. We assume that q is taken from a family of distributions parameterized differentiably by parameters ϕ∈^m.We can follow an unbiased estimator of the ELBO's gradient by sampling ∼ q_ϕ( | ) and updating the parameters by θ^' = θ + η∇_θlog p_θ(, ) and ϕ^' = ϕ + η (log p_θ(, ) - log q_ϕ(| )) ∇_ϕlog q_ϕ(| ), where the gradients are computed conditional on the sampleand η is a learning rate. Such estimators follow the ELBO's gradient in expectation, but variance reduction techniques are usually necessary <cit.>. A lower variance gradient estimator can be derived if q_ϕ is a reparameterizable distribution <cit.>. Reparameterizable distributions are those that can be simulated by sampling from a distribution ϵ∼ d(ϵ), which does not depend on ϕ, and then applying a deterministic transformation = f_ϕ(, ϵ). When p_θ, q_ϕ, and f_ϕ are differentiable, an unbiased estimator of the ELBO gradient consists of sampling ϵ and updating the parameter by (θ^', ϕ^') = (θ,ϕ) + η∇_(θ,ϕ) (log p_θ(, f_ϕ(, ϵ)) - log q_ϕ(f_ϕ(, ϵ) | )). Given ϵ, the gradients of the sampling process can flow through = f_ϕ(, ϵ).Unfortunately, when the variational family of q_ϕ is restricted, following gradients of -q_ϕ( | )p_θ( | ) tends to reduce the capacity of the model p_θ to match the assumptions of the variational family. This KL penalty can be “removed” by considering generalizations of the ELBO whose tightness can be controlled by means other than the closenesss of p and q, e.g., <cit.>. We consider this in the next section. § MONTE CARLO OBJECTIVES (MCOS) Monte Carlo objectives (MCOs) <cit.> generalize the ELBO to objectives defined by taking the log of a positive, unbiased estimator of the marginal likelihood. The key property of MCOs is that they are lower bounds on the marginal log-likelihood, and thus can be used for MLE. Motivated by the previous section, we present results on the convergence of generic MCOs to the marginal log-likelihood and show that the tightness of an MCO is closely related to the variance of the estimator that defines it.One can verify that the ELBO is a lower bound by using the concavity of log and Jensen's inequality,_q(| )[logp( , )/q( | )] ≤ log∫p( , )/q( | ) q( | )d = log p().This argument only relies only on unbiasedness of p( , ) / q(|) when z ∼ q( | ). Thus, we can generalize this by considering any unbiased marginal likelihood estimatorp̂_N() and treating [logp̂_N()] as an objective function over models p. Here N ∈ℕ indexes the amount of computation needed to simulate p̂_N(), e.g., the number of samples or particles. Monte Carlo Objectives. Let p̂_N() be an unbiased positive estimator of p(),[p̂_N()]= p(), then the Monte Carlo objective N(, p) over p ∈𝒫 defined by p̂_N() isN(, p) = [logp̂_N()] For example, the ELBO is constructed from a single unnormalized importance weight p̂() = p( , ) / q(|). The IWAE bound <cit.> takes p̂_N() to be N averaged i.i.d. importance weights,N(x, p , q) = _q(^i | )[log(1/N∑_i=1^N p( , ^i)/q(^i|))]We consider additional examples in the Appendix. To avoid notational clutter, we omit the arguments to an MCO, e.g., the observationsor model p, when the default arguments are clear from context. Whether we can compute stochastic gradients of N efficiently depends on the specific form of the estimator and the underlying random variables that define it.Many likelihood estimators p̂_N() converge to p() almost surely as N →∞ (known as strong consistency). The advantage of a consistent estimator is that its MCO can be driven towards log p() by increasing N. We present sufficient conditions for this convergence and a description of the rate:Properties of Monte Carlo Objectives.Let N(, p) be a Monte Carlo objective defined by an unbiased positive estimator p̂_N() of p(). Then, *(Bound) N(, p) ≤log p().*(Consistency) If logp̂_N() is uniformly integrable (see Appendix for definition) and p̂_N() is strongly consistent, then N(, p) →log p() as N →∞.*(Asymptotic Bias) Let g(N) = [(p̂_N() - p())^6] be the 6th central moment. If the 1st inverse moment is bounded, lim sup_N →∞[p̂_N()^-1]<∞, thenlog p() - N(, p) = 1/2(p̂_N()/p()) + 𝒪(√(g(N))). See the Appendix for the proof and a sufficient condition for controlling the first inverse moment when p̂_N() is the average of i.i.d. random variables. In some cases, convergence of the bound to log p() is monotonic, e.g., IWAE <cit.>, but this is not true in general. The relative variance of estimators, (p̂_N() / p()), tends to be well studied, so property <ref> gives us a tool for comparing the convergence rate of distinct MCOs. For example, <cit.> study marginal likelihood estimators defined by particle filters and find that the relative variance of these estimators scales favorably in comparison to naive importance sampling. This suggests that a particle filter's MCO, introduced in the next section, will generally be a tighter bound than IWAE.§ FILTERING VARIATIONAL OBJECTIVES (FIVOS)The filtering variational objectives (FIVOs) are a family of MCOs defined by the marginal likelihood estimator of a particle filter. For models with sequential structure, e.g., latent variable models of audio and text, the relative variance of a naive importance sampling estimator tends to scale exponentially in the number of steps. In contrast, the relative variance of particle filter estimators can scale more favorably with the number of steps—linearly in some cases <cit.>. Thus, the results of Section <ref> suggest that FIVOs can serve as tighter objectives than IWAE for MLE in sequential models.Let our observations be sequences of T -valued random variables denoted , where _i:j≡ (_i, …, _j). We also assume that the data generation process relies on a sequence of T unobserved -valued latent variables denoted . We focus on sequential latent variable models that factor as a series of tractable conditionals, p(, ) = p_1(_1, _1) ∏_t=2^T p_t(_t, _t | t-1, t-1).A particle filter is a sequential Monte Carlo algorithm, which propagates a population of N weighted particles for T steps using a combination of importance sampling and resampling steps, see Alg. <ref>. In detail, the particle filter takes as arguments an observation , the number of particles N, the model distribution p, and a variational posterior q( | ) factored over t, q( | ) = ∏_t=1^T q_t(_t | t, t-1) .The particle filter maintains a population{w_t-1^i, t-1^i}_i=1^N of particles t-1^i with weights w_t-1^i. At step t, the filter independently proposes an extension _t^i ∼ q_t(_t | t,t-1^i) to each particle's trajectory t-1^i. The weights w_t-1^i are multiplied by the incremental importance weights,α_t(t^i) = p_t(_t, _t^i | t-1, t-1^i)/q_t(_t^i | t, t-1^i),and renormalized. If the current weights w_t^i satisfy a resampling criteria, then a resampling step is performed and N particles t^i are sampled in proportion to their weights from the current population with replacement. Common resampling schemes include resampling at every step and resampling if the effective sample size (ESS) of the population (∑_i=1^N (w_t^i)^2)^-1 drops below N/2 <cit.>. After resampling the weights are reset to 1. Otherwise, the particlest^i are copied to the next step along with the accumulated weights. See Fig. <ref> for a visualization.Instead of viewing Alg. <ref> as an inference algorithm, we treat the quantity [logp̂_N()] as an objective function over p. Because p̂_N() is an unbiased estimator of p(x_1:T), proven in the Appendix and in <cit.>, it defines an MCO, which we call FIVO:Filtering Variational Objectives. Let logp̂_N() be the output of Alg. <ref> with inputs (x_1:T, p, q, N), then N(, p, q) = [logp̂_N()] is a filtering variational objective.p̂_N() is a strongly consistent estimator <cit.>. So if logp̂_N() is uniformly integrable, then N(, p, q) →log p() as N →∞. Resampling is the distinguishing feature of N; if resampling is removed, then FIVO reduces to IWAE. Resampling does add an amount of immediate variance, but it allows the filter to discard low weight particles with high probability. This has the effect of refocusing the distribution of particles to regions of higher mass under the posterior, and in some sequential models can reduce the variance from exponential to linear in the number of time steps <cit.>. Resampling is a greedy process, and it is possible that a particle discarded at step t, could have attained a high mass at step T. In practice, the best trade-off is to use adaptive resampling schemes <cit.>. If for a given , p, q a particle filter's likelihood estimator improves over simple importance sampling in terms of variance, we expect N to be a tighter bound thanorN.§.§ OptimizationThe FIVO bound can be optimized with the same stochastic gradient ascent framework used for the ELBO. We found in practice it was effective simply to follow a Monte Carlo estimator of the biased gradient [∇_(θ,ϕ)logp̂_N()] with reparameterized _t^i. This gradient estimator is biased, as the full FIVO gradient has three kinds of terms: it has the term [∇_θ,ϕlogp̂_N()], where ∇_θ,ϕlogp̂_N() is defined conditional on the random variables of Alg. <ref>; it has gradient terms for every distribution of Alg. <ref> that depends on the parameters; and, if adaptive resampling is used, then it has additional terms that account for the change in FIVO with respect to the decision to resample. In this section, we derive the FIVO gradient when _t^i are reparameterized and a fixed resampling schedule is followed. We derive the full gradient in the Appendix.In more detail, we assume that p and q are parameterized in a differentiable way by θ and ϕ. Assume that q is from a reparameterizable family and that _t^i of Alg. <ref> are reparameterized. Assume that we use a fixed resampling schedule, and let resampling at stept be an indicator function indicating whether a resampling occured at step t. Now, N depends on the parameters via logp̂_N() and the resampling probabilities w_t^i in the density. Thus, ∇_(θ,ϕ)N =[∇_(θ,ϕ)logp̂_N() + ∑_t=1^T∑_i=1^Nresampling at steptlogp̂_N()/p̂_N(t)∇_(θ, ϕ)log w_t^i]Given a single forward pass of Alg. <ref> with reparameterized _t^i, the terms inside the expectation form a Monte Carlo estimator of Eq. (<ref>). However, the terms from resampling events contribute to the majority of the variance of the estimator. Thus, the gradient estimator that we found most effective in practice consists only of the gradient ∇_(θ, ϕ)logp̂_N(), the solid red arrows of Figure <ref>. We explore this experimentally in Section <ref>.§.§ Sharpness As with the ELBO, FIVO is a variational objective taking a variational posterior q as an argument. An important question is whether FIVO achieves the marginal log-likelihood at its optimal q. We can only guarantee this for models in which t-1 and _t are independent given t-1.Sharpness of Filtering Variational Objectives.Let N(, p, q) be a FIVO, and q^*(, p) = _q N(, p, q). If p has independence structure such that p(t-1 | t) = p(t-1 | t-1) for t∈{2, …, T}, thenq^*(, p)() = p(|)andN(, p, q^*(, p)) = log p() .See Appendix.Most models do not satisfy this assumption, and deriving the optimal q in general is complicated by the resampling dynamics. For the restricted the model class in Proposition <ref>, the optimal q_t does not condition on future observations x_t+1:T. We explored this experimentally with richer models in Section <ref>, and found that allowing q_t to condition on x_t+1:T does not reliably improve FIVO. This is consistent with the view of resampling as a greedy process that responds to each intermediate distribution as if it were the final. Still, we found that the impact of this effect was outweighed by the advantage of optimizing a tighter bound.§ RELATED WORK The marginal log-likelihood is a central quantity in statistics and probability, and there has long been an interest in bounding it <cit.>. The literature relating to the bounds we call Monte Carlo objectives has typically focused on the problem of estimating the marginal likelihood itself. <cit.> use Jensen's inequality in a forward and reverse estimator to detect the failure of inference methods. IWAE <cit.> is a clear influence on this work, and FIVO can be seen as an extension of this bound. The ELBO enjoys a long history <cit.> and there have been efforts to improve the ELBO itself. <cit.> generalize the ELBO by considering arbitrary operators of the model and variational posterior. More closely related to this work is a body of work improving the ELBO by increasing the expressiveness of thevariational posterior. For example, <cit.> augment the variational posterior with deterministic transformations with fixed Jacobians, and <cit.> extend the variational posterior to admit a Markov chain. Other approaches to learning in neural latent variable models include <cit.>, who use importance sampling to approximate gradients under the posterior, and <cit.>, who use sequential Monte Carlo to approximate gradients under the posterior. These are distinct from our contribution in the sense that for them inference for the sake of estimation is the ultimate goal. To our knowledge the idea of treating the output of inference as an objective in and of itself, while not completely novel, has not been fully appreciated in the literature. Although, this idea shares inspiration with methods that optimize the convergence of Markov chains <cit.>. We note that the idea to optimize the log estimator of a particle filter was independently and concurrently considered in <cit.>. In <cit.> the bound we call FIVO is cast as a tractable lower bound on the ELBO defined by the particle filter's non-parameteric approximation to the posterior. <cit.> additionally derive an expression for FIVO's bias as the KL between the filter's distribution and a certain target process. Our work is distinguished by our study of the convergence of MCOs in N, which includes FIVO, our investigation of FIVO sharpness, and our experimental results on stochastic RNNs.§ EXPERIMENTSIn our experiments, we sought to: (a) compare models trained with ELBO, IWAE, and FIVO bounds in terms of final test log-likelihoods, (b) explore the effect of the resampling gradient terms on FIVO, (c) investigate how the lack of sharpness affects FIVO, and (d) consider how models trained with FIVO use the stochastic state. To explore these questions, we trained variational recurrent neural networks (VRNN)<cit.> with the ELBO, IWAE, and FIVO bounds using TensorFlow <cit.> on two benchmark sequential modeling tasks: natural speech waveforms and polyphonic music. These datasets are known to be difficult to model without stochastic latent states <cit.>.The VRNN is a sequential latent variable model that combines a deterministic recurrent neural network (RNN) with stochastic latent states _t at each step. The observation distribution over _t is conditioned directly on _t and indirectly on t-1 via the RNN's state h_t(_t-1, _t-1, h_t-1). For a length T sequence, the model's posterior factors into the conditionals ∏_t=1^T p_t(_t | h_t(_t-1, _t-1,h_t-1))g_t(_t | _t, h_t(_t-1, _t-1,h_t-1)), and the variational posterior factors as ∏_t=1^T q_t(_t | h_t(_t-1, _t-1,h_t-1), _t). All distributions over latent variables are factorized Gaussians, and the output distributions g_t depend on the dataset. The RNN is a single-layer LSTM and the conditionals are parameterized by fully connected neural networks with one hidden layer of the same size as the LSTM hidden layer. We used the residual parameterization <cit.> for the variational posterior. For FIVO we resampled when the ESS of the particles dropped below N/2. For FIVO and IWAE we used a batch size of 4, and for the ELBO, we used batch sizes of 4N to match computational budgets (resampling is 𝒪(N) with the alias method). For all models we report bounds using the variational posterior trained jointly with the model. For models trained with FIVO we report 128. To provide strong baselines, we report the maximum across bounds, max{, 128, 128}, for models trained with ELBO and IWAE. Additional details in the Appendix. §.§ Polyphonic Music We evaluated VRNNs trained with the ELBO, IWAE, and FIVO bounds on 4 polyphonic music datasets: the Nottingham folk tunes, the JSB chorales, the MuseData library of classical piano and orchestral music, and the Piano-midi.de MIDI archive <cit.>. Each dataset is split into standard train, valid, and test sets and is represented as a sequence of 88-dimensional binary vectors denoting the notes active at the current timestep. We mean-centered the input data and modeled the output as a set of 88 factorized Bernoulli variables. We used 64 units for the RNN hidden state and latent state size for all polyphonic music models except for JSB chorales models, which used 32 units. We report bounds on average log-likelihood per timestep in Table <ref>. Models trained with the FIVO bound significantly outperformed models trained with either the ELBO or the IWAE bounds on all four datasets. In some cases, the improvements exceeded 1 nat per timestep, and in all cases optimizing FIVO with N=4 outperformed optimizing IWAE or ELBO for N={4,8,16}. §.§ Speech The TIMIT dataset is a standard benchmark for sequential models that contains 6300 utterances with an average duration of 3.1 seconds spoken by 630 different speakers. The 6300 utterances are divided into a training set of size 4620 and a test set of size 1680. We further divided the training set into a validation set of size 231 and a training set of size 4389, with the splits exactly as in <cit.>. Each TIMIT utterance is represented as a sequence of real-valued amplitudes which wesplit into a sequence of 200-dimensional frames, as in <cit.>. Data preprocessing was limited to mean centering and variance normalization as in <cit.>. For TIMIT, the output distribution was a factorized Gaussian, and we report the average log-likelihood bound per sequence relative to models trained with ELBO. Again, models trained with FIVO significantly outperformed models trained with IWAE or ELBO, see Table <ref>. §.§ Resampling Gradients All models in this work (except those in this section) were trained with gradients that did not include the term in Eq. (<ref>) that comes fromresampling steps. We omitted this term because it has an outsized effect on gradient variance, often increasing it by 6 orders of magnitude. To explore the effects of this term experimentally, we trained VRNNs with and without the resampling gradient term on the TIMIT and polyphonic music datasets. When using the resampling term, we attempted to control its variance using a moving-average baseline linear in the number of timesteps. For all datasets, models trained without the resampling gradient term outperformed models trained with the term by a large margin on both the training set and held-out data. Many runs with resampling gradients failed to improve beyond random initialization. A representative pair of train log-likelihood curves is shown in Figure <ref> — gradients without the resampling term led to earlier convergence and a better solution. We stress that this is an empirical result — in principle biased gradients can lead to divergent behaviour. We leave exploring strategies to reduce the variance of the unbiased estimator to future work. §.§ Sharpness FIVO does not achieve the marginal log-likelihood at its optimal variational posterior q^*, because the optimal q^* does not condition on future observations (see Section <ref>). In contrast, ELBO and IWAE are sharp, and their q^*s depend on future observations. To investigate the effects of this, we defined a smoothing variant of the VRNN in which q takes as additional input the hidden state of a deterministic RNN run backwards over the observations, allowing q to condition on future observations. We trained smoothing VRNNs using ELBO, IWAE, and FIVO, and report evaluation on the training set (to isolate the effect on optimization performance) in Table <ref> . Smoothing helped models trained with IWAE, but not enough to outperform models trained with FIVO. As expected, smoothing did not reliably improve models trained with FIVO. Test set performance was similar, see the Appendix for details. §.§ Use of Stochastic StateA known pathology when training stochastic latent variable models with the ELBO is that stochastic states can go unused. Empirically, this is associated with the collapse of variational posterior q( | ) network to the model prior p() <cit.>. To investigate this, we plot the KL divergence from q(T | T) to p(T) averaged over the dataset (Figure <ref>). Indeed, the KL of models trained with ELBO collapsed during training, whereas the KL of models trained with FIVO remained high, even while achieving a higher log-likelihood bound.§ CONCLUSIONSWe introduced the family of filtering variational objectives, a class of lower bounds on the log marginal likelihood that extend the evidence lower bound. FIVOs are suited for MLE in neural latent variable models. We trained models with the ELBO, IWAE, and FIVO bounds and found that the models trained with FIVO significantly outperformed other models across four polyphonic music modeling tasks and a speech waveform modeling task. Future work will include exploring control variates for the resampling gradients, FIVOs defined by more sophisticated filtering algorithms, and new MCOs based on differentiable operators like leapfrog operators with deterministically annealed temperatures. In general, we hope that this paper inspires the machine learning community to take a fresh look at the literature of marginal likelihood estimators—seeing them as objectives instead of algorithms for inference. §.§.§ AcknowledgmentsWe thank Matt Hoffman, Matt Johnson, Danilo J. Rezende, Jascha Sohl-Dickstein, and Theophane Weber for helpful discussions and support in this project. A. Doucet was partially supported by the EPSRC grant EP/K000276/1.Y. W. Teh's research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 617071. unsrt§ APPENDIX TO FILTERING VARIATIONAL OBJECTIVES §.§ Other Examples of MCOs.There is an extensive literature on marginal likelihood estimators <cit.>. Each defines an MCO, and we consider two in more detail, annealed importance sampling <cit.> and multiple importance sampling <cit.>. Letdenote an observation of an 𝒳-valued random variable generated in a process with an unobserved 𝒵-valued random variable . Let p(, ) be the joint density. Annealed Importance Sampling MCO. Annealed importance sampling (AIS) is a generalization of importance sampling <cit.>. We present an MCO derived from a special case of the AIS algorithm. Let q( | ) be a variational posterior distribution and let β_i be a sequence of real numbers for i ∈{1, …, N+1} such that 0 ≤β_i ≤ 1 and β_1 = 0 and β_N+1 = 1. Let T_i(^' | , ) be a Markov transition distribution whose stationary distribution is proportional to q( | )^1-β_ip(, )^β_i. Then for _1 ∼ q( | ) and _i ∼ T_i(^' | _i-1, ) for i ∈{2, …, N} we have the following unbiased estimator,[p̂_N(x)] = [∏_i=1^N ( p(, _i)/q(_i | ))^β_i+1 - β_i] = p()Notice two things. First, there is no assumption that the states _i are at equilibrium, and second, we did not require a transition operator keeping p(, ) as an invariant distribution. All together, we can define the AIS MCO,N(x, q, {T_i}_i=2^N, p) = [∑_i=1^N (β_i+1 - β_i)logp(, _i)/q(_i | )]This is a sharp objective, if we take q as the true posterior, q( | ) = p( | ), and T_i(^' | , ) = δ(^' - ) to be the Dirac delta copy operator. The difficulty in applying this MCO is finding T_i, which are scalable and easy to optimize. Generalizations of the AIS procedure have been proposed in <cit.>. The resulting Sequential Monte Carlo samplers procedures also provide an unbiased estimator of the marginal likelihood and are structurally identical to the particle algorithm presented in this paper. Multiple Importance Sampling MCO. Multiple importance sampling (MIS) <cit.> is another generalization of importance sampling. Let q_i( | ) be N possibly distinct variational posterior distributions and w_i() ≥ 0 be such that ∑_i=1^N w_i() = 1. There are a variety of distinct estimators that could be formed from the q_i <cit.>. We present just one. Let_i ∼ q_i( | ), then we have the following unbiased estimator[p̂_N(x)] = [∑_i=1^N w_i() p(, _i)/∑_j=1^N w_j() q_j(_i | )] = p()Notice that the latent sample _i ∼ q_i( | ) is evaluated under all q_i's. One can view this as a Rao-Blackwellized estimator corresponding to the mixture distribution ∑_i=1^N w_i() q_i( |). All together,N(x, {q_i}_i=1^N, {w_i}_i=1^N, p) = [log(∑_i=1^Nw_i() p(, _i)/∑_j=1^N w_j() q_j(_i | ))]Again, this objective is sharp, if we take any q_i( | ) = p( | ) and w_i() = 1. The difficulty in making this objective more useful is optimizing it in a way that distinguishes the q_i and assigns the appropriate w_i. §.§ Proof of Proposition 1. Let [p̂_N(x)] = p(x) and define ℒ_N(x,p) = 𝔼[logp̂_N(x)] as the Monte Carlo objective defined by p̂_N(x). * By the concavity of log and Jensen's inequality,ℒ_N(x,p) = 𝔼[logp̂_N(x)] ≤log𝔼[ p̂_N(x)] = log p(x) * Assume * p̂_N(x) is strongly consistent, i.e. p̂_N(x) a.s.⟶ p(x) as N →∞.* logp̂_N(x) is uniformly integrable. That is, let (Ω, ℱ, μ) be the probability space on which logp̂_N(x) is defined. The random variables {logp̂_N(x)}_N=1^∞ are uniformly integrable if [|logp̂_N(x) |] < ∞ and if for any ϵ > 0, there exists δ > 0, such that for all N and E ∈ℱ, μ(E) < δ implies [|logp̂_N(x)| E] < ϵ, where E is an indicator function of the set E.Then by continuity of log, logp̂_N(x) converges almost surely to log p(x). By Vitali's convergence theorem (using the uniform integrability assumption), we get ℒ_N(x,p) = [logp̂_N(x)]→log p(x) as N→∞.* Let g(N) = 𝔼[( p̂_N(x) - p(x))^6], and assume lim sup_N →∞𝔼[( p̂_N(x))^-1] < ∞. Define the relative errorΔ = p̂_N(x) - p(x)/p(x)Then the bias log p(x) - ℒ_N(x,p) = - 𝔼[log(1 + Δ)]. Now, Taylor expand log(1 + Δ) about 0,log(1 + Δ)= Δ - 1/2Δ^2 + ∫_0^Δ(1/1+x - 1 + x)dx= Δ - 1/2Δ^2 + ∫_0^Δ(x^2/1+x)dxand in expectation- 𝔼[log(1 + Δ)]= 1/2Δ^2 - 𝔼[∫_0^Δ(x^2/1+x)dx]Our aim is to show| 𝔼[∫_0^Δx^2/1+x dx]| ∈𝒪(g(N)^1/2)In particular,by Cauchy-Schwarz| 𝔼[∫_0^Δ(x^2/1+x)dx]|≤𝔼[|∫_0^Δ1/(1+x)^2 dx|^1/2|∫_0^Δ x^4dx|^1/2]=𝔼[| Δ/1+ Δ|^1/2|Δ^5/5|^1/2]=𝔼[| 1/1+ Δ|^1/2|Δ^6/5|^1/2] and again by Cauchy-Schwarz ≤(𝔼[| 1/1+ Δ|])^1/2(𝔼[Δ^6/5])^1/2.This concludes the proof.§.§ Controlling the first inverse moment. We provide a sufficient condition that guarantees that the inverse moment of the average of i.i.d. random variables is bounded, a condition used in Proposition <ref> (c). Intuitively, this is a fairly weak condition, because it only requires that the mass in an arbitrarily small neighbourhood of zero is bounded.Let w_i be i.i.d. positive random variables and p̂_N(x) = 1/N∑_i=1^N w_i. If there exist M, C, ϵ > 0 such that (w_i < w) ≤ Cw^1+ϵ for w ∈ [0, M), then [p̂_N(x)^-1] ≤ CM^ϵ/ϵ + 1/M. Let M, C, ϵ > 0 be such that (w_i < w) ≤ Cw^1+ϵ for w ∈ [0, M). We proceed in two cases. If N = 1, then[p̂_N(x)^-1]= ∫_0^∞(w_1^-1 > u)du= ∫_0^∞(w_1 < 1/u)du= ∫_0^M(w_1 < w)/w^2 dw +∫_M^∞(w_1 < w)/w^2 dw≤∫_0^MCw^1+ϵ/w^2 dw +∫_M^∞1/w^2 dw= CM^ϵ/ϵ + 1/MFor N > 1, we show that [p̂_N(x)^-1] ≤[p̂_1(x)^-1], so the same condition is sufficient for any N. The AM-GM inequality tells us that∑_i=1^N w_i/N ≥(∏_i=1^N w_i)^1/Nso[p̂_N(x)^-1]≤[(∏_i=1^N w_i)^-1/N]= ∏_i=1^N [ w_i^-1/N]= [ w_1^-1/N]^N and by Lyapunov's inequality, we have ≤[ (w_1^-1/N)^N] = [p̂_1(x)^-1]This concludes the proof. §.§ Unbiasedness of p̂_N() from the particle filter. We sketch an argument that the random variable p̂_N() defined by Algorithm <ref> is an unbiased estimator of the marginal likelihood p(). This is a well-known fact <cit.>, and our sketch is based on <cit.>. The strategy is to cast the particle filter's estimator p̂_N() as a single importance weight over an extended space. The lack of bias in the particle filter therefore reduces to the unbiasedness of importance sampling. Key to this is identifying the target and proposal distributions in the extended space. The target distribution is called “conditional sequential Monte Carlo”, Algorithm <ref>. The proposal distribution is the particle filter itself, Algorithm <ref>.We argue that it is enough to consider just an arbitrary fixed (non-adaptive) resampling schedule that always resamples at step T. First, consider adaptive resampling criteria, i.e. criteria that are deterministic functions of the weights w_t^i. For such criteria the joint density of random variables in Algorithm <ref> will be piecewise continuous, composed of 2^T regions corresponding to a sequence of resample/no-resample decisions. This density has a form on each piece that is exactly the same as the density for some fixed resampling schedule. Moreover, it is globally normalized, because of the sequential structure of the filter. Because Algorithm <ref> makes the same decisions, it also is partitioned along the same sets and each piece has the same fixed resampling schedule. Thus, it is enough to consider only a fixed resampling schedule. Second, notice that in the final step, step T, of Algorithms <ref> and <ref> resampling has no effect on p̂_N(). Thus, we assume that the resampling criteria of Algorithms <ref> and <ref> at step T is set to always resample.All together it is safe to assume a fixed resampling schedule with R resampling events, 1 ≤ R ≤ T, at steps k_r ∈{1, …, T} for r ∈{0, …, R} with k_R = T and k_0 = 0. Now we derive the joint density of Algorithm <ref> and <ref> taken at each iteration after possibly resampling. To avoid notational clutter we let g, f (omitting their arguments) represent the densities of the variables in Algorithms <ref> and <ref>. Technically, we should also be keeping track of the indices that indicate the inheritance of the resampling step. So, let the random variables {{w_t^i, t^i}_i=1^N}_t=1^T be the particles before resampling and s(i) ∈{1, …, N} be the index that is selected for inheritance of the ith particle after resampling. Then the density corresponding to Algorithm <ref> isg = ∏_r=1^R∏_i=1^N w_k_r^s(i)∏_k=k_r-1 + 1^k_r q_k(_k^i | k, k-1^i)For Algorithm <ref>,f = ∏_r=1^R(∏_i≠ jw_k_r^s(i)∏_k=k_r-1 + 1^k_r q_k(_k^i | k, k-1^i)) (1/N∏_k=k_r-1 + 1^k_r p(_k^j | , k-1^j) )These densities are normalized, so _g[f/g] = 1. Thus, our goal is to show p̂_N() = p()f/g.p()∏_r=1^R( ∏_i≠ j w_k_r^s(i)∏_k=k_r-1 + 1^k_r q_k(_k^i | k, k-1^i)) (N^-1∏_k=k_r-1 + 1^k_r p(_k^j | , k-1^j)) /∏_i=1^N w_k_r^s(i)∏_k=k_r-1 + 1^k_r q_k(_k^i | k, k-1^i)=p()∏_r=1^RN^-1∏_k=k_r-1 + 1^k_r p(_k^j | , k-1^j)/w_k_r^j ∏_k=k_r-1 + 1^k_r q_k(_k^j | k, k-1^j) = and pushing in the marginal likelihood ∏_r=1^R N^-1∏_k=k_r-1 + 1^k_r p(_k^j, _k | k-1, k-1^j) /w_k_r^j ∏_k=k_r-1 + 1^k_r q_k(_k^j | k, k-1^j)Now, letting α_k(k^i) =p(_k^i, _k | k-1, k-1^i)/q_k(_k^i | k, k-1^i) one can show that for this sequence of resampling times the weight w_k_r^j telescopes intow_k_r^j = ∏_k=k_r-1 + 1^k_rα_k(k^j)/∑_i=1^N∏_k=k_r-1 + 1^k_rα_k(k^i)and the estimator p̂_N() = ∏_t=1^T p̂_t telescopes intop̂_N() = ∏_r=1^R (1/N∑_i=1^N∏_k=k_r-1 + 1^k_rα_k(k^i))and thusp()f/g = ∏_r=1^R N^-1∏_k=k_r-1 + 1^k_r p(_k^j, _k | k-1, k-1^j)/w_k_r^j ∏_k=k_r-1 + 1^k_r q_k(_k^j | k, k-1^j)= ∏_r=1^R N^-1/w_k_r^j∏_k=k_r-1 + 1^k_rα_k(k^j) = ∏_r=1^R 1/N(∑_i=1^N∏_k=k_r-1 + 1^k_rα_k(k^i)) = p̂_N().The result follows. An intuitive way to understand this result is the following: Algorithm <ref> matches the distribution of every random variable in the particle filter except it interleaves a true posterior sample into the set of particles with uniform probability. The only mismatch in the densities are the normalization terms of the resampling probabilities of that privileged posterior sample, with terms ∑_i=1^N∏_k=k_r-1 + 1^k_rα_k(k^i) coming from the filter's resampling and terms N from conditional SMC's resampling. Of course, we never run Algorithm <ref>, it just serves to define the target density. §.§ Gradients of N(, p, q).We formulate unbiased gradients of N(, p, q) by considering Algorithm <ref> as a method for simulating FIVO. We consider the cases when the sampling of _t^i is and is not reparameterized. We also consider the case where we make adaptive resampling decisions.First, we assume that the decision to resample is not adaptive (i.e., depends in some way on the random variables already produced until that point in Algorithm <ref>), and are fixed ahead of time. When the sampling _t^i is not reparameterized there are three terms to the gradient: (1) the gradients of logp̂_N() with respect to the parameters conditional on the latent states, (2) gradients of the densities q_t with respect to their parameters, and (3) gradients of the resampling probabilities with respect to the parameters. All together, the following is a gradient of FIVO,[∇_θ,ϕlogp̂_N() + ∑_t=1^T ∑_i=1^N(logp̂_N()/p̂_N(t-1)∇_ϕlog q_t,ϕ(_t^i|t, _1:t-1^i)+resampling at steptlogp̂_N()/p̂_N(t)∇_θ, ϕlog w_t^i)]where A is an indicator function. If _t^i is reparameterized, then the first and third terms suffice for an unbiased gradient,[∇_θ,ϕlogp̂_N() + ∑_t=1^T∑_i=1^Nresampling at steptlogp̂_N()/p̂_N(t)∇_θ, ϕlog w_t^i]In this work we only considered reparameterized q_ts, and we dropped the terms of the gradient that arise from resampling.Second, when the decision to resample is adaptive, the domain of the random variables involved in simulating logp̂_N() can be partitioned into 2^T regions, over each of which the density is differentiable. Between those regions, the density experiences a jump discontinuity. Thus, there are additional terms to the gradient of N(, p, q) that correspond to the change in the regions of continuity as the parameters change. These terms can be written as surface integrals over the boundaries of the regions. We drop these terms in practice. §.§ Proof of Proposition 2. Assume p(t-1 | t) = p(t-1 | t-1) for all t ∈{2, …, T}. We will show N(, p, q) = log p() at q(_t | t-1, t) = p(_t | t-1, t). We will do this by induction, showing that every particle has a constant weight and that p̂_N() = p() is a constant. For t=1 we haveα_1^i(_1) = p_1(_1, _1)/p(_1 | _1) = p_1(_1)Thus, all particles have the same weight and p̂_1 = p_1(x_1). Now for any t we have that the weights must be 1/N since the particles all have the same weight andα_t^i(t)= p_t(_t, _t | t-1, t-1)/p(_t | t-1, t)= p(t, t)/p( t-1, t-1)p(_t | t-1, t)= p(t)/p(t-1)p(t | t)/p( t-1 | t-1)p(_t | t-1, t)= p(t)/p(t-1)p(t | t)/p( t-1 | t)p(_t | t-1, t)= p(t)/p(t-1)and thus,p̂_N() =p_1(x_1) ∏_t=2^Tp(t)/p(t-1)= p(x_1:T)§.§ Implementation detailsWe initialized weights using the Xavier initialization<cit.> and used the Adam optimizer <cit.> with a batch size of 4.During training, we did not truncate sequences and performed full backpropagation through time for all datasets. For the results presented in Sections <ref> and <ref> we performed a grid search over learning rates {3 × 10^-4,1× 10^-4, 3 × 10^-5, 1 × 10^-5} and picked the run and early stopping step by the validation performance. §.§ Evaluation and Comparison of BoundsComparing models trained with different log-likelihood lower bounds is challenging because calculating the actual log-likelihood is intractable. Burda et al. <cit.> showed that the IWAE bound is at least as tight as the ELBO and monotonically increases with N. This suggests comparing models based on the IWAE bound evaluated with a large N. However, we found that IWAE and ELBO bounds tended to diverge for models trained with FIVO. Although FIVO is not provably a tighter bound than the ELBO or IWAE, our experiments suggest that this tends to be the case in practice. In Figure <ref>, we plotted all three bounds over training for a representative experiment. All plots use the same model architecture, but the training objective changes in each panel. For the model trained with IWAE, the FIVO and IWAE bounds are tighter than their counterparts on the model trained with ELBO, suggesting that the model trained with IWAE is superior. The ELBO bound evaluated on the model trained with IWAE, however, is lower than its counterpart on the model trained with the ELBO. For the model trained with FIVO, both IWAE and ELBO bounds seem to diverge, but the FIVO bound outperforms the FIVO bounds on both of the other models. As in the figure, we generally found that the same model evaluated with FIVO, IWAE, and ELBO produced values descending in that order. We suspect that q distributions trained under the FIVO bound are more entropic than those trained under ELBO or IWAE because of the resampling operation. During training under FIVO, q is able to propose state transitions that could poorly explain the observations because the bad states will be resampled away without harming the final bound value. Then, when a FIVO-trained q is evaluated with ELBO or IWAE it proposes poor states that are not resampled away, leading to a poor final bound value. Conversely, qs trained with ELBO and IWAE are not able to fully leverage the resampling operation when evaluated with the FIVO bound.Because of this behavior, we chose to optimistically evaluate models trained with IWAE and ELBO by reporting the maximum across all the bounds. For models trained with FIVO, we reported only the FIVO bound. We felt this evaluation scheme provided the strongest comparison to existing bounds.§.§ Evaluating TIMIT Log-Likelihoods We reported log-likelihood scores for TIMIT relative to an ELBO baseline instead of raw log-likelihoods. Previous papers (e.g., <cit.>) report the log-likelihood of data that have been mean centered and variance normalized, but it would be more proper to report the results on the un-standardized data. Specifically, if the training set has mean μ and variance σ^2 and the model outputs μ̂ and σ̂^2, then the un-standardized test data would be evaluated under a 𝒩(μ̂σ + μ, σ̂^2σ^2) distribution.Log-likelihoods produced by these approaches differ by a constant offset that depends on σ. Because the offset is a function of only training set statistics, it does not affect relative comparison between methods. Because of this we chose to report log-likelihoods relative to a baseline instead of absolute numbers. Absolute numbers calculated on standardized data are reported in Tables 3, 4, and 5 to allow for comparisons with other papers. | http://arxiv.org/abs/1705.09279v3 | {
"authors": [
"Chris J. Maddison",
"Dieterich Lawson",
"George Tucker",
"Nicolas Heess",
"Mohammad Norouzi",
"Andriy Mnih",
"Arnaud Doucet",
"Yee Whye Teh"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.NE",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170525175241",
"title": "Filtering Variational Objectives"
} |
Thermal light curves of Haumea, 2003 VS_2 and 2003 AZ_84 with Herschel Space Observatory-PACSHerschel is an ESA space observatory with science instruments provided by European–led Principal Investigator consortia and with important participation from NASA. PACS: The Photodetector Array Camera and Spectrometer is one of Herschel's instruments. P. Santos-Sanz: [email protected] de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, 18008-Granada, Spain. [email protected] de Paris, CNRS, UPMC Univ. Paris 6, Univ. Paris-Diderot, France. Aix Marseille Université, CNRS, LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326, 13388, Marseille, France. Astrophysics Research Centre, Queen's University Belfast, Belfast BT7 1NN, United Kingdom. Max–Planck–Institut für extraterrestrische Physik (MPE), Garching, Germany. Konkoly Observatory of the Hungarian Academy of Sciences, Budapest, Hungary. Max–Planck-Institut für Sonnensystemforschung (MPS), Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germany. Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA. Univ. Paris Diderot, Sorbonne Paris Cité, 4 rue Elsa Morante, 75205 Paris, France. Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, Arizona, 86001, USA.Time series observations of the dwarf planet Haumea and the Plutinos 2003 VS_2 and 2003 AZ_84 with Herschel/PACS are presented in this work. Thermal emission of these trans-Neptunian objects (TNOs) were acquired as part of the TNOs are Cool Herschel Space Observatory key programme.We search for the thermal light curves at 100 and 160 μm of Haumea and 2003 AZ_84, and at 70 and 160 μm for 2003 VS_2 by means of photometric analysis of the PACS data. The goal of this work is to use these thermal light curves to obtain physical and thermophysical properties of these icy Solar System bodies.When a thermal light curve is detected, it is possible to derive or constrain the object thermal inertia, phase integral and/or surface roughness with thermophysical modeling. Haumea's thermal light curve is clearly detected at 100 and 160 μm. The effect of the reported dark spot is apparent at 100 μm. Different thermophysical models were applied to these light curves, varying the thermophysical properties of the surface within and outside the spot. Although no model gives a perfect fit to the thermal observations, results implyan extremely low thermal inertia (< 0.5 J m^-2 s^-1/2 K^-1, hereafter MKS) and a high phase integral (>0.73) for Haumea's surface. We note that the dark spot region appears to be only weakly different from the rest of the object, with modest changes in thermal inertia and/or phase integral. The thermal light curve of 2003 VS_2 is not firmly detected at 70 μm and at 160 μm but a thermal inertia of (2±0.5) MKS can be derived from these data. The thermal light curve of 2003 AZ_84 is not firmly detected at 100 μm. We apply a thermophysical model to the mean thermal fluxes and to all the Herschel/PACS and Spitzer/MIPS thermal data of 2003 AZ_84, obtaining a close to pole-on orientation as the most likely for this TNO.For the three TNOs, the thermal inertias derived from light curve analyses or from the thermophysical analysis of the mean thermal fluxes confirm the generally small or very small surface thermal inertias of the TNO population, which is consistent with a statistical mean value Γ _mean = 2.5 ± 0.5 MKS.PACS thermal light curves of Haumea, 2003 AZ_84 and 2003 VS_2 TNOs are Cool: a survey of the Transneptunian Region XII P. Santos-Sanz1E. Lellouch2O. Groussin3P. Lacerda4T.G. Müller5J.L. Ortiz1C. Kiss6 E. Vilenius5,7J. Stansberry8R. Duffard1S. Fornasier2,9L. Jorda3A. Thirouin10 Received/ Accepted======================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONThe study of the visible photometric variation ofSolar System minor bodies enables us to determine optical light curves (flux or magnitude versus time), for which essential parameters are the peak-to-peak amplitude and the rotational period of the object. Short-term photometric variability of TNOs and Centaurs can be shape-driven <cit.> or be causes by albedo contrasts on the surface of a Maclaurin-shaped rotating spheroid (e.g. the Pluto case). Combinations of shape and albedo effects are also possible and very likely occur within TNOs (e.g. Makemake) and Centaurs. Contact-binaries can also produce short-term photometric variability within the TNO and Centaur populations. The largest amplitudes are usually associated with Jacobi shapes and the smallest ones with Maclaurin shapes with highly variegated surfaces. Large amplitudes can also be associated with contact-binaries and small amplitudes with objects with rotational axes close to pole-on. If we know the rotational properties (i.e. rotation period and amplitude) of a Jacobi shaped object, it is possible to derive the axes ratio of the ellipsoid (i.e. a shape model) and also a lower limit for the density <cit.>, assuming the object is in hydrostatic equilibrium <cit.> with a certain aspect angle. On the other hand, if we suspect that the object has a Maclaurin shape we can derive a shape model from the rotational period, but it is needed to assume a realistic density in this case. The majority of the TNOs/Centaurs (∼ 70%) present shallow light curves (amplitudes less than 0.15 magnitudes), which indicates that most of them are Maclaurin-shaped bodies <cit.>. For a couple of special cases (only for Centaurs until now) where more information is available (i.e. long-term changes in light curve amplitudes), the position of the rotational axis can be derived or at least constrained <cit.>.Complementary to optical light curves, thermal light curves are a powerful tool to obtain additional information about physical and, in particular, thermal properties of these bodies. At first order, immediate comparison of the thermal and optical light curves enables us to differentiate between shape-driven light curves (the thermal light curve is then correlated with the optical one) and light curves that are the result ofalbedo markings (the two light curves are anti-correlated). Furthermore, quantitative modeling of the thermal light curve enables us to constrain the surface energetic and thermal properties, namely its bolometric albedo, thermal inertia, and surface roughness <cit.>. In a more intuitive way, in the case of positively correlated thermal and optical light curves, it is also possible to constrain the thermal inertia using the delay between the thermal and optical light curves and their relative amplitudes.Untilrecently, only a few thermal light curves of outer Solar System minor bodies (or dwarf planets) have been obtained. Pluto's thermal light curve was detected with ISO/PHOT, Spitzer/MIPS, and /IRC at a variety of wavelengths longwards of 20 μm <cit.>. More recently, Pluto was also observed with the Herschel Space Observatory <cit.> Photodetector Array Camera and Spectrometer <cit.> and with the Spectral and Photometric Imaging Receiver <cit.>, which provides thermal light curves at six wavelengths: 70, 100, 160, 250, 350, and 500 μm <cit.>.A marginal thermal light curve of dwarf planet Haumea was reported with Spitzer-MIPS <cit.>. The definite detection of Haumea's thermal light curve was obtained with Herschel/PACS within the Science Demonstration Phase <cit.>. Other tentative thermal light curves of TNOs/Centaurs that were observed with Herschel/PACS (i.e. Varuna, Quaoar, Chiron, Eris) were also presented for the first time <cit.> and will be published separately <cit.>.Here, we present thermal time series photometry of the dwarf planet Haumea and the Plutinos 2003 VS_2 and 2003 AZ_84 that were taken with Herschel/PACS using its 3-filter bands, which are centred at 70, 100, and 160 μm (hereafter blue, green, and red bands, respectively). In the case of Haumea, we present additional and improved data and we merge them with the SDP observations that were originally presented in <cit.>. The thermal time series of 2003 VS_2 and 2003 AZ_84 are presented here for the first time. The Herschel/PACS observations are described in Sect. <ref>, the data reduction and photometry techniques applied are detailed in Sect. <ref>. Data are analyzed, modeled, and interpreted in Sect. <ref>. Finally, the major conclusions of this work are summarized and discussed in Sect. <ref>. § OBSERVATIONSThe observations presented here are part of the project TNOs are Cool: a survey of the trans-Neptunian region, a Herschel Space Observatory open time key programme <cit.>. This programmeused ∼ 372 hours of Herschel time (plus ∼ 30 hours within the SDP) to observe 130 TNOs/Centaurs, plus two giant planet satellites (Phoebe and Sycorax), with Herschel/PACS; 11 of these objects were also observed with Herschel/SPIRE at 250, 350, and 500 μm <cit.>, with the main goal of obtaining sizes, albedos, and thermophysical properties of a large set of objects that are representative of the different dynamical populations within the TNOs. For PACS measurements, we used a range of observation durations from about 40 to 230 minutes based on flux estimates for each object. Results of the PACS and SPIRE measurements to date have been published in <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>[All the TNOs are Cool results (and additional information) are collected in the public web page:<http://public-tnosarecool.lesia.obspm.fr>]. In addition, four bright objects (Haumea, 2003 VS_2, 2003 AZ_84 and Varuna) were re-observed long enough to search for thermal emission variability related to rotation (i.e. thermal light curve). This paper presentsresults for the first three. Other objects (Pluto, Eris, Quaoar, Chiron)were also observedoutside of the TNOs are Cool programme to search for their thermal light curve. The general strategy we used to detect a thermal light curve with Herschel/PACS was to perform a long observation covering most of the expected light curve duration, followed by a shorter follow-on observation, which enabled us toclean the images' backgrounds, as explained in Sect <ref>. Dwarf planet (136108) Haumea was observed twice with Herschel/PACS in mini scan maps mode at 100 and 160 μm. The first visit was performed on 23 December 2009, which covers 86% of its 3.92 h rotational period followed by a shorter follow-on observation on 25 December 2009. The second one was obtained on 20 June 2010 (follow-on observations on 21 June 2010) using the same detector and bands and covering 110% of its rotational period.The Plutino (84922) 2003 VS_2 was observed with Herschel/PACS in mini scan-maps mode at 70 and 160 μm on 10 August 2010, covering 106% of its 7.42 hr rotational period. Follow-on observations were performed on 11 August 2010.The binary Plutino (208996) 2003 AZ_84 was observed with Herschel/PACS in mini scan maps mode at 100 and 160 μm on 26-27 September 2010 (with follow-on observations on 28 September 2010). The observation lasted ∼7.4 h, i.e 110% of an assumed single-peak rotational period of 6.79 h (or 55 % of a double-peaked period of 13.58 h).All observations were made using only one scanning direction(see e.g. for a detailed description of the mini scan maps mode in the case of TNOs[The observing mode itself is described in the technical note PACS Photometer-Point/Compact Source Observations: Mini Scan-Maps & Chop-Nod, 2010, PICC-ME-TN-036, custodian T. Muller]).Table <ref> shows the orbital parameters, B-R colors, absolute magnitudes, rotational properties, taxonomy, and dynamical classification of the observed objects. This table also includes the radiometric solutions of these three objects (equivalent diameter for an equal-area sphere, geometric albedo, beaming factor) previously published as part of the TNOs are Cool project. Table <ref> shows the observational circumstances of each one of these TNOs. Owing to the spatial resolution of Herschel/PACS, the satellites of the binaries/multiple systems (i.e. Haumea and 2003 AZ_84) are not resolved, and their thermal fluxes are merged with the thermal flux of the main body.§ DATA REDUCTION AND PHOTOMETRY §.§ Data reductionPACS images obtained from the Herschel Space Observatory were processed using the Herschel Interactive Processing Environment (HIPE[HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia members, see: http://herschel.esac.esa.int/DpHipeContributors.shtml]) and our own adapted pipelines developed within the TNOs are Cool project. The application of the pipeline provides individual or single maps, each one of these single images covering ∼4.7 minutes. The re-sampled pixel scale of the single maps is 1.1/pixel, 1.4/pixel, and 2.1/pixel for the 70 μ m (blue), 100 μ m (green), and 160 μ m (red) bands, respectively. Apparent motion over the duration of an individual map is negligible compared to the PACS PSF (FWHM in radius is 5.2/7.7/12 in blue/green/red bands, respectively) and does not need to be corrected.These single maps are combined afterwards, using ephemeris-based recentering processes within HIPE, to obtain enough signal-to-noise ratio (S/N) to perform a good photometry, while keeping enough time resolution to resolve the thermal light curve, in a similar way to <cit.>. The exact number of individual and combined maps for each target is detailed in Sect. <ref>.To minimize at best contamination by background sources, all light curve data are associated with complementary observations acquired one or few days later (follow-on observations), where the target has moved enough that a background map can be determined and subtracted from each combined map of the light curve. The method was demonstrated in Spitzer/MIPS TNOs/Centaurs observations <cit.>. However, this technique to remove background sources fails when trying to remove some background features in the 2003 AZ_84 images. In this case another technique, known as double-differential background subtraction, is applied. A complete and detailed description of the data reduction process, the background subtraction and the double-differential techniques applied to the Herschel/PACS images can be found in <cit.> and in <cit.>. Figures <ref>, <ref>, and <ref> illustrate the advantages of these background-removing techniques for the three targets respectively.§.§ PhotometryAs indicated above, single maps obtained with a time resolution of 4.7 min were combined to improve image quality for photometry. Typically, the number of single images to be combined was larger at 160 μm than at 70/100 μm, owing to lower S/N and larger sky residuals. The details can be found below:Haumea data were taken in two epochs, each time using the green (100 μm)/red (160 μm) filter combination. For the first (resp. second) epoch, the total number of single images is 40 (resp. 55) per filter. These images were grouped by 4 (18.8 minutes time resolution) in the green and by 6 (28.2 minutes per data point) in the red.The choice of this particular grouping of single images for the green (by 4) and the red (by 6) for theHaumea data is based on a compromise between obtaining enough S/N to extract a reliable photometry and having enough timeresolution to properly sample the light curve, as mentioned above. Usually, more single images must be grouped for the red bandbecause those images are normally noisier than images at other shorter wavelengths (even after the application of background-removing techniques).The different grouping elections for 2003 VS_2 and 2003 AZ_84 are based on the same described compromise between S/N and time resolution.After removing clear outliers in the Haumea data, 37 images remain for the first epoch (resp. 53 for the second epoch) at green band, and 35images for the first epoch (resp. 50 ) at red band (see Table <ref> in Appendix <ref>). 2003 VS_2 was observed in blue/red combination, with 96 single maps for each color. We grouped these single images by 5 for the blue (23.5 minutes per data point) and by 10 for the red band (47.0 minutes per data point). Each blue data point is independent of the previous and following point (no time overlap) while, for the red, each data point has a time overlap of 23.5 minutes with the previous and following point. Consecutive points for both bands have a separation of ∼ 0.05 in rotational phase, clear data outliers have been removed. At the end, 18 data points remain for the blue band, and 18 for the red one (see Table <ref> in Appendix <ref>).Similarly, 95 single maps of 2003 AZ_84 were acquired in green/red combination. They were grouped by 6 in the green without time overlap between previous and following point, and with a separation between consecutive points of ∼ 0.07 in phase. Final data points for the green are 15 (see Table <ref> in Appendix <ref>). Red images were discarded for a thermal light curve analysis because they are very contaminated by background sources (even after the application of background removing techniques) and the final photometry on these images is very noisy. We still use the mean value of the red band flux (see Table <ref>) for thermophysical model analysis.Photometry was performed on the combined maps. The flux of the objects was obtained using DAOPHOT <cit.> routines adapted to IDL[Interactive Data Language, Research Systems Inc.]to perform synthetic-aperture photometry on the final images. The object is usually located at the centre or very close to the centre of the images. Since our targets are bright enough, we do not need to use ephemeris coordinates to find them: photocentre routines are then used to obtain the best coordinates to place the centre of the circular aperture. Once the photocentre is obtained, we performed aperture photometry for radii that span from 1 to 15 pixels. We applied the aperture correction method <cit.> for each aperture radius using the tabulated encircled energy fraction for a point-source observed with PACS[Müller et al. (2011): PACS Photometer -Point-Source Flux Calibration, PICC-ME-TN-037, Version 1.0; Retrieved November 23, 2011; <http://herschel.esac.esa.int/twiki/pub/Public/PacsCalibrationWeb/pacs_bolo_fluxcal_report_v1.pdf>]. Uncertainties on the fluxes are estimated by means of a Monte-Carlo technique, in which artificial sources are randomly implanted on the images. We obtain and correct by aperture the fluxes of these artificial sources using a median optimum aperture radius. Uncertainties are computed as the standard deviation of these implanted fluxes. These photometric techniques and uncertainty estimations used to extract the PACS fluxes are further described in <cit.> and <cit.>.In addition to the random photometric errors, the data may suffer from a systematic flux calibration uncertainty, which is estimated to be ∼5 %. As the latter affects all points of a given light curve in an identical way, it is not included in theindividual error bars. Color corrections, which are ∼1-2% for Herschel/PACS data, were applied to the fluxes (see caption in Tables <ref>, <ref>, and <ref> for the exact value of the color-correction factors applied).Finally, time-phasing of all the images was computed using the preferred rotational periods (see Table <ref>) of the objects. For Haumea, the relative phasing of data taken at two epochs separated by six months did not pose any problem, thanks to the highly accurate knowledge of the period[We re-determined this period as P= 3.915341 ± 0.000005 h using additional optical data from January 2010 combined with data from 2007 <cit.>. A detailed description of these observations and of the technique used to derive this rotation period can be found in <cit.>.]. A running mean of the Haumea thermal fluxes in each filter was applied with a bin of 0.05 in rotational phase (= 11.75 minutes of time), which finally obtained20 points at green band and 19 at red band. The final thermallight curves are shown in Figs. <ref>, <ref>, and <ref> for Haumea, 2003 VS_2 and 2003 AZ_84, respectively.§ RESULTS AND ANALYSIS §.§ (136108) HaumeaHaumea's optical light curve is one of the best studied among TNOs <cit.>. Besides its strong amplitude (0.28 magnitudes), its most remarkable feature is its asymmetric character, exhibiting two unequal brightness maxima, which cannot be explained by a pure shape effect and is interpreted as being due to the presence of a darker (and redder)region on the object's surface <cit.>.Figure <ref> shows the thermal data for Haumea at 100 and 160 μm, respectively, as a function of rotational phase, using a period of 3.915341 h. The zero phase epoch is JD = 2455188.720000 (uncorrected for light-time) and phases are calculated using light-time corrected julian dates and light-time corrected zero date. In addition to the thermal fluxes, Fig. <ref> displays the Scaled optical LC, which represents the optical fluxes rescaled by some multiplicative factor to match the mean thermal flux level. The part of the optical light curve that is affected by the dark spot is outlined in yellow. The overall positive correlation between the thermal and optical light curves indicate that both are mostly caused by the object elongated shape, as already noted in <cit.>, hereafter Paper I. However, the highest quality 100 μm data further indicates an asymmetry in the two thermal flux maxima,with the strongest occurring near phase ∼0.75, i.e. in the part of the optical light curve that is affected by the dark spot. The possible influence of the spot could not be discerned inPaper I, and the present data are of higher quality. On the other hand, the 160 μm data do not show evidence for an enhanced thermal flux associated with the dark spot. A Fourier fit of the thermal data permits us to determine the amplitude of the thermal light curve (defined as the difference between maximum and minimum fluxes in the Fourier fit), as well as its phasing relative to the optical light curve. In both filters, the thermal light curve amplitude is larger than its optical counterpart and diminishes with increasing wavelength; these behaviors are in accordance with thermophysical model expectations. However, the two thermal filters do not give fully consistent information of the phase shift between the thermal and optical data: 100 μm data appear well in phase with the optical light curve, while 160 μm data appear shifted by 0.06 in phase (i.e. by about 21 degrees: see Table <ref>). Finally, we perform a consistency check of the fluxes obtained in the thermal light curves. To do this, we run a Near Earth Asteroid Thermal Model <cit.> for the green and red fluxes at the minimum and maximum of the thermal light curves. We assume H_V = 0.43 ± 0.01 mag as in <cit.> and the geometric albedo and beaming factor derived in that work (p_V = 80.4%, η = 0.95 ). Under these assumptions we estimate the area-equivalent diameter from the NEATM for the minimum of the thermal light curves, using an absolute magnitude of (H_V + 0.21/2) mag, where 0.21/2 is the semi-amplitude from the optical light curve, obtaining D_min = 1173 km.Running a NEATM in the same way for the maximum, for an absolute magnitude (H_V - 0.21/2) mag, we obtain D_max = 1292 km. These diameters are consistent, within error bars, with the best equivalent diameter obtained in <cit.>. §.§.§ Haumea modelingFollowing our work in <cit.>, modeling of the thermal light curve was performed using OASIS <cit.>. OASIS is a versatile tool in which an object is described by triangular facets. The orientation of each facet with respect to pole orientation, Sun direction, observer direction, and time as the object rotates, is calculated. OASIS therefore requires a shape model and an assumed aspect angle (i.e., pole orientation). For the shape of Haumea, we used an ellipsoid made of 5 120 triangles. For the aspect angle, the large amplitude of Haumea's optical light curve favors a large angle and,here, weassumed an equator-on geometry (aspect angle θ = 90^∘). A spectral and bolometric emissivity of 0.9 in all filters is assumed as well.In Paper I,two shape models for Haumea(defined by the a, b, and c semi-major axes of the ellipsoid)were used, based on optical light curves observations by <cit.> and assuming a Jacobi hydrostatic equilibrium figure (a > b > c). The two shape models were derived by considering two different scattering properties for Haumea's surface (Lambertian reflectivity, shape model 1, and Lommel-Seelinger reflectance properties, shape model 2), leading to slightly different values of b/a, c/a, and the object density. For a given shape model, knowledge of the object mass <cit.> provided the absolute values of a, b, and c, which in turn provided the object mean geometric albedo, based on its H_v magnitude. All these parameters were then implementedin a NEATM thermal model <cit.>, and the only free parameter in fitting the thermal light curve was the so-called beaming factor, η. In this process, a phase integral q = 0.7 was adopted; this value is reasonable for a high albedo object (see, Fig. 7 and) but admittedly uncertain. Considering mostly an aspect angle θ = 90^∘, the main conclusion of Paper I was that η = 1.15 satisfies the mean thermal flux constraint for both shape models, but matches the light curve amplitude only for model 2, which was therefore favored. The relatively low η value (for an object at this distance from the Sun) pointed to a generally low thermal inertia for the surface and significant surface roughness effects <cit.>. Paper I also briefly explored the effect of aspect angle by considering the case θ = 75^∘, and found that such a model could be valid, but using a slightly larger η value (e.g. η = 1.35 instead of η = 1.15). However, the modest quality of the thermal light curve in Paper I did not warrant the use of more elaborate models.Given the improved data quality in the current work, including the apparent detection of increased thermal emission at the expected location of the dark spot, we now improve these early models by (i) considering thermophysical models (TPM); (ii) exploring in some detail the effect of a surface spot. The essential physical parameter to constrain is now the surface thermal inertia, Γ. To make allowances for possible surface roughness effects, however, the TPM can also include an η factor, but which in this formalism is by definition ≤ 1 (see e.g., Eq 3;, Eq 2). Unlike NEATM, which for a uniform (constant albedo) elliptical surface, calculates (by construction) a thermal light curve in phase with the optical light curve, the TPM approach enables us to investigate temperature lags owing to thermal inertia. Thus, in principle, the thermal inertia can be derived by investigating the relative phase of the thermal and optical light curves, as constrained by the observations. Once Γ is determined, η and q may be adjusted so as to match the mean flux level and the amplitude of the light curve. However, the problem may be underconstrained, i.e. η and q cannot necessarily be determined separately. If the surface includes a spot of known albedo and spatial extent, the parameters (i.e. Γ, and η and/or q) may be adjusted separately in the spot region and outside. In what follows, and given the low thermal inertias we inferred (see below), we found that the observed flux levels did not require to be enhanced by surface roughness, so we simply assumed η = 1, recognizing that some degeneracy exists betweenη and q.<cit.> show that the asymmetry in Haumea's optical light curve can be interpreted with different spot models, characterized by the albedo contrast of the spot with respect of its surroundings and its spatial extent. In all cases, the spot is assumed to be centred on Haumea's equator and to lead one of the semi-major axes by 45^∘. Possible models range from a very localized (6% of Haumea maximum cross section) and low albedo spot (30 % of the non-spot albedo) to much more extended spot (hemispheric) and subdued in contrast (95 % of non-spot albedo). In a brief study of the spot effect on the thermal light curve, Paper I considered a spot covering 1/4 of Haumea's maximum projected cross section, with an albedo contrast (about 80 %) of the non-spot albedo, as prescribed by the <cit.> results. The same spot description was adopted here. Being relatively limited in extent, the spot has negligible effect on about half of the thermal light curve (from phase ∼ 0.0 to ∼ 0.5 with the adopted phase convention). Therefore, as a first step, we focus on the part of the light curve that is not affected by the spot. Fig. <ref> shows the comparison of the observed 100 μm light curve with several homogeneous (no spot) models differing by their surface thermal inertia (Γ = 0.0 to 0.5 MKS by steps of 0.1). Here, and throughout the following, shape model 2 is adopted <cit.> with an aspect angle θ = 90^∘, giving a = 961 km, b = 768 km, c = 499 km, and p_v = 0.71. These a, b, c values lead to a mean area-equivalent diameter D_equiv= 2· a^1/4· b^1/4· c^1/2 = 1309 km, within the error bars of the area-equivalent diameters obtained from radiometric techniques for this object (1324±167 km from; 1240^+69_-58 km from). Using a shape model that is consistent with the optical light curve is preferable to a radiometric solution that may include measurements at different light curve phases. For each value of Γ, the phase integral (q) is adjusted to provide the best fit to the mean flux level and amplitude of the light curve outside the spot region (i.e. at phases 0.0–0.5). For Γ = 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 MKS, respectively, the required values of q are 0.851, 0.778, 0.712, 0.660, 0.613, and 0.578, respectively. As can be seen in Fig. <ref>, Γ = 0.0 provides the best fit to the part of the light curve not affected by the spot, while larger Γ values progressively lead to larger delays of the thermal emission, which are not observed in the 100 μm data. Detailed comparisons of the data show that the reduced χ^2 is minimum for Γ = 0.0 MKS and larger but still reasonable for Γ = 0.2 MKS. From this we conclude that Γ values in the range 0.0–0.2 MKS are consistent with the data. The associated phase integral values are in the range 0.851–0.712. These are generally consistent but still somewhat larger than the 0.7 value assumed in Paper I. We note that using the <cit.> empirical relationship between geometric albedo and phase integral, phase integrals of 0.851 (obtained for Γ = 0 MKS) – 0.712 (for Γ = 0.2 MKS) would imply geometric albedos in the range 1.05–0.64. The latter value is more consistent with p_v= 0.71, as indicated by shape model 2. From this point of view, Γ = 0.1–0.2 MKS would seem a more plausible solution but, as indicated above, the associated fits are worse than with Γ = 0 MKS. We also note that these q values hold for our assumption of η = 1 (no surface roughness), and that for a given Γ, any finite surface roughness would require an even larger value of q.We now turn to constraints on the dark spot. Continuing with the assumption of no surface roughness, we investigate the effect of a specific phase integral or thermal inertia in the spot. In all models, the spot has an equivalent geometric albedo of 79% of its surroundings (p_v = 0.71), i.e. p_v= 0.56. Figure <ref> shows the effect of changing the phase integral in the spot (q_spot) by steps of 0.05, maintaining its thermal inertia to Γ_spot = 0 as in the best fit solution of the no-spot region (Fig. <ref>). As is clear from Fig. <ref>, mild changes in q_spot, e.g. from ∼0.85 to ∼1.05 produce flux variations at the adequate level.The flux excess near phase 0.75 would point to q_spot = 0.85, essentially identical to the non-spot region (meaning that the flux excess is purely an effect of the spot darker albedo). However this model (blue curve in Fig. <ref>) clearly overpredicts the fluxes beyond the peak phase, where data are best fit with q_spot = 0.95–1.05. The best overall fit of the light curve using an χ^2 criterion is achieved with q_spot = 1.00, and the model light curve shows only a very weak flux excess in the spot region (meaning that the effect of the darker spot albedo is essentially compensated for by the effect of a larger phase integral). A phase integral of 1.00 seems very high (and not in line with the relationship), but it has been observed in other solar system objects <cit.>. However, while the fit is satisfactory (first best-fit model in Fig. <ref>), we note that it might not be a very realistic solution, since there is a general positive correlation between albedo and phase integral on airless bodies ( , Fig. 7;), in contradiction with the dark (low albedo) spot, which has a larger phase integral (higher albedo) than its surroundings.Figure <ref> shows the effect of changing the spot thermal inertia, maintaining its phase integral to the value of its surroundings. For this exercise, we first consider Γ = 0.1 MKS in the no-spot region, associated with q= 0.778. We maintain the q_spot at this value and show models with Γ_spot = 0.0, 0.1, and 0.2 MKS. These models show that variations of Γ within this range produce flux changes at the appropriate level. In this family of models, the best fit is achieved with Γ_spot = 0.2 MKS (blue curve in the Fig.), which again suggest that, to avoid too large fluxes past the peak, the thermal effect of darker albedo needs to be partly compensated for by a larger thermal inertia. As discussed above, however, Γ = 0.1 MKS is not an optimum fit of the no-spot part of the light curve. The red curve in Fig. <ref> shows a model with Γ = 0.0 MKS and q = 0.851 outside of the spot, andΓ_spot = 0.05 MKS (and still q_spot = 0.851) in the spot (second best fit model). This model fit is worse than the first best fit model (Γ_spot = 0.0 MKS, q_spot = 1.00) and, in fact, no better than the no-spot model: the fit improves in the region of flux maximum around phase 0.75, but the non-zero thermal inertia in the spot tends to delay the model too much in the 0.85–1.0 phase region. Thus the models of Fig. <ref> indicate that, at most, the thermal inertia in the spot region is slightly higher than in its surroundings. This type of behaviour might be expected. A statistical study of TNO thermal properties<cit.> suggests that the highest albedo TNOs generally exhibit particularly low thermal inertia. A plausible explanation is that in addition to a specific composition (e.g. pure ice), a higher albedo may reflect a smaller grain size. In fact, the generally low thermal inertia of TNOs points to a regime where radiative conductivity (i.e. in surface pores) is important in the overall heat transfer, thus high albedo objects might plausibly be associated with low thermal inertia, and the association of lower albedo with larger thermal inertia is also plausible. Fig. <ref> shows the application of the above two best fit models to the Haumea's 160 μm light curve. In both cases, the agreement with the observed 160 μm mean flux, and especially light curve amplitude is reasonable, but the models fail to match the data in two respects: (i) the modeled flux levels are on average too high (by ∼ 10%), (ii) the models are somewhat out of phase with the observations. The first problem may suggest a calibration error in the 160 μm data (more subject to sky contamination). The second one is related to the fact that the 160 μm data appear to be shifted by about 0.06 ± 0.01 in rotational phase (∼ 21 degrees or ∼ 13 minutes of time, see Table <ref>) with respect to the 100 μm data. Optimum phasing of the model with respect to the 160 μm data would require a thermal inertia Γ ∼ 0.5 MKS, and fitting the mean flux levels would then require a phase integral q = 0.68. This model tailored to the 160 μm data does not include any specific values of Γ or q in the dark spot region, as this is not required by these data. Overall a simultaneous fit of the 100 and 160 μm data within error bars does not appear possible. Althoughso-called compromise parameters could be formally found by performing an χ^2 minimization on both datasets simultaneously, we feel that separate modeling is preferable since it provides a handle on model limitations and realistic range of solution parameters. Giving more weight to the higher quality 100 μm data, we favor Γ = 0.0–0.2 MKS but, based on 160 μm modeling, we regard Γ = 0.5 MKS as an upper limit to Haumea's thermal inertia. In summary, thermophysical modeling of the Haumea light curves indicates that: (i) the object's thermal inertia Γ is extremely small (less than 0.5 MKS and probably less than 0.2 MKS) and its phase integral is high (at least 0.8 for Γ = 0.1 MKS and probably even higher if surface roughness is important); (ii) only small changes in the surface properties of the dark spot (e.g. changes in the thermal inertia by ∼ 0.1 MKS or in the phase integral by ∼ 0.1) are required to significantly affect the emitted fluxes on the hemisphere where the spot resides; larger changes are excluded by the data; (iii) the most plausible scenario may invoke a slightly higher thermal inertia in the dark spot compared to its surroundings, but a fully consistent picture is still not found, since the ∼ 21 degrees shift in phase between the 100 and 160 μm data is difficult to understand. Finally, we note that in all of our Haumea models we ignored the possible contribution of its satellites Hi'iaka and Namaka. Although their albedos are unknown, these moons are thought to have been formed by a catastrophic impact that excavated them from the proto-Haumea ice mantle and led to the Haumea family <cit.> or from rotational fission <cit.>. As such, their albedos are probably comparable to, or even higher than, Haumea's itself. Assuming a 0.70 geometric albedo, Hi'iaka and Namaka diameters are ∼320 and ∼160 km, respectively <cit.>. Furthermore, assuming identical thermophysical properties, they would contribute in proportion of their projected surfaces, i.e. 6 % and 1.5 % of Haumea's thermal flux. Although this is not negligible, we did not include this contribution owing to its uncertain character. Should the Haumea's thermal fluxes to be modeled decrease by ∼20 mJy x 7.5 % ∼ 1.5 mJy, this could be taken care of in the models by a slight increase of the phase integral for a given thermal inertia, without any changes to the conclusions on the object thermal inertia and the dark spot properties. §.§ (84922) 2003 VS_2 2003 VS_2 is a Plutino without known satellites. Near infrared spectra of this body shows the presence of exposed water ice <cit.>, which probably increases the geometric albedo of this Plutino <cit.>, compared with the mean albedo of TNOs without water ice. This object presents an optical light curve with moderately large peak-to-peak amplitude ∼ 0.21 ± 0.01 mag and a double-peaked rotational period ∼ 7.42 hours <cit.>. To fold the Herschel data to the rotation period with enough precision, we refined the knowledge of the rotational period. To achieve this goal, we used optical observations taken on 4-8 September 2010 by means of the 1.5-m telescope at Sierra Nevada Observatory (OSN, Spain) using a 2k × 2k CCD with a FOV of 7.8 × 7.8 and 2 × 2 binning mode (image scale = 0.46/pixel). Then we merged these observations with old optical light curves obtained also at OSN on 22, 26, 28 December 2003 and 4, 19-22 January 2004, and with 2003 observations published in <cit.>. No filter was used to perform the OSN observations. We reduced and analysed all these data asdescribed in <cit.> to finally obtain an accurate rotational period of 7.4175285 ± 0.00001 h (Δm = 0.21 ± 0.01 mag.). A further description of the observations and techniques leading to this very accurate rotational period are detailed in <cit.>. The thermal light curves at 70 μm and 160 μm are not firmly detected with only a 1.7 σ and 1.5 σ confidence levels respectively (see Table <ref> and Fig. <ref>). A Fourier fit of the 70 μm data indicates a mean flux of 14.16 mJy and an amplitude of 1.73 mJy, which is slightly smaller than the optical light curve amplitude. The same analysis yields a negligible shift in time of the 70 μm data relative to the optical (Fig. <ref>). The lower quality 160 μm data would suggest a -0.054 ± 0.102 phase shift (see Table <ref>). We do not consider this last negative shift significant since it is well below the estimated error bars. In what follows, we pursue with our thermophysical modeling, focusing on the mean thermal flux and light curve amplitude at 70 μm. §.§.§ 2003 VS_2 modeling The large amplitude of the optical light curve and its double peaked nature indicates that the main cause for the variability is a triaxial shape <cit.>. Then, if we assume that the optical light curve is entirely shape-driven, its period and amplitude can be used to derive a shape model under the assumption of hydrostatic equilibrium. As for Haumea, the large object size (∼500 km) makes the assumption of a Jacobi hydrostatic equilibrium figure (semi-major axes a> b> c) reasonable. Further assuming an equator-on viewing geometry (aspect angle = 90^∘), the Δ m = 0.21 mag light curve amplitude is related to shape by Δ m = 2.5 · log ( a/b). Using the rotation period of 7.42 h and the Chandrasekhar figures of equilibrium tables for the Jacobi ellipsoids <cit.>, we obtain b/a = 0.82, c/a = 0.53 and ρ = 716 kg/m^3, where ρ is a lower limit to the density because, if the object is not observed equator-on, the true a, b axial ratios may be higher and the implied density would be also higher. Using the area-equivalent radiometric diameter (D_equiv = 523 km) derived from earlier thermal modeling of Herschel and Spitzer data <cit.>, defined as D_equiv= 2· a^1/4· b^1/4· c^1/2, we obtain the values of the semi-major axes of 2003 VS_2: a= 377 km, b= 310 km, and c= 200 km. We further adopt a phase integral q = 0.53 and a V geometric albedo p_V = 0.147 from the previous papers and, as for Haumea, we do not consider surface roughness (i.e. η = 1). All these values are used as input parameters to the OASIS code for modeling the 70 μm thermal light curve of 2003 VS_2. With this approach, the only free parameter is the thermal inertia Γ. Figure <ref> shows model results forΓ = 1.0, 2.0 and 3.0 MKS. In this figure, the phase of the thermal models is determined by requiring that the model with Γ = 1.0 matches the observed phase of the thermal data (i.e. a maximum at phase 0.55, see top panel of Fig.<ref>). We note that, unlike in the Haumea case (see Fig. <ref>), models with the various thermal inertias all appear to be approximately in phase. The difference in behaviour is caused by the combination of the longest period and warmer temperatures at 2003 VS_2 versus Haumea. For a given thermal inertia, this causes a much smaller value of the thermal parameter for 2003 VS_2 <cit.>. A thermal inertia of 2.0 MKS matches reasonably well the mean flux levels and the light curve amplitude, while the other two models significantly over- or underestimate the mean flux. Thus, we conclude to a thermal inertia Γ = (2.0 ± 0.5) MKS for 2003 VS_2. This value is fully consistent with the mean thermal inertia for TNOs and centaurs derived statistically from the Herschel TNOs are Cool sample <cit.>, but significantly above that for Haumea. As indicated in the above paper, high-albedo objects seem to have preferentially low thermal inertias. §.§ (208996) 2003 AZ_842003 AZ_84 is a binary Plutino with a shallow optical light curve (Δm ∼ 0.07 mag) and with a rotational period ∼ 6.79 hours, assuming a single-peaked light curve <cit.>. Nonetheless, the double-peaked solution, which corresponds to a rotational period ∼ 13.58 hours, cannot be totally discarded <cit.>. Near infrared spectra have detected water ice on its surface <cit.>. As for the other two TNOs, we acquired additional time series images of 2003 AZ_84 on 4-5 February 2011 with the 1.23-m telescope at Calar Alto Observatory (CAHA, Spain), equipped with a 2k × 2k CCD camera in the R filter. These data are merged with old CAHA and OSN data from 2003 and 2004 to refine the rotational period, obtaining P = 6.7874 ± 0.0002 h (Δm = 0.07 ± 0.01 mag), which we nominally use to fold the Herschel/PACS data and compare them with the visible light curve. A more detailed description of the observations and techniques of analysis leading to this rotation period are included in <cit.>.The thermal light curve of 2003 AZ_84 is not firmly detected in the PACS data at 100 μm (see Fig. <ref>). While a Fourier fit to the thermal data formally provides a best fit amplitude of 1.97 ± 1.40 mJy at 100 μm, its significance is thus at the 1.4 σ level. By making a visual comparison of the Fourier fit to the thermal data in Fig. <ref> with the Fourier fit to the optical data (shown with a dashed line), it appears that there could be a weak anticorrelation of the 100 μm data with the visible data. This would give confidence to the interpretation that the thermal light curve could be generated by enhanced thermal emission in the darker spots or darker terrains that give rise to the optical light curve, in the same way as the dark spot in Haumea generates enhanced thermal emission. However a Spearman test to analyze a possible anticorrelation of the thermal data with the optical light curve gave a non-significant result. Moreover, in the regime of low albedo (∼ 10% for 2003 AZ_84), the thermal emission is essentially albedo-independent, so that a thermal light curve resulting from optical markings would have an undetectable amplitude, barely above 0.05 mJy.§.§.§ Analysis of 2003 AZ_84 results2003 AZ_84 is a large enough TNO <cit.> so that it is very likely to be in hydrostatic equilibrium. This means that the expected 3D shape of this TNO should be a figure of equilibrium: either a rotationally symmetric Maclaurin spheroid or a triaxial Jacobi body.2003 AZ_84 has a low light curve amplitude in the visible, which means that either this TNO is a Maclaurin object with small albedo variability on its surface, or it is seen nearly pole on, or both. Recent results on stellar occultations by 2003 AZ_84 <cit.> have shown an equal-area diameter of D_equiv = 766 km and a small projected flattening of only 0.05. The small flattening has two possible extreme explanations: The object has a typical density of a TNO of its size but it is seen nearly pole-on so that the large flattening of the body becomes a small projected flattening, or 2003 AZ_84 has an exceptionally high density for its size. The density required for a Maclaurin body with flattening of 0.05 and a rotation period of 6.79 h is 5 500 kg/m^3. This huge density is not feasible in the Transneptunian region. Assuming that 2003 AZ_84 could have a density of ∼2500 kg/m^3, which is already too high for a TNO of its size, its true oblateness would be 0.12. This would require an aspect angle < 45 degrees for the Maclaurin spheroid to give rise to the projected oblateness of 0.05 seen in the occultation. We note that densities of around 2 500 kg/m^3 have only been measured for the very largest TNOs, such as Pluto, Eris and Haumea whose internal pressures do not allow for the macroporosity that can exist in bodies of smaller size <cit.>. So it is extremely unlikely that 2003 AZ_84 could have such a high density of 2 500 kg/m^3. Hence we are confident that the aspect angle of 2003 AZ_84 must be smaller than 45 degrees. Therefore the low light curve amplitude in the visible, the small thermal variability, and the occultation results are reasons to believe that 2003 AZ_84 could be close to pole-on (have a small aspect angle). To further constrain the spin axis orientation, we run a thermophysical model <cit.>. The model takes into account the thermal conduction and surface roughness for objects of arbitrary shapes and spin properties. The model was extensively tested against thermal observations of near Earth asteroids (NEAs), Main Belt asteroids (MBAs), and TNOs over the last two decades. Recent works with this TPM code <cit.> have shown that a combined analysis of Herschel and Spitzer thermal measurements enable us to constrain the spin-axis orientation of TNOs. Following up on this expertise, we also applied this code in the case of 2003 AZ_84, assuming a spherical shape model and a constant emissivity of 0.9 at all MIPS and PACS wavelengths <cit.>. We used all available Herschel/PACS and Spitzer/MIPS thermal data (except the Spitzer/MIPS data point at 71.42 μm, which is affected by a background source). The Spitzer MIPS data were presented in <cit.>. We re-analysed the two MIPS observations of 2003 AZ_84. The 71.42 μm measurements are problematic owing to contaminating background sources, but the 23.68 μm points are clean and the object's point-spread-function is as expected. Table <ref> shows the thermal measurements used in the thermophysical modeling. Using H_V = 3.78 ± 0.05 mag and the stellar occultation size D = 766 ± 16 km <cit.>, we check models with a range of thermal inertias from 0.0 to 100 MKS and a range of different levels of surface roughness (rms of surface slopes of 0.1, 0.3, 0.5, 0.7, and 0.9). The biggest issue with the radiometric analysis is that the MIPS and the PACS data do not match very well. A standard D-p_V radiometric analysis for only the PACS data favors the pole-on solutions (combined with low surface roughness - rms of surface slopes < 0.5) and provide the correct occultation size (760-790 km), for the two possible rotation periods (P = 6.7874 h and P = 13.5748 h) and almost independent of thermal inertia. The overall fit to only PACS observations is excellent. Equator-on solutions can also provide correct sizes, but only under the assumptions of extremely low thermal inertias far below 1.0 MKS and combined with extremely high surface roughness (rms of surface slopes > 0.7), which looks very unrealistic. The very best solutions are found for a spin-axis orientation 30^∘ away from pole-on, intermediate levels of surface roughness (i.e. realistics values), and acceptable values for the thermal inertia (Γ = 0.5-3.0 MKS). Overall, the pole-on ±30^∘ configuration explains very well the PACS fluxes, but slightly overestimates the MIPS 24 μm within the 2 σ level (see Fig. <ref>). This difference between MIPS flux and model could be due to some light curve effects at the moment of the Spitzer/MIPS observations. The equator-on geometry only works when using very extreme settings, which seem very unrealistic. Summarizing the combined thermal and occultation analysis, we find that its spin-axis is very likely close to pole-on (±30^∘). § SUMMARY AND BRIEF DISCUSSIONTime series thermal data of three bright TNOs (Haumea, 2003 VS_2 and 2003 AZ_84) have been acquired with Herschel/PACS in search of thermal light curves, with thefollowing main results: * The thermal light curve of Haumea is clearly detected at 100 and 160 μm, superseding the early results of <cit.> with Herschel/PACS.Both light curves are correlated with the optical one, implying primarily shape-driven light curves. Nonetheless, the 100-μm data indicates a small extra flux at rotational phases affected by the optical dark spot. * Thermophysical modeling of the Haumea thermal light curves indicates an overall surfacewith an extremely small thermal inertia (Γ < 0.5 MKS and probably Γ < 0.2 MKS) and high phase integral (q ∼ 0.8 for Γ = 0.1 MKS and no surface roughness), which will be even higher if surface roughness is present. * The energetic and thermophysical properties of Haumea's dark spot appear to be only modestly different from the rest of the surfaces, with changes of only ∼ +0.05–0.1 MKS in thermal inertia or ∼ +0.1 in phase integral. We favor the case for a small increase of thermal inertia in the dark region.* The thermal light curve of 2003 VS_2 is not firmly detected at 70 μm and at 160 μm. However, Fourier fits to the thermal data are correlated with the optical light curve. The amplitude and mean flux of 2003 VS_2's 70 μm light curve indicate a thermal inertia Γ = (2.0±0.5) MKS. * The thermal light curve of 2003 AZ_84 at 100 μm is not firmly detected. A thermophysical model applied to the mean thermal light curve fluxes and to all the Herschel/PACS and Spitzer/MIPS thermal data favors a close to pole-on (±30^∘) orientation. * Our conclusion of extremely small thermal inertias for 2003 VS_2 and even smaller for Haumea statistically nicely matchesinferences on the TNO/Centaurs population based on Spitzer/Herschel radiometry <cit.>, including an albedo dependence of the thermal inertia. These authors interpreted their results in terms of highly porous surfaces, in which the heat transferefficiency is affected by radiative conductivity within pores and increases with depth in the subsurface. For heat conduction dominated by radiation, the thermal inertia is essentially proportional to r_h^-3/4 <cit.>, or to r_h^-(0.9-1.0) if the temperature dependence of the specific heat of ice is taken into account <cit.>. Our thermal inertia for the three objects (2.0 ± 0.5 MKS for 2003 VS_2 at r_h = 36.5 AU, ∼ 0.2 MKS for Haumea at r_h = 51 AU, and 0.7-2.0 MKS for 2003 AZ_84 at r_h = 45 AU) convert into Γ = 10–90 MKS and Γ = 4–35 MKS at 1 AU for the two temperature-dependence cases, respectively. While somewhat even lower, these numbers compare generally well with the thermal inertias of large (> 100 km) asteroids <cit.>, where the smallest values are indicative of fine grain regolith. Recently, <cit.> re-addressed the general issue of low thermal inertias in outer solar system bodies (including icy satellites), and pointed out several other important factors, in addition to surface porosity, affecting surface effective thermal inertias. One such factor is the quality of grain contact (i.e. tight or loose) in determining solid-state conductivity. For H_2O-ice covered surfaces, another factor, already recognized by <cit.>, is the amorphous vs. crystalline state of water, as the two states are associated with different bulk conductivities (and different temperature dependence thereof). On the basis of a detailed physical model of conductivity, including radiative conductivity, <cit.> were able to reproduce the order of magnitude and heliocentric dependence of the thermal inertias measured by <cit.>, by invoking loose contacts in a moderately porous regolith of sub-cm-sized grains made of amorphous ice. Since water ice, when detected on the surface of TNOs, is usually in cristalline form, this scenario implies the presence of amorphous ice at cm depths below a thin layer of crystalline ice.Herschel is an ESA space observatory with science instruments provided by European led Principal Investigator consortia and with important participation from NASA. Herschel data presented in this work were processed using HIPE, a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia. This research is partially based on observations collected at Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut fur Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). This research is partially based as well on observations carried out at the Observatorio de Sierra Nevada (OSN) operated by Instituto de Astrofísica de Andalucía (CSIC). The research leading to these results has received funding from the European Union's Horizon 2020 Research and Innovation Programme, under Grant Agreement no 687378. P. Santos-Sanz and J.L. Ortiz would like to acknowledge financial support by the Spanish grant AYA-2014-56637-C2-1-P and the Proyecto de Excelencia de la Junta de Andalucía J.A. 2012-FQM1776. C. Kiss acknowledges financial support from NKFIH grant GINOP-2.3.2-15-2016-00003. E. Vilenius was supported by the German DLR project number 50 OR 1108. R. Duffard acknowledges financial support from the MINECO for his Ramon y Cajal Contract.aa 68 natexlab#1#1[Barkume et al.(2008)Barkume, Brown, & Schaller]2008AJ....135...55B Barkume, K. M., Brown, M. E., & Schaller, E. L. 2008, , 135, 55[Barucci et al.(2011)Barucci, Alvarez-Candal, Merlin, Belskaya, de Bergh, Perna, DeMeo, & Fornasier]2011Icar..214..297B Barucci, M. 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L. 2007, , 655, 1172[Vilenius et al.(2012)Vilenius, Kiss, Mommert, Müller, Santos-Sanz, Pal, Stansberry, Mueller, Peixinho, Fornasier, Lellouch, Delsanti, Thirouin, Ortiz, Duffard, Perna, Szalai, Protopapa, Henry, Hestroffer, Rengel, Dotto, & Hartogh]2012A A...541A..94V Vilenius, E., Kiss, C., Mommert, M., et al. 2012, , 541, A94[Vilenius et al.(2014)Vilenius, Kiss, Müller, Mommert, Santos-Sanz, Pál, Stansberry, Mueller, Peixinho, Lellouch, Fornasier, Delsanti, Thirouin, Ortiz, Duffard, Perna, & Henry]2014A A...564A..35V Vilenius, E., Kiss, C., Müller, T., et al. 2014, , 564, A35§ TABLESIn this appendix, we include the light curve photometric results obtained from the Herschel/PACS observations of Haumea, 2003 VS_2 and 2003 AZ_84.ccccHaumea thermal time series photometry results from Herschel/PACS observations at green (100 μm) and red (160 μm) bands (some clear outliers have been removed in the table). Each data point in the green band spans around 18.8 minutes and is the combination of four single images, there is an overlap of around 14.1 minutes between consecutive data points with the same OBSID (except when some outliers have been removed). For the red band each data point spans around 28.2 minutes and is the combination of six single frames, there is an overlap of around 23.5 minutes between consecutive data points with the same OBSID (except when some outliers have been removed). Thermal light curves in Fig. <ref> have been obtained from these data folding the dates with the Haumea's rotational period and computing a running mean with a temporal bin of 0.05 in rotational phase. OBSID are the Herschel internal observation IDs, JD are the julian dates at the middle of the integration uncorrected for light-time (the mean one-way light-time for OBSID 1342188470 is 426.329455 min, and 421.967915 min for OBSID 1342198851), Band are the different filters (green or red) used to observe with PACS, Flux/unc are the in band fluxes and 1-σ associated uncertainties expresed in millijansky (mJy), these values must be divided by the factors 0.98 and 0.99 for the green and red bands, respectively, to obtain color corrected fluxes/uncertainties. The zero time used to fold the rotational light curves in Fig. <ref> is JD = 2455188.720000 days (uncorrected for light-time, the one-way light-time for this date is 426.308537 min).OBSID JDBandFlux/unc [days] [mJy] 4c– Haumea thermal time series with Herschel/PACS Continued from previous pageOBSID JDBandFlux/unc [days] [mJy]4cContinued on next page1342188470 2455188.7448774963 green 23.61 ± 2.38 1342188470 2455188.7537522637 green 24.44 ± 2.36 1342188470 2455188.7579938867 green 24.94 ± 2.39 1342188470 2455188.7622028766 green 26.64 ± 2.33 1342188470 2455188.7663792432 green 27.80 ± 2.51 1342188470 2455188.7705882331 green 26.12 ± 2.66 1342188470 2455188.7740794080 green 24.05 ± 2.46 1342188470 2455188.7775705927 green 22.78 ± 2.43 1342188470 2455188.7810617676 green 20.10 ± 2.41 1342188470 2455188.7845529523 green 18.72 ± 2.29 1342188470 2455188.7880441272 green 15.87 ± 2.29 1342188470 2455188.7915353021 green 14.35 ± 2.37 1342188470 2455188.7950591105 green 16.87 ± 2.26 1342188470 2455188.7985502952 green 16.35 ± 2.38 1342188470 2455188.8020414701 green 15.89 ± 2.41 1342188470 2455188.8055326547 green 16.49 ± 2.43 1342188470 2455188.8090238296 green 17.20 ± 2.42 1342188470 2455188.8125150045 green 15.67 ± 2.71 1342188470 2455188.8160061892 green 13.90 ± 2.60 1342188470 2455188.8188121826 green 14.09 ± 2.64 1342188470 2455188.8209003611 green 14.30 ± 2.65 1342188470 2455188.8286331687 green 16.71 ± 2.22 1342188470 2455188.8307213471 green 20.04 ± 2.18 1342188470 2455188.8334947070 green 23.30 ± 2.19 1342188470 2455188.8369858819 green 26.45 ± 2.24 1342188470 2455188.8404770764 green 27.89 ± 2.19 1342188470 2455188.8440008848 green 28.96 ± 2.07 1342188470 2455188.8474920597 green 29.05 ± 2.17 1342188470 2455188.8509832346 green 27.61 ± 2.21 1342188470 2455188.8544744095 green 27.16 ± 2.29 1342188470 2455188.8579655844 green 22.66 ± 2.52 1342188470 2455188.8614567788 green 22.28 ± 2.76 1342188470 2455188.8649479537 green 22.19 ± 2.84 1342188470 2455188.8684717622 green 22.00 ± 2.86 1342188470 2455188.8719629371 green 20.67 ± 2.79 1342188470 2455188.8754541120 green 23.04 ± 2.83 1342188470 2455188.8835458173 green 19.90 ± 2.821342198851 2455368.3686949732 green 24.11 ± 2.01 1342198851 2455368.3700889852 green 23.14 ± 2.00 1342198851 2455368.3720406014 green 23.52 ± 1.95 1342198851 2455368.3740121326 green 22.45 ± 1.93 1342198851 2455368.3759239200 green 22.81 ± 2.02 1342198851 2455368.3778954512 green 23.87 ± 2.00 1342198851 2455368.3804644155 green 23.92 ± 1.98 1342198851 2455368.3830532948 green 23.42 ± 2.03 1342198851 2455368.3856620882 green 22.77 ± 2.01 1342198851 2455368.3889081441 green 21.12 ± 2.03 1342198851 2455368.3921741145 green 21.51 ± 2.12 1342198851 2455368.3954400853 green 21.56 ± 2.25 1342198851 2455368.3987060557 green 20.84 ± 2.26 1342198851 2455368.4019720261 green 20.37 ± 2.43 1342198851 2455368.4052379965 green 21.14 ± 2.31 1342198851 2455368.4085039669 green 19.01 ± 2.49 1342198851 2455368.4117500233 green 17.80 ± 2.28 1342198851 2455368.4150159936 green 17.76 ± 2.18 1342198851 2455368.4182819640 green 18.46 ± 2.15 1342198851 2455368.4215479344 green 18.03 ± 2.29 1342198851 2455368.4248139048 green 18.24 ± 2.26 1342198851 2455368.4287171378 green 19.91 ± 2.44 1342198851 2455368.4319831086 green 21.60 ± 2.66 1342198851 2455368.4345919020 green 22.69 ± 2.65 1342198851 2455368.4431551173 green 23.93 ± 2.05 1342198851 2455368.4451266481 green 24.54 ± 1.89 1342198851 2455368.4477752708 green 23.71 ± 1.91 1342198851 2455368.4510810701 green 23.69 ± 1.92 1342198851 2455368.4543271260 green 24.32 ± 2.00 1342198851 2455368.4575930964 green 25.63 ± 2.24 1342198851 2455368.4608590668 green 25.58 ± 2.46 1342198851 2455368.4641250377 green 26.49 ± 2.53 1342198851 2455368.4673910080 green 25.53 ± 2.28 1342198851 2455368.4706569784 green 23.38 ± 2.25 1342198851 2455368.4739229488 green 22.80 ± 2.17 1342198851 2455368.4771690047 green 21.66 ± 2.07 1342198851 2455368.4804349756 green 19.40 ± 1.89 1342198851 2455368.4837009460 green 17.78 ± 2.03 1342198851 2455368.4869669164 green 17.91 ± 2.08 1342198851 2455368.4902328867 green 17.52 ± 2.12 1342198851 2455368.4934988571 green 14.78 ± 2.27 1342198851 2455368.4967449135 green 16.90 ± 2.43 1342198851 2455368.5000108839 green 16.57 ± 2.37 1342198851 2455368.5032768543 green 15.93 ± 2.25 1342198851 2455368.5065428247 green 15.26 ± 2.28 1342198851 2455368.5098087951 green 19.69 ± 2.13 1342198851 2455368.5130747659 green 20.06 ± 2.23 1342198851 2455368.5163208218 green 20.92 ± 2.24 1342198851 2455368.5195867922 green 22.20 ± 2.28 1342198851 2455368.5227531902 green 24.30 ± 2.21 1342198851 2455368.5253619840 green 22.66 ± 2.37 1342198851 2455368.5279508629 green 23.96 ± 2.27 1342198851 2455368.5299024796 green 24.67 ± 2.141342188470 2455188.7545345956 red 23.15 ± 4.10 1342188470 2455188.7583467197 red 24.92 ± 4.20 1342188470 2455188.7618420459 red 22.16 ± 3.10 1342188470 2455188.7653373731 red 24.36 ± 2.30 1342188470 2455188.7688332782 red 25.14 ± 2.50 1342188470 2455188.7723280261 red 24.99 ± 2.50 1342188470 2455188.7758239312 red 22.66 ± 2.40 1342188470 2455188.7793186796 red 19.05 ± 3.20 1342188470 2455188.7828145847 red 17.91 ± 2.60 1342188470 2455188.7863093326 red 18.62 ± 4.20 1342188470 2455188.7898052381 red 16.14 ± 4.50 1342188470 2455188.7933005644 red 18.69 ± 4.20 1342188470 2455188.7967958916 red 18.69 ± 4.10 1342188470 2455188.8002912179 red 20.25 ± 3.70 1342188470 2455188.8037865451 red 18.13 ± 3.50 1342188470 2455188.8072818713 red 22.59 ± 2.80 1342188470 2455188.8107771981 red 16.71 ± 3.50 1342188470 2455188.8142731031 red 17.21 ± 3.90 1342188470 2455188.8177678520 red 20.53 ± 3.80 1342188470 2455188.8212637566 red 20.75 ± 4.50 1342188470 2455188.8247585054 red 21.38 ± 4.30 1342188470 2455188.8282544101 red 21.67 ± 4.30 1342188470 2455188.8317491589 red 20.32 ± 4.10 1342188470 2455188.8352450635 red 19.97 ± 4.00 1342188470 2455188.8387403907 red 22.59 ± 4.00 1342188470 2455188.8422357170 red 20.82 ± 3.30 1342188470 2455188.8457310442 red 22.37 ± 3.30 1342188470 2455188.8492263705 red 24.07 ± 2.40 1342188470 2455188.8527216977 red 23.93 ± 2.50 1342188470 2455188.8562170244 red 25.70 ± 3.10 1342188470 2455188.8597129295 red 22.59 ± 2.80 1342188470 2455188.8632076778 red 24.92 ± 2.40 1342188470 2455188.8667035829 red 24.07 ± 3.30 1342188470 2455188.8701983313 red 22.73 ± 3.30 1342188470 2455188.8736044089 red 21.31 ± 4.001342198851 2455368.3739723968 red 25.55 ± 3.30 1342198851 2455368.3775566947 red 24.42 ± 3.00 1342198851 2455368.3808211582 red 24.10 ± 4.20 1342198851 2455368.3840844650 red 24.42 ± 4.00 1342198851 2455368.3873489285 red 23.37 ± 5.00 1342198851 2455368.3906122353 red 23.29 ± 4.90 1342198851 2455368.3938766997 red 23.37 ± 4.30 1342198851 2455368.3971400065 red 23.86 ± 3.50 1342198851 2455368.4004044710 red 23.78 ± 2.90 1342198851 2455368.4036677782 red 24.42 ± 2.80 1342198851 2455368.4069322422 red 25.07 ± 2.40 1342198851 2455368.4101955500 red 24.99 ± 2.50 1342198851 2455368.4134600144 red 23.13 ± 2.30 1342198851 2455368.4167233212 red 20.94 ± 2.30 1342198851 2455368.4199877856 red 19.00 ± 3.10 1342198851 2455368.4232510934 red 14.31 ± 3.10 1342198851 2455368.4265155573 red 12.37 ± 3.10 1342198851 2455368.4297788651 red9.22 ± 3.20 1342198851 2455368.4330433300 red 11.48 ± 2.30 1342198851 2455368.4363066372 red 11.97 ± 2.60 1342198851 2455368.4395711031 red 14.88 ± 2.30 1342198851 2455368.4428344108 red 17.63 ± 3.10 1342198851 2455368.4460988762 red 16.82 ± 2.60 1342198851 2455368.4493621849 red 19.08 ± 3.50 1342198851 2455368.4526266507 red 18.92 ± 3.40 1342198851 2455368.4558899594 red 21.03 ± 3.60 1342198851 2455368.4591544257 red 20.94 ± 3.80 1342198851 2455368.4624177348 red 20.22 ± 3.40 1342198851 2455368.4656822011 red 21.11 ± 3.60 1342198851 2455368.4689455107 red 22.56 ± 3.30 1342198851 2455368.4722099770 red 23.37 ± 3.10 1342198851 2455368.4754732866 red 20.78 ± 3.20 1342198851 2455368.4787377538 red 19.81 ± 3.40 1342198851 2455368.4820010634 red 16.42 ± 3.90 1342198851 2455368.4852655306 red 17.63 ± 4.30 1342198851 2455368.4885288402 red 17.55 ± 5.00 1342198851 2455368.4917933075 red 17.47 ± 3.40 1342198851 2455368.4950566171 red 18.68 ± 3.60 1342198851 2455368.4983210848 red 19.49 ± 4.20 1342198851 2455368.5015843948 red 17.95 ± 3.20 1342198851 2455368.5048488625 red 15.45 ± 2.60 1342198851 2455368.5081121726 red 18.20 ± 2.70 1342198851 2455368.5113766398 red 16.25 ± 3.00 1342198851 2455368.5146399504 red 16.82 ± 3.20 1342198851 2455368.5179026825 red 17.14 ± 2.90 1342198851 2455368.5211665714 red 19.81 ± 3.00 1342198851 2455368.5244304603 red 19.41 ± 4.30 1342198851 2455368.5276943492 red 18.20 ± 3.50 1342198851 2455368.5309582381 red 17.87 ± 2.80 1342198851 2455368.5341299847 red 15.36 ± 2.80cccc2003 VS_2 thermal time series observations with Herschel/PACS at blue (70 μm) and red (160 μm) bands (some clear outliers have been removed in the table). Each data point in the blue band spans around 23.5 minutes and is the combination of five single images, there is no time overlap between consecutive images. For the red band each data point spans around 47 minutes and is the combination of ten single frames, there is an overlap of around 23.5 minutes between consecutive data points (except for the outliers removed). Thermal light curves in Fig. <ref> are obtained folding these data with the 2003 VS_2 rotational period. OBSID is the Herschel internal observation ID, JD are the julian dates at the middle of the integration uncorrected for light-time (the mean one-way light-time is 306.228046 min), Band are the different filters (blue or red) used to observe with PACS, Flux/unc are the in band fluxes and 1-σ associated uncertainties expressed in millijansky (mJy), these values must be divided by the factors 0.98 and 1.01 for the blue and red bands, respectively, to obtain color corrected fluxes/uncertainties. The zero time used to fold the rotational light curves in Fig. <ref> is JD = 2452992.768380 days (uncorrected for light-time, the one-way light-time for this date is 296.244149 min). OBSID JDBandFlux/unc [days] [mJy] 4c– 2003 VS_2 thermal time series with Herschel/PACS Continued from previous pageOBSID JDBandFlux/unc [days] [mJy]4cContinued on next page1342202371 2455418.9107945664 blue 12.50 ± 1.801342202371 2455418.9303779081 blue 13.60 ± 1.701342202371 2455418.9466973604 blue 14.40 ± 1.701342202371 2455418.9630168136 blue 15.90 ± 1.501342202371 2455418.9793362664 blue 13.10 ± 1.901342202371 2455418.9956557201 blue 13.20 ± 1.601342202371 2455419.0152390650 blue 13.50 ± 2.001342202371 2455419.0413501919 blue 14.60 ± 1.801342202371 2455419.0576696461 blue 12.90 ± 1.601342202371 2455419.0739890989 blue 15.30 ± 1.801342202371 2455419.0903085498 blue 13.90 ± 1.701342202371 2455419.1098918878 blue 15.80 ± 1.501342202371 2455419.1262113363 blue 14.00 ± 1.701342202371 2455419.1425307849 blue 14.00 ± 1.401342202371 2455419.1588502331 blue 13.30 ± 1.501342202371 2455419.1751696821 blue 12.70 ± 1.801342202371 2455419.1914891317 blue 13.80 ± 1.201342202371 2455419.2078085812 blue 14.20 ± 1.701342202371 2455418.9254817832red8.30 ± 2.501342202371 2455418.9418018144red9.60 ± 1.801342202371 2455418.9581206883red 13.70 ± 1.901342202371 2455418.9744407199red 13.30 ± 1.801342202371 2455418.9907595953red 10.80 ± 3.10 1342202371 2455419.0070796274red9.80 ± 1.401342202371 2455419.0201351903red9.00 ± 2.201342202371 2455419.0364540662red 11.10 ± 3.101342202371 2455419.0527740987red 10.20 ± 2.901342202371 2455419.0690929731red 11.70 ± 2.401342202371 2455419.0854130024red 12.00 ± 1.601342202371 2455419.1017318745red 14.50 ± 1.901342202371 2455419.1180519015red 11.50 ± 2.401342202371 2455419.1343707712red 10.00 ± 1.401342202371 2455419.1506907986red 12.00 ± 2.001342202371 2455419.1670096684red 12.20 ± 2.301342202371 2455419.1833296963red 12.60 ± 1.601342202371 2455419.1996485675red 14.70 ± 2.50 cccc2003 AZ_84 thermal time series observations with Herschel/PACS at green (100 μm) band. Each data point spans around 28.2 minutes and is the combination of six single images, there is no time overlap between consecutive data points. Thermal light curve in Fig. <ref> is obtained folding these data with the 2003 AZ_84 rotational period. OBSID are the Herschel internal observation IDs, JD are the julian dates at the middle of the integration uncorrected for light-time (the mean one-way light-time is 379.855579 min), Band is the filter used to observe with PACS, Flux/unc are the in band fluxes and 1-σ associated uncertainties expressed in millijansky (mJy), these values must be divided by the factor 0.98 to obtain color corrected fluxes/uncertainties. The zero time used to fold the light curves in Fig. <ref> is JD = 2453026.546400 days (uncorrected for light-time, the one-way light-time for this date is 373.264731 min).OBSID JDBandFlux/unc [days] [mJy] 4c– 2003 AZ_84 thermal time series with Herschel/PACS Continued from previous pageOBSID JDBandFlux/unc [days] [mJy]4cContinued on next page1342205152 2455466.5342869759 green 24.95 ± 2.17 1342205152 2455466.5538702821 green 27.07 ± 2.26 1342205152 2455466.5734535865 green 26.93 ± 1.98 1342205152 2455466.5930368891 green 28.81 ± 2.05 1342205152 2455466.6126201935 green 24.93 ± 2.40 1342205152 2455466.6224124241 green 27.77 ± 2.49 1342205152 2455466.6419957289 green 26.28 ± 2.43 1342205152 2455466.6615790343 green 29.85 ± 3.05 1342205152 2455466.6811623396 green 24.50 ± 2.76 1342205152 2455466.7007456459 green 26.36 ± 2.33 1342205152 2455466.7203289527 green 31.56 ± 1.71 1342205152 2455466.7399122599 green 27.40 ± 2.75 1342205152 2455466.7594955675 green 27.24 ± 3.43 1342205152 2455466.7790788747 green 26.24 ± 2.44 1342205152 2455466.7969396170 green 27.96 ± 2.76 | http://arxiv.org/abs/1705.09117v1 | {
"authors": [
"P. Santos-Sanz",
"E. Lellouch",
"O. Groussin",
"P. Lacerda",
"T. G. Mueller",
"J. L. Ortiz",
"C. Kiss",
"E. Vilenius",
"J. Stansberry",
"R. Duffard",
"S. Fornasier",
"L. Jorda",
"A. Thirouin"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170525101409",
"title": "TNOs are Cool: a survey of the Transneptunian Region XII. Thermal light curves of Haumea, 2003 VS2 and 2003 AZ84 with Herschel Space Observatory-PACS"
} |
The Coronal Loop Inventory Project: Expanded Analysis and ResultsJ.T. Schmelz1,2,3,G.M. Christian3,R.A. Chastain3 December 30, 2023 ================================================================= The subgraph enumeration problem asks us to find all subgraphs of a target graph that are isomorphic to a given pattern graph. Determining whether even one such isomorphic subgraph exists is -complete—and therefore finding all such subgraphs (if they exist) is a time-consuming task. Subgraph enumeration has applications in many fields, including biochemistry and social networks, and interestingly the fastest algorithms for solving the problem for biochemical inputs are sequential.Since they depend on depth-first tree traversal, an efficient parallelization is far from trivial. Nevertheless, since important applications produce data sets with increasing difficulty, parallelism seems beneficial. We thus present here a shared-memory parallelization ofthe state-of-the-art subgraph enumeration algorithms RI and RI-DS (a variant of RI for dense graphs) by Bonnici et al. [BMC Bioinformatics, 2013]. Our strategy uses work stealing and our implementation demonstrates a significant speedup on real-world biochemical data—despite a highly irregular data access pattern.We also improve RI-DS by pruning the search space better; this further improves the empirical running times compared to the already highly tuned RI-DS. Keywords: subgraph enumeration; subgraph isomorphism; parallel combinatorial search; graph mining; network analysis§ INTRODUCTION Graphs are used in a plethora of fields to model relations or interactions between entities. One frequently occurring (sub)task in graph-based analysis is the subgraph isomorphism problem (SGI). It requires finding a smaller pattern graph G_p in a larger target graph G_t or, equivalently, finding an injection from the nodes of G_p to the nodes of G_t such that the edges of G_p are preserved.The SGI decision problem is -complete <cit.>. Finding all isomorphic subgraphs of G_p in G_t is commonly referred to as the subgraph enumeration (SGE) problem;this is the problem we deal with in this paper. Note that SGE algorithms require exponential time in the worst case, as there may be exponentially many matches to enumerate.Efficient algorithms for SGI exist only for special cases such as planar graphs <cit.>, where a linear time algorithm exists for constant query graph size; this method can further be used to count the number of occurrences of the query graph in linear time. Consequently, the most successful tools in practice for more general graphs are quite time-consuming when one or all exact subgraphs are sought <cit.>. At the same time, the data volumes in common SGI and SGE applications are steadily increasing. Example fields are life science <cit.>,complex network analysis <cit.>,decompilation of computer programs <cit.>,and computer vision <cit.>.A problem related to SGE is motif discovery, where the goal is to find allfrequent subgraphs up to a very small size <cit.>; note that in our problem, queries consist of only one subgraph. Similar to motif search, works dealing with massive target graphs often focus on very small pattern graphs <cit.>, often with only up to 10-20 vertices. In contrast, we enumerate pattern graphs with several dozens of vertices and hundreds of edges.In bioinformatics and related life science fields, graphs are used among other things for analyzing protein-protein interaction networks and finding chemical similarities <cit.>. Such data is often labeled, which further speeds up SGE algorithms by excluding from search those vertex pairs with different labels. The fastest algorithms for SGE on these graphs use a backtracking approach based on depth-first search (DFS) of the search space <cit.>, where often pruning rules are employed to reduce the search space.The fastest such algorithm is RI by Bonnici <cit.>, as evidenced by a recent study by Carletti <cit.>. RI, as well as other state-of-the-art algorithms, are not parallel and thus there is potential for solving large and/or hard instances faster by parallelization.Distributed SGE algorithms exist (for MapReduce <cit.>). However, when the target graph fits into main memory (which is typically the case for the mentioned life science applications), a shared-memory parallelization is much more promising. Parallelizing DFS-based backtracking efficiently is not trivialthough <cit.>. As in our case, this usually stems from a highly irregular data access pattern, which makes load balancing difficult (in particular when enumerating all isomorphic subgraphs).*Contribution Our contribution is twofold. First, we present a shared memory parallelization of subgraph enumeration algorithmsRI and RI-DS (the version of RI for dense graphs) using work stealing with private double-ended queues (see Section <ref>). We conduct a detailed experimental analysis on three data collections, consisting of fifty target graphs and thousands of pattern graphs from the original RI paper <cit.>. On these three data collections we achieve parallel speedups of 5.96, 5.21, and 9.49 with 16 workers on long running instances, respectively (see Section <ref>). The maximum speedup for any single instance is 13.40.Moreover, on one data collection we manage to reduce the number of instances not solved within the time limit of 180 seconds by more than 50%.Second, we introduce an improved version of RI-DS that makes better use of available data by further constraining the search space with minimal overhead (Section <ref>). It decreases search space size and variability considerably and somewhat reduces the average running time on the three data collections.Our combined improvements give considerable performance gains over state-of-the-art implementations for longer running instances:we achieve speedups of 7.75 (RI), 4.37 (RI-DS) and 13.67 (RI-DS), respectively, over the original implementations on the three data collections used in our experiments.§ PRELIMINARIES§.§ Definitions and Notation *Basics A graph G = (V,E) consists of a set of n vertices V and a set of m edges E ⊆ V × V. Unless stated otherwise, we assume all graphs to be directed. We refer to the set of nodes that have an edge starting or ending at v by N(v) ⊆ V or by neighborhood of v. For directed graphs one can also distinguish betweenincoming and outgoing neighbors. In an undirected graph the degree of a node is the number of edges incident to that node. For the directed case we define the indegree of a node v as ^-(v) = |{u | (u,v) ∈ E}|and the outdegree as ^+(v) = |{w | (v, w) ∈ E}|.*Labels Graphs can be annotated with semantic information by adding labels to nodes or edges. Let L_V and L_E be the sets of possible node and edge labels, respectively. We define the node label function lab: V → L_V that associates each node with a label. Similarly, we define the edge label function β: E → L_E.We say two nodes u,v are equivalent and write u ≡ v if lab(u) = lab(v). Two edges e,f are compatible if β(e) = β(f). We assume strict equality for labels but there may beother application-specific equivalence functions. *(Sub)Graph isomorphismTwo graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2) are considered isomorphic, denoted by G_1 ≅ G_2, if there is a bijective function f mapping all nodes of G_1 to nodes of G_2 such that the edges of the graph are preserved and compatible, and nodes in G_1 are mapped onto equivalent nodes in G_2.More formally, G_1 ≅ G_2 if and only if a bijection f: V_1 → V_2 exists that fulfills the following properties. ∀ u,v ∈ V_1: (u,v) ∈ E_1(f(u), f(v)) ∈ E_2 ∀ v ∈ V_1: v ≡ f(v) ∀ (u,v) ∈ E_1: (u,v) ≡ (f(u), f(v)) In subgraph enumeration we must list all (in our case non-induced) subgraphs in a target graph G_t thatare isomorphic to a pattern graph G_p.Clearly, subgraph enumeration is noteasier than subgraph isomorphism. In practice one can even expect a significant slowdown, even when working in parallel. After all, we cannot stop after a potentially early first hit, but have to explore the search space exhaustively. §.§ Related Work§.§.§ Overview According to Carletti <cit.>, most SGI and SGE algorithms can be categorized as follows. State space based A common way to model the SGI/SGE search space is to use a state space representation (SSR) <cit.>, modeling the search space as a tree. Finding a subgraph isomorphism can then be viewed as the problem of finding a mapping M: V_p → V_t that maps nodes of the pattern graph onto nodes of the target graph. If the mapping is injective and does not violate the isomorphism constraints (<ref>) to (<ref>), it yields a subgraph G_s ⊆ G_t isomorphic to G_p. Every node in the tree represents a state of the (partial) mapping; the root represents the empty mapping. A branch taken represents extending the partial mapping by a certain pair of nodes. To find asubgraph isomorphism, one thus needs to find a path in the state space tree of length |V(G_p)| beginning at the root, such that the equivalent mapping does not violate the graph isomorphism constraints. The key to doing that efficiently is ignoring parts of the state space tree early which cannot be extended to a valid solution. This is especially critical when enumerating all isomorphic subgraphs, since otherwise states at depth |V(G_p)| in the state space tree would need to be visited.Typically, state space based approaches use depth first search (DFS) in order to search the state space tree and make use of pruning rules in order toremove parts of the search space that do not contain valid solutions <cit.>.In an independent experimental comparison, Carletti <cit.> concluded that “RI seems to be currently the best algorithm for subgraph isomorphism” for sparsebiochemical graphs. Other popular algorithms of this category besides RI <cit.> are VF2 <cit.> and VF2 Plus <cit.> (VF2 was also part of Carletti 's comparison). One important distinction between the different state space exploration approaches is the order in which the nodes of the pattern graph are processed.Algorithms like VF2 use a dynamic variable ordering: they decide at every state, based on the nodes already mapped onto the target graph, which node of the pattern graph to examine next. This allows them a greater freedom in eliminating unfruitful branches of the search space but comes with the additional cost of running whatever logic is used to pick the next pattern graph node in every step of the search process.Algorithms like RI and VF2 Plus use a static ordering fixed before the search. This reduces the amount of work done during the search process but means that at any step the selection of the next variable may not be optimal regarding search space size <cit.>.Since we base our parallelization on RI, a more detailed explanation of the sequential algorithm is worthwhile. As already mentioned, RI uses state space exploration with a search strategy that is static and depends only on the pattern graph. The general idea is to order the nodes of the pattern graph in a way that ensures the next node visited is always the one being most constrained by already matched nodes, while introducing additional constraints as early as possible. During the search no expensive pruning or inference rules are used, trading faster comparisons for a larger search space. The pattern graph nodes are ordered by starting with highly connected nodes and then greedily adding nodes that are connected with already selected nodes. For the actual search process, a set of increasingly expensive rules are checked for each candidate extension. There is also a version of RI called RI-DS <cit.> that computes an initial list of possible target nodes for all nodes in G_p and is better on medium to large dense graphs <cit.>. RI and RI-DS participated in the ICPR2014 contest on graph matching algorithms for pattern search in biological databases, “outperforming all other methods in terms of running time and memory consumption” <cit.>. Index based Index-based approaches employ an indexing data structure to store features (such as distance in the graph) that can then queried to quickly find a valid extension of a partial mapping. This reduces the time to find an isomorphic subgraph if one exists (called the matching time). Index-based approaches spend more time up front preprocessing the target graph, and are therefore typically used for larger target graphs and very small pattern graphs. Examples include QuickSI <cit.>, GraphQL <cit.>, GADDI <cit.> and STwig <cit.>. Constraint propagation based Modeling the subgraph isomorphism problem as a constraint satisfaction problem (CSP) means that each variable/node in G_p has a domain (a set of candidate nodes in G_t) and a set of constraints that ensure edge preservation and injectivity. Tools in this category focus on reducing the search space at the cost of spending more time to achieve that reduction. The search space reduction is dependent on both the pattern and target graph <cit.>.LAD <cit.> is a CSP approach in which initial domains can be based solely on node degrees, but can also incorporate label compatibility.In addition to the constraints above, LAD uses a constraintwhich results in the removal of values from the domain if using them would imply remaining nodes cannot all be mapped to different nodes. Also, LAD propagates constraints after each assignment to further reduce domains of remaining variables.These groups are not disjoint; LAD for example uses a state space representation with depth first search, but it reduces the search space by using constraint propagation <cit.>.§.§.§ Parallelism Parallel backtracking has been considered in recent years both in a generic manner (<cit.>) and for specific combinatorial search problems such as maximal clique enumeration <cit.>. To the best of our knowledge, there are no parallel versions of the newer state-of-the-art algorithms for subgraph enumeration or isomorphism like RI and VF2 Plus. Some work has been done in order to accelerate the search process using GPUs, with an algorithm named GPUSI, but here the focus is on tiny pattern graphs <cit.>. The work by Shahrivari and Jalili <cit.> is also for shared memory, but targets motif search. Thus, there are a few critical differences: (i) they search for all subgraphs of a certain size k, (ii) this size k is tiny, and (iii) their parallelization approach is simpler due to the problem structure.There is also a CSP based parallel approach for SGI that beats VF2 and LAD on some graphs, but it was not compared to RI nor RI-DS <cit.>. Of the backtracking approaches there is one parallelization of VF2 <cit.> using Cilk++. Its authors note that the amount of state copied to enable work stealing results in a lot of overhead. To remedy this, our parallelization copies partial solutions only for stolen tasks,not those that remain private.RI's static node ordering, and focus on speed of exploration, makes it a good candidate for parallelization. The static order and lack of complex state should allow for reasonable overhead when distributing work among workers. § RIWe begin by discussing several critical features of RI, which we use in our parallelization.RI performs fast backtracking search by choosing vertices for inclusion in the solution by using a static constraint-first variable ordering of pattern graph nodes, and by employing efficient rules for pruning the search space.In particular, RI repeatedly expands a mapping M: G_p → G_t that reflects the current (partial) solution.And the vertex selection order ensures the most constrained nodes are chosen first, and that additional constraints are introduced as soon as possible—which significantly reduce the search space. We denote a (partial) ordering by μ = [μ_0 …μ_|E(G_p)| ] where μ_i ∈ G_p and μ_i μ_j if ij.The nodes of G_p will be inserted into M in this order. The search begins with an empty mapping M. The first pattern node in μ, μ_0, is checked against a node v_t ∈ G_t. This check is performed using a number of pruning rules (see next paragraph) to ascertain whetherexpanding M to include μ_0 → v_t would violate anyof the subgraph isomorphism constraints. If that is not the case M is expanded accordingly and the next node to be processed is μ_1.We can illustrate how search proceeds using the vertex ordering by growing a mapping M: G_p → G_t.RI begins by selecting the node with the highest degree and adding it to μ resulting in μ = [μ_0]. Afterwards, it greedily adds nodes until all nodes of G_p are in μ. Let μ be a partial variable ordering and v_p be a pattern node. We denote by w_m(μ, v_p) the number of neighbors of v_p in μand by w_n(μ, v_p) the number nodes in μ reachable via nodes not in μ. Formally we define w_m(μ, v_p) = |N(v_p) ∪μ| andw_n(μ, v_p) = |{w ∈μ|∃ x ∉μ: {v_p, w}⊆ N(x)}|. For the purpose of the ordering we do not care about the directionality of edges. In each iteration the algorithm picks the unprocessed node with the most neighbors in μ, that is the one with the highest value w_m. This is the fail-first principle in action: picking the most constrained node first. If multiple nodes have the same w_m, ties are broken first by the highest number of nodes in μ that are reachable via nodes not yet in μ, then by highest degree.[This differs from the description in <cit.>, but matches the implementation of RI 3.6.] That is the one with the highest w_n is picked—maximizing the number of further constraints applied.During the construction process the algorithm also keeps track of, for each node, the first node in the ordering from which it can be reached. The resulting parent mapping P is used in the depth first search to select candidate nodes for a given pattern node.At any point in the search process, when processing μ_i, the candidate node v_t ∈ G_t is one of the neighbors of M(P[μ_i]) in G_t.By definition, P(μ_i) is the first node in μ from which μ_i can be reached in G_p. This means that if constraint (<ref>) holds M(μ_i) will have to be reachable from M(P(μ_i)) in G_t.For the first node, and nodes not connected to already mapped nodes in M, all nodes of G_t are candidates.If none of the pruning rules are violated the mapping is expanded so that M(μ_i) = v_t. Whenever the mapping reaches the size of V(G_p) it induces a valid subgraph and a match is reported. §.§.§ Pruning rulesThe rules for pruning infeasible paths are kept very simple in order to minimize the time spent exploring a single state. There are four rules and they are executed in order, from cheapest to most costly and evaluation stops as soon as one fails. For a mapping M, a pattern node μ_i and a target node v_t they ensure the following. * v_t is not used in the mapping M.* μ_i ≡ v_t verifying constraint (<ref>).* Indegree and outdegree of μ_i are not bigger than indegree and outdegree of v_t.* Constraint (<ref>), that is the structure of G_p, is preserved and the edge labels are compatible as per constraint (<ref>).§ PARALLELIZING RIPerhaps the biggest challenge for fine-grained parallelism is to keep workers from starving. For subgraph isomorphism and enumeration, starvation is even more of a concern, as the search space is highly irregular <cit.>—few long overlapping search paths exist that lead to solutions. However, there is another challenge: assuming we can find enough tasks for all workers, we need to efficiently communicate tasks, which include a potentially large mapping M. We address both of these challenges by combining the work stealing strategy of Acar <cit.> and task coalescing. §.§ Task RepresentationThe search phase of RI iteratively expands a mapping M of nodes from G_p onto nodes from G_t by choosing vertices according to a static ordering μ = [μ_1,…,μ_V(G_P)] of the pattern nodes. A natural way to represent a task then would be as a tuple (M, μ_i, v_t) with the current partial mapping M, and a check to perform: can we map pattern node μ_i to target node v_t? If the new mapping is consistent, then the task spawns new tasks (M^', μ_i+1, v_c) to check if the next pattern node μ_i+1 maps to each remaining candidate target node v_c. While this seems like the ideal representation, it has serious drawbacks. First, the mapping M is potentially large, as it must include every mapped node of G_p. Second, when executing a task, a worker explores only one state in the search space. Thus, this task representation comes with large overhead: copying M for each new task is too time consuming.For that reason, we design our tasks with several built-in optimizations. First, we do not explicitly store a partial mapping with a task, we instead communicate partial mappings as needed. Therefore, we effectively represent a task by the node pair (μ_i, v_t). Second, when expanding a mapping M to include pattern node μ_i and target node v_t, we first check the consistency of each new task before spawning it. This reduces the risk of a worker stealing a dead-end task.§.§ Work stealing with private dequesOur solution is to use the work stealing strategy of Acar <cit.>, which gives each worker i a private double-ended queue (deque) q_i. Worker i adds and removes from the front of its private deque q_i, and workers request to steal from the back of other workers' deques. The private deque serves two purposes for our problem: (i) Tasks at the front of worker i's deque are added and removed in depth-first search order, and therefore worker i always has a correct partial mapping M_i. Thus, for private tasks, a partial mapping is never copied.(ii) Tasks at the back of a worker's deque are closer to the root of the search space tree than those near the front. Therefore, when stealing, a worker receives a task with larger part of the tree below them than nodes closer to the leaves. Thus, we expect stolen tasks to be relatively long-running, reducing the number of steals overall. An example of workers stealing and executing tasks can be seen in <ref>.We quickly point out that an alternative solution would be to use lock-free data structures, which ensure efficient access to tasks for all workers with low overhead <cit.>.However, they have two drawbacks: first they are notoriously difficult to implement <cit.>, and second, they still require copying a partial map with every task. The straightforward methods to address these drawbacks are not ideal.Synchronizing access to the mapping is not a viable option, as the mapping is used in every iteration of the search, and would add high synchronization overhead <cit.>.Another option is to remove the mapping M from the task, which would reduce the task to tuple of nodes (μ_i, v_t). To check if μ_i may be mapped to v_t, a worker would still need access to M. One way to solve this is by creating copies of M at each depth of the search and copying those again when transferring tasks between workers. This is the approach used by Blankstein andGoldstein in their VF2 parallelization <cit.>. A further option is to increase the task size so that a worker explores more states (e.g., one thousand) to combat excessive copying. This would however force us to have less granularity, which is less desirable given the irregular search space <cit.>. While there are implementations in frameworks like OpenMP that allow easy parallelization of task-based problems, they can exhibit poor performance with fine grained tasks <cit.> and their performance characteristics can be hard to understand <cit.>.In order to give us full control over all parameters we choose to implement work stealing ourselves. Many work stealing algorithms rely on lock-free data structures to ensure efficient access to tasks for all workers with low overhead <cit.>. These data structures are however notoriously difficult to implement <cit.>.Load balancing could either be sender or receiver initiated and is performed byall workers explicitly calling communication methods in their work loops. We implement a receiver-initiated private deque work stealing since its performance is comparable to classic work stealing. Aside from a private deque for each worker, this method requires three shared data structures:* —an array of Boolean values, one for each worker, indicating if that worker currently has tasks in its queue.* —an array of worker ids, used by workers to request a task from another worker.* —an array of tasks, used to transfer tasks between workers. The main loop of a worker consists of taking a task from the deque, updating the entry in . It then checks for a work request in , answering that viafrom the back of its queue if possible and concludes by executing the task it took at the start of the loop. Once it runs out of tasks, it repeatedly requests work from a random worker until it receives a task or is terminated. The main loop can be seen in <ref>. For synchronization the arraycontains C++11's s to store the worker ids andis used to ensure only one request is placed for one worker at any time. No other synchronization efforts are required. Functiontransfers partial mappings between workers.This design keeps tasks small and reduces overhead when creating tasks. Except for the [For , we use C++11'sfor each worker id andto ensure only one request is placed for one worker at any time.], all data structures are completely unsynchronized. §.§ Initial work distributionWe initially create tasks corresponding to states directly below the root of the search space tree. Each taskmaps the node μ_1 of the node ordering onto one node of the target graph. At the beginning of the search process, each worker creates an equal number of those tasks and places them in its private deque. §.§ Task coalescingOne further reason to use work stealing with private deques, is the ease with which we can perform task coalescing <cit.>—that is, grouping together tasks into task groups that can each be processed as a single unit of work. In general, grouping together tasks that have short execution time can reduce the number of steals—and the number of times we must copy a worker's partial mapping. This design makes it easy to experiment with the size of task groups to strike a balance between overhead and granularity.§.§ Termination detectionFinally, we implement a termination detection algorithm, since there is no central scheduler, and we do not know the number of tasks in advance.To detect when all workers terminate, we implement a variant of Dijkstra's popular termination algorithm <cit.> described by Schnitger <cit.>: workers are arranged in a ring and when a worker becomes idle, it passes a token to the next worker. Initially the token is colored white; if a worker is busy, it colors the token black. If the token makes it around the ring and is still white, then all workers terminate.The algorithm has termination delay proportional to the number of workers. As we use no more than 16 workers for our shared memory implementation, this simple approach is sufficient. More efficient algorithms exist for systems with many more workers or more complex topologies <cit.>.§ SPEEDING UP RI-DS RI-DS is a variation of RI that is faster on dense graphs <cit.> anddiffers from RI by precomputing sets of compatible nodes for each pattern node. These sets are incorporated into the initial node ordering and consistency checks. Before computing the node ordering, RI-DS first computes for each pattern node v_p a set of compatible target nodes D(v_p) ⊆ G_t called the domain of v_p. This process is called domain assignment.§.§ Domains in RI-DSInitially, the domain of v_p, denoted by D(v_p), is set to all pattern nodes with compatible degrees and equivalent labels. That is, all nodes with in- and outdegree at least that of v_p's, and with labels that match v_p's.We then remove each v_t from D(v_p) whose neighborhood is not consistent with the neighborhood of v_p. For example, let v_t ∈ D(V_p) and suppose we have an edge (v_p, w_p) ∈ E(G_p); then we want at least one w_t ∈ D(w_p) such that (v_p, w_p) ≅ (v_t, w_t) ∈ E(G_t). If no such edge exists, then mapping v_p onto v_t cannot lead to a solution and we can remove v_t from D(v_p).If any domain becomes empty, then there are no isomorphic subgraphs to enumerate. This step is based on arc-consistency (AC) from constraint programming <cit.>.Domains are used in both the preprocessing and search phases. First, when computing the node ordering μ, all pattern nodes with domain size one (called singleton domains) are placed at the beginning of the ordering. Further, when initializing the search, RI-DS uses domains as candidates for the root node of the search space (unlike RI, which considers V(G_t)). Lastly, during the search, to determine if we can map μ_i∈ V(G_p) onto v_t∈ V(G_t), we first check if v_t∈ D(μ_i). §.§ RI-DS with improved tie-breaking and forward checking We integrate two improvements into the preprocessing phase of RI-DS:we use domain size to break ties in the node ordering μ, and we further reduce domain sizes with forward checking. §.§.§ Node orderingBonnici and Giugno <cit.> provide an extensive comparison of node ordering strategies. They show that,while other domain-based orderings occasionally reduce the search space further than RI-DS, RI-DS consistently outperforms other methods.Our goal is to strengthen the node ordering of RI-DS by further using domains, without introducing unacceptable overhead. RI's node ordering (on which RI-DS is based) is constructed by greedily selecting pattern nodes according to the number of neighbors in the partial ordering (denoted by w_m), the number of nodes in the ordering reachable via nodes not in the ordering (denoted w_n), and the degree.We propose to further break ties when two nodes have the same degree, in favor of the node with the smaller domain. That is, when deciding how to order two nodes with identical values w_m and w_n and identical degrees, we select the one with the smaller domain to appear first in the node ordering. This is a continuation of the constraint-first principle: by preferring the node with the smaller domain, we pick the node which is most constrained first.§.§.§ Forward checkingIn forward checking, a concept in constraint programming, once we assign a value to a variable (in our case, mapping a pattern node onto a target node), we place additional restrictions on the remaining unassigned variables. After assigning a variable, we can remove from the domains of all unassigned variables those values that will violate a constraint because of this assignment <cit.>.In RI-DS, assignments take place only during the search phase—not while computing domains. Yet, we observe that pattern nodes with singleton domains can only be assigned to a single target node. Wetherefore perform forward checking for all pattern nodes with singleton domains, aseach pattern node will be assigned to its target node in the future.The constraint we verify is injectivity. For each pattern node with a singleton domain, we remove the target node in that domain from the domains ofall other pattern nodes. We repeat this procedure for any newly introducedsingleton domains. In RI, domains are implemented as bitmasks, which we use to quickly remove singleton domains' contents from all other domains.§.§ Other approaches consideredAside from RI and RI-DS as well as our improved version we alsodid some preliminary experiments with several other techniques hoping that any of them might show good behavior when parallelized. Among the things we tried where domain size and degree based node orderings as well as a parallel implementation of VF2 Plus.Unfortunately none of the orderings showed any real promise and including them all would have led to a pointless duplication of Bonnici and Giugno's excellent paper on variable ordering <cit.>.Our implementation of VF2 Plus did not show promising results either.We implemented VF2 Plus based on the publicly available version of VF2 and the paper in which Carletti <cit.> introduced VF2 Plus. On longer running instances (which we focus on) the performance wasmuch worse than both RI and RI-DS. This is in no way indicative of the potential performance of VF2 Plus and may be solely due to deficiencies in our implementation. This may warrant further investigation, but we focus here on parallelizing RI and RI-DS. § EXPERIMENTAL EVALUATIONWe now give an in-depth experimental evaluation of our SGE algorithms. Our experimental evaluation is split into two parts. In the first one we evaluate the effectiveness of our parallelization. We do so by comparing our parallel RI against our sequential RI implementation and against the original RI implementation (RI version 3.6).In the second part we compare RI-DS (version 3.51) against our new variants with domain size ordering (RI-DS-SI) and with domain size ordering and forward checking (RI-DS-SI-FC). §.§ Data collections We select a subset of the six biochemical data collections tested by Bonnici <cit.> for the original experiments with RI. We focus on data collections with large graphs that contain hard, long-running instances, as we do not expect easy instances to benefit from parallelism.We select three data collections: PDBSv1, on which RI is more efficient than RI-DS, and PPIS32 and GRAEMLIN32 on which RI-DS is more efficient.Properties of these data collections are listed in Table <ref> and described next. *PPI The PPI data collection consists of large, dense protein-protein interaction networks, which have either 32, 64, 28, 256, 512, 1024, 2048, or unique labels; For each label count there is a version with a uniform and one with a normal (Gaussian) distribution. We run our experiments on the variant with 32 normally distributed labels (which we call PPIS32) since it contains the highest number of unsolved instances. PPIS32 has ten target graphs;there are also 420 pattern graphs with 4, 8, 16, 32, 64, 128, and 256 edges,which are are classified as either being dense, semi-dense, or sparse <cit.>.*PDBSv1 The PDBSv1 data collection consists of large, sparse graphs with data from RNA, DNA, and proteins.The 30 target graphs have between 240 and 330067 nodes. There are 1760 pattern graphs have 4, 8, 16, 32, 64 or 128 edges. Note that RI could not solve about half of the 128-edge pattern graphsin the 3-minute time limit of the original experiments <cit.>.*Graemlin The Graemlin data collection consists of medium sized and large graphs from microbial networks. The ten target graphs have between 1081 and 6726 vertices. There are 420 pattern graphs with 4, 8, 16, 32, 64, 128, and 256 edgesthat are grouped into dense, semi-dense and sparse groups. There are different versions of this data collection. One is labeled with unique labels, while the others are labeled with 32, 64, 128, 256, 512, 1024 and 2048 different labels using a uniform distribution. The low label versions especially have a high percentage of unsolved instances and longer running times <cit.>. We use the 32-label version since it is hard and still contains many solvable instances. We refer to it as GRAEMLIN32. §.§ MetricsIn order to gain insight into different facets of the algorithms' behaviorwe want to measure different aspects of the algorithm.*Preprocessing time The time required for creating the node ordering and, in the case of RI-DS, the initial domain assignment.This is in contrast to Bonnici where the domain assignment time is included in the matching time. Because our parallelization only affects the actual search it makes more sense to split the linear processing from the parallel processing part. *Matching time The time required for searching the state space representation. This includes the time to start worker threads but does not include the preprocessing. *Total time The time required for preprocessing an instance and subsequently solving it. This does not include the time required to load the instance from disk. *Search space size The number of pattern node and target node pairs considered during the search process. This shows the effects on search space size of our improvements to RI-DS. *Match count The number of matches found for a certain instance. We compare all match counts against the results of the reference implementation of RI 3.6 and RI-DS 3.51 to ensure the validity of our results. *Steal attempts The number of steal attempts by any individual worker. We record both failed and successful steal attempts. *Peak memory usage The peak memory usage reached while solving an instance.§.§ Experimental platform and results§.§ Experimental setupWe measure the running times with C++11's .Memory consumption is recorded using GNU time[https://www.gnu.org/software/time/]. Search space size and steal attempts are recorded by the workers and reported at the end of a run. We run our experiments on a dedicated machine with 256 GB RAM and 2×8 Intel® Xeon® E52680 cores, runningopenSUSE 13.1 with Linux version 3.12.6252 (x86_64). Our code was compiled using GCC 4.8.1 (gcc-4_8-branch revision 202388) and optimization flag . We use TCMALLOC[] for memory allocation andfor threading. §.§ Experimental resultsIn the detailed comparison of the algorithms below,we are chiefly concerned with the matching time of the algorithms—the time it takes to enumerate all isomorphic subgraphs.To measure the efficiency of our parallel implementation, we report the speedup of our algorithm as we increase the number of workers.One challenge to measuring the speedup is that the data collections contain many more short running instances than long running instances. Since our goal is to solve long running instances quickly, we must prevent short running instances (which have little speedup) from dominating our measurements. First, we compute the speedup as an arithmetic mean over the total runtime to process all instances of a particular data collection (avg in our tables). We report the standard error of the mean with red bars in our point plots. We further compute the geometric mean (gmean in our tables) of the speedup of each instance. Moreover, we split instances into short and long running groups either by the time required in the original RI/RI-DS implementation or against a single worker of our parallel implementation, depending on the comparison.First we have a few small experiments that motivate different design aspects of our algorithm. Secondly we determine the right parameters for task granularity in the parallel implementation. Thirdly we aim to understand how well our proposed improvements to RI-DS work at reducing search space size. Finally we compare the best version of our implementation against the original RI and RI-DS implementations and analyze how well parallelization worksfor our problem. §.§ Search space parallelismIn <ref> we show the search space of four sample instances (target graphs from PPI are Homo_sapiens.gfd, Drosophila_melanogaster.gfd, Rattus_norvegicus.gfd and Saccaromyces_cerevisiae.gfd).We see for every depth of the search space, the number of large branches, that is branches with at least 5% of the total search space below them. We can see that, while there may at times be limited parallelism, for the majority of the states to be explored at least some parallelism can be found.It should be noted that having phases with just a single large branch followed by an increase in large branches is not necessarily a problem. A single worker may only need to explore a few states to reach a depth where the number of large branches increases.These results were obtained by instrumenting the original implementation of RI to record all states visited during the search process. They can of course only serve to give some intuition as to the structure of the search space.§.§.§ Work stealingWe show the effect of work stealing on our parallelization by measuring the matching time for a sample of PPI with and without work stealing. In <ref>, we see that work stealing reduces the time required to find a solution for 16 workers by a factor of 1.65. Without work stealing, the number of states explored by all workers has a high standard deviation, indicating that work is unevenly distributed. §.§.§ Task coalescingTo measure the effect of task coalescing, we vary the task group size. We experiment on a sample of short running instances of PPIS32, PDBSv1 and GRAEMLIN32 using 2, 4, 8, and 16 workers. We repeat each run 15 times to reduce variability and experiment with task group sizes of 1, 2, 4, 8 and 16 tasks. Results with a single worker are not included, as larger task group sizes obviously reduce overhead in that case.In <ref> we can see that for PDBSv1 a task group size of four is ideal.For GRAEMLIN32 task group sizes oftwo and four yield the best running time for four or more workers.We note that PPIS32 (not shown here) has similar running times for all task group sizes, except that size 16 is much worse.Furthermore, the two rightmost plots in <ref> clearly show that less work stealing takes place for a task group size of two or four.Notice that large task group sizes (16 for GRAEMLIN32 and 8-16 for PDBSv1) have many more steals than smaller task group sizes. This behavior is due to the irregular search space: few states (and few tasks) have much work below them in the state space tree.For large task group sizes, these tasks end up in single task groups, which are stolen by a single worker. If multiple such task groups are held by a single worker, then other workers can only steal small tasks, which finish quickly and lead to more steals.For our remaining experiments, we use task group size four.§.§.§ Performance of parallel RIWe measure the performance of parallel RI on PDBSv1 with 1, 2, 4, 8 and 16 workers.Like Bonnici et al. <cit.> we do not include RI-DS in the comparison on PDBSv1 because it was designed for dense graphs. Likewise we do not test parallel RI on the dense GRAEMLIN32 and PPIS32 sets but rather the versions designed for dense graphs: parallel RI-DS and our improvements. As can be seen in <ref>, multithreading significantly reduces the number of unsolved instances for long running instances. While RI 3.6 fails to solve 90 instances in the time limit, parallel RI leaves only 38 instances unsolved with 16 workers.For our time comparison for parallel RIin <ref>, we only include instances solved by RI 3.6 within the time limit. We do this to prevent skewing the results towards the algorithm that solves the fewest hard instances.Surprisingly, RI 3.6 is slower than parallel RI with a single worker, which we believe is due to subtle implementation differences. Since it is faster, we therefore compare our parallel RI implementation to itself with one worker. However, before this comparison, we briefly mention how parallel RI compares to RI 3.6. On instances solved by RI 3.6 within the time limit, parallel RI with 16 threads beats RI 3.6 by an average factor of 3.07 on short running instances with 4 workers, and a speedup of 7.75 with 16 workers on long running instances. We can see in <ref> that most of the benefits of parallelization occur with up to four workers. This is likely caused by the large number of easy to solve instances in PDBSv1. Of the 1760 query graphs in the PDBSv1 data collection, only 160 have a match time of longer than one second with our single threaded implementation. Since short instances do not benefit much fromparallelism, we only briefly discuss them. <ref> we can see that for short running instances using more than four workers actually results in an increase in match time.parallelism (as can be seen in Table <ref>),on average, using more than four workers increases match time.This is not surprising, as easily solved instanceshave a smaller search space than hard ones, offering little opportunity for improved performance with parallelism. This behavior mirrorsthe slowdown observed by Acar et al. <cit.>—where private deque based work stealing reduced search space parallelism.However, for long running instances, increasing the number of workers always improves the speedup. As seen in Table <ref>, parallel RI reaches an average speedup of 5.96 when running with 16 workers–and even achieves a maximum speedup of 10.01 on some long running instances.This is further corroborated in <ref>, where we see thatthe average match time decreases as we increase the number of workers. Thenumber of unsolved instances decreases similarly, as seen in <ref>. We do, however, note that the number of unsolved instances does not decrease linearly with the number of workers.Since a higher speedup of up to 8.91 was achieved on some instances,we may assume that the limited speedup here is at least in part due to a lack ofparallelism in the search space of most PDBSv1 instances. The overall speedup compared to RI 3.6 is given in Table <ref>.On the short running instances we achieve the best speedup of 3.07 using 4 threads and on long running instances we achieve a speedup of 7.75 using 16 threads. In <ref> we can clearly see that RI 3.6 has much lower memory consumption. The higher consumption of parallel RI compared to RI 3.6 is probably in part due to increased usage of the C++ standard library and larger binary size (the binary size of RI 3.6 is 48,461 bytes compared to 118,544 bytes for the parallel implementation). No efforts were undertaken to lower base memory consumption. We can see that increases in parallelism result in a very moderate increase in memory usage. §.§.§ RI-DS-SI and RI-DS-SI-FC with domain size ordering and forward checking We measure the search space size and total time on PDBSv1, PPIS32 and GRAEMLIN32 with and without our improvements (see Section <ref>) on short running instances with a time limit of one second and on a sample of longer running instances of PPIS32 and GRAEMLIN32. *Search space size In <ref> we give the search space size and total time for domain size ordering (RI-DS-SI) and for domain size ordering with forward checking (RI-DS-SI-FC). Domain size ordering clearly reducessearch space size and total time for all data collections.However, forward checking is less clear. On PDBSv1, forward checking clearly reduces search space size as well as total time.For PPIS32 we see a slight improvement in search space sizebut no change in time, while for GRAEMLIN32 the search space size remains essentially the same. However, in <ref> we can seethat on long running instances of PPIS32, RI-DS-SI clearly reduces thesearch space size, while RI-DS-SI-FC does not seem to affect search space size at all. Yet, on long running instances of GRAEMLIN32, RI-DS-SI-FC shows a clear improvement over RI-DS-SI. From <ref> we can see that for PPIS32,RI-DS-SI and RI-DS-SI-FC have similar total time and match time and both show improvements in total time over RI-DS. On GRAEMLIN32, RI-DS-SI-FC beats RI-DS-SI and RI-DS in total time, albeit not as strongly as in search space.Mean search space on the GRAEMLIN32 sampleis nearly halved from RI-DS to RI-DS-SI-FC, while the mean match time changes from 14.54 (σ = 13.3) to 13.53 (σ = 13.86). This behavior can also be seen in <ref>, where the number of explored states per second drops betweenRI-DS to RI-DS-SI.One possible explanation for these results may be that, while we manage to reducemore search space, we do not change the memory access pattern required for solving an instance.During search, we mustiterate over relatively short adjacency lists, implemented as arrays. Skipping a few entries will not lead to large changes in running time, since the running time is dominated by loading the array into memory. Since RI-DS-SI-FC is at least as fast as RI-DS-SI, and further has a smaller search space for the GRAEMLIN32 data collection (shown in <ref>(a)), we compare RI-DS-SI-FC with RI-DS in our remaining experiments.§.§.§ Parallel RI-DS-SI-FC We compare parallel RI-DS-SI-FC (our improved version of RI-DS) against our own implementation of RI-DS and the original RI-DS implementation, RI-DS 3.51. We measure the performance on PPIS32 and GRAEMLIN32 with 1, 2, 4, 8 and 16 workers.In <ref> and <ref> we show the total time, including the time for domain assignment and preprocessing.To explain the source of limited scalability, we provide not only the speedups for all instances in <ref>; in <ref> we also consider short running instances and long running instances (with less and more than one second total time, respectively) separately.In <ref> and <ref> we see that our parallelization of RI-DS (parallel RI-DS) with one worker is slightly faster than RI-DS 3.51.For short running instances, RI-DS 3.5.1 is faster on GRAEMLIN32 but slower on PPIS32. However, on long running instances, the trend is reversed; RI-DS 3.51 is slightly faster than parallel RI-DS on GRAEMLIN32 and slower on PPIS32. We conclude that these differences are caused by implementation differences, since the search strategy of parallel RI-DS is the same as that of RI-DS 3.51.As for the performance of our improved version RI-DS-SI-FC, <ref> shows a reduction in total time of about 38% on GRAEMLIN32 with the mean total time improving from 4.50 seconds (σ = 19.25) to 3.26 seconds (σ = 14.16). Looking at <ref>, the improvement is especially pronounced on long running instances. For PPIS32 the situation is reversed. There are some improvements, but in contrast to GRAEMLIN32 on PPIS32 the numbers are improved for the short running instances, not the long running ones. This results in a (small) improvement of about 2% for the overall data collection with RI-DS having a mean total time of 6.04 seconds (σ = 20.45) and RI-DS-SI-FC having a mean total time of 5.93 seconds (σ = 20.44).The reduction in search space size seen in <ref>reflects this as well. The mean search space size for long running PPIS32 instances is small, while for the long running GRAEMLIN32 instances the search space size is reduced to less than a third. Notably, for both data collections, RI-DS-SI-FC reduces variability in search space size. Overall these results reflect the trends in <ref> where speedups for GRAEMLIN32 on long running instances are much stronger than those for PPIS32.As an aside, in Appendix <ref> we also give time and search space size for RI-DS-SI-FC on PDBSv1 but as RI-DS is weaker than RI on sparse graphs <cit.> we do not include it here. Still it should be noted that RI-DS-SI-FC leads to noticeable improvements in search space size and total time (about 42%). RI-DS-SI-FC even beats RI on the long running instances of PDBSv1. On short running instances however RI remains much stronger.As can be seen in Table <ref>, speedup is greater on GRAEMLIN32 than PPIS32. The average speedup with 16 threads on long running instances for GRAEMLIN32 is 9.49 while it is 5.21 for PPIS32. Still, both speedups are higher than the mean speedup for parallel RI on PDBSv1 in Table <ref>. The differences between data collections suggest that inherent parallelism of the instance may be crucial for achieving high speedup. Also, the maximum speedup of 13.40 on GRAEMLIN32 shows that, for some inputs, our implementation is capable of speedups close to the optimum.Interestingly, while the speedup on the denser PPIS32 and GRAEMLIN32 is higher than the speedup on PDBSv1, the number of unsolved instancesin <ref>does not improve as much. The number of unsolved instances in PPIS32 drops from 212 with RI-DS 3.51 to 197 with RI-DS-SI-FC and 16 threads and from 171 to 158 on GRAEMLIN32 respectively—suggesting that GRAEMLIN32 and PPIS32 contain many difficult instances.The overall speedup compared to RI-DS 3.51 can be seen in Table <ref>.We achieve the best speedup for short and long running instances of GRAEMLIN32 and PPIS32 with 16 threads. For GRAEMLIN32 we achieve speedups of 6.67 onshort running instances and 13.67 on long running instances. With PPIS32 we have stronger speedup on short running instances. We achieve a speedups of 5.12 on short running instances and 4.37 on long running instances of PPIS32. Finally, we briefly summarize how our parallel implementation compares to RI-DS 3.51. We achieve the best speedups for instances of GRAEMLIN32 and PPIS32 with 16 threads. For GRAEMLIN32, we achieve speedups of 6.67 onshort running instances and 13.67 on long running instances. With PPIS32, we have stronger speedup of 5.12 on short running instances when compared to 4.37 for long running instances. With respect to memory usage, in <ref> we see the same behavior as with parallel RI. Our implementation has a much larger base consumption, mostly due to more extensive use of the C++ standard library and larger binary size. The increase in memory usage with more threads is very moderate and we notice that RI-DS-SI-FC has a slightly lower memory usage than RI-DS.Table <ref> entries are suspicious. § CONCLUSIONSIn this paper we have presented a shared-memory parallelization forthe state-of-the-art subgraph enumeration algorithms RI and RI-DS.Besides a parallelization based on work stealing, we also contribute improved search state pruning using techniques from constraint programming.On the long running instances of the biochemical data collections we used in our experiments, we achieve notable (although not perfect) speedups, considering the highly irregular data access. We conclude that parallelization often yields a significant acceleration of state space representation based subgraph isomorphism/enumeration algorithms, individual results depend on the input being considered.Future work should thus address a dynamic strategy for determining the optimal level of parallelism during the search process.§ ACKNOWLEDGMENTThis work was partially supported by DFG grant ME 3619/3-1 within Priority Programme 1736.abbrv | http://arxiv.org/abs/1705.09358v1 | {
"authors": [
"Raphael Kimmig",
"Henning Meyerhenke",
"Darren Strash"
],
"categories": [
"cs.DC",
"cs.DS",
"F.2.2; G.2.2"
],
"primary_category": "cs.DC",
"published": "20170525205248",
"title": "Shared Memory Parallel Subgraph Enumeration"
} |
89.75.Hc, 64.60.ah, [email protected] Department of Theoretical Physics,Budapest University of Technology and Economics,H-1111 Budafoki út 8., Budapest, [email protected] Center for Network Science, Central European University, H-1051 Nádor u. 9., Budapest, Hungary Department of Theoretical Physics,Budapest University of Technology and Economics,H-1111 Budafoki út 8., Budapest, Hungary Cascading failures may lead to dramatic collapse in interdependent networks, where the breakdown takes place as a discontinuity of the order parameter. In the cascading failure (CF) model with healing there is a control parameter which at some value suppresses the discontinuity of the order parameter. However, up to this value of the healing parameter the breakdown is a hybrid transition, meaning that, besides this first order character, the transition shows scaling too. In this paper we investigate the question of universality related to the scaling behavior. Recently we showed that the hybrid phase transition in the original CF model has two sets of exponents describing respectively the order parameter and the cascade statistics, which are connected by a scaling law. In the CF model with healing we measure these exponents as a function of the healing parameter. We find two universality classes: In the wide range below the critical healing value the exponents agree with those of the original model, while above this value the model displays trivial scaling meaning that fluctuations follow the central limit theorem.Universality and scaling laws in the cascading failure model with healing János Kertész December 30, 2023 ========================================================================= § INTRODUCTIONCoupled infrastructural networks are extremely vulnerable to cascading failures <cit.>. Buldyrev et al. introduced the concept of interdependent networks <cit.> in order to elucidate the mechanism behind this observation. This cascading failure (CF) model consists of two layers of networks, and, in addition to the intra-layer connectivity links, it introduces so-called dependency links that model the need for resources or services coming from the other the layer.This model exhibits rich behavior, among others it shows a hybrid phase transition (HPT), where the order parameter has both a jump and critical scaling. The CF model has been extended in several ways, e.g. dependency links were limited to finite range <cit.> to capture the cost one must pay for long-range connections, multiple layers were considered <cit.> to model a variety of interconnected infrastructures and partial dependence <cit.> allowed some nodes to be autonomous on some resources. These earlier works focused only on the necessary conditions for the breakdown but now the repairing of interdependent networks is getting into focus too.Some repairing strategies involve the random recovery of a portion of the existing infrastructures. This has been modelled in a probabilistic cellular automaton for interdependent networks <cit.> and in the original CF model <cit.>. These strategies assume that the original components and links are repairable and they act respectively on the two layers in a mean-field way and at the mutual boundary of the still functional component. Crisis situations, in contrast, where the original components cannot be recovered, are better described by the dynamic reorganization of existing components. A dynamical stochastic healing rule was introduced <cit.> that describes the efforts spent on repairing the network via new links on longer timescales.Although the original model exhibits HPT, it was shown that control parameters, such as the range of dependency links or the healing probability, allows for eliminating the jump in the order parameter, i.e., changing the transition from hybrid to second order <cit.>.Recently it was found that the hybrid transition can be characterized by two sets of exponents, one for order parameter and another one for the statistics of finite cascades <cit.>. These are related by a scaling law, which connects the exponent of the order parameter to that of the first moment of the cascade size distribution. Calculations were carried out on the square lattice and the Erdős–Rényi network (corresponding to the mean field case). A very efficient algorithm is needed to calculate these critical parameters <cit.>. This is probably the reason, why little effort has been devoted to the problem of universality in such systems. In this paper we study an extension of the original CF model. First, we extend the simulation algorithm to interdependent networks with healing. Next, we identify two universality classes separated the critical healing probability: One bearing a hybrid phase transition and a continuous one described by trivial scaling exponents. Finally, we argue that networks close to the critical healing probability are mixtures of network realizations from below and above the critical healing therefore their behavior is ambiguous and not well characterized by scaling exponents.§ THE CASCADING FAILURE MODELInterdependent networks mutually supply and depend on each other. This is captured by the cascading failure (CF) model <cit.> that is built of two topologically identical starting networks A and B with usual connectivity links. These networks are basically graphs: both of them have the same number of nodes N and the usual edges are termed connectivity links. Here we consider special graphs organized in periodic structure forming square lattice networks with nearest-neighbor connectivity links within each layer.The relationship between the layers is expressed by random dependency links producing a one-to-one mapping of the nodes of the two layers. Dependency link means that if a node fails, its dependent fails too and is their links are removed. Only mutually connected components (MCC) are viable that is nodes belonging to a MCC must be connected via connectivity links in layer A and their dependents in layer B. The nodes of each MCC form a subset of a usual connected component in layer A and, similarly, in layer B too. Due to this restriction, the failure of a single node may result in fragmentation of connected components in a layer that triggers splits in the other layer, and so on, the failure might propagate back and forth between the layers. This iterative process resulting in the MCCs is referred to as failures cascading between the layers.The robustness of the network is studied, as follows. At the beginning one layer is subject to a random attack in which failure is induced externally in 1-p fraction of the original nodes. Then the mutually connected components fulfilling the above restrictions are determined. Finally, the size m of the remaining giant (largest) mutually connected component is measured relative to the original network. The iterative steps of determining the mutually connected components can be viewed as a critical branching process as described in <cit.>. Although this is a simple dynamic interpretation of the original model, in the next section we define a more complex dynamics which takes into account the whole history of the network that tries to adapt and reorganize itself due to the external failures. §.§ CF model with healing The cascading failure model with healing <cit.> is a dynamic version of the CF model in the sense that there is an external source of failures which targets the nodes one-by-one. This is a one-way process and the nodes of network layer A are targeted gradually, in a random order. In each step the next node in the ordering is “targeted” and a successful “attack” is carried out if the node is part of the giant mutually connected component. (The fraction of targeted nodes is 1-q while the fraction of attacked nodes is 1-p, it follows from the definition that 1-q>1-p, therefore p>q. The difference between p and q is ∫_q^1 ((q̃-m(q̃))/q̃) dq̃ as derived in <cit.>.) The unavoidable failure of the targeted node is induced by removing all of its connectivity links. After that the following dynamics is applied to relax the network and form the new mutually connected components with healing links added:* Make a list of the nodes that lost any links. For each node in the list take each pair of its previous neighbors that survived the removal and connect them with independent probability w if there is no link between them and their dependents are in a connected component in the other layer.* Then, in the other network, remove those links of dependent counterparts that run between nodes that are no longer connected.*Remove the links running between nodes whose dependent counterparts are no longer in the same connected component.*Make a list of the nodes that lost any links in since the previous iteration. For each node in the list take each pair of its previous neighbors and propose them as a healing link with independent probability w. Realize the proposed links if they do not exist and their dependents are in the same connected component in the other layer. *Repeat <ref> to <ref> on all layers until no more links are removed. The creation of new links in Step <ref> can be interpreted as the effort to find new partners which replace the failed ones. These healing links change the original topology of the network <cit.>. Note that this is an extended version of the original healing algorithm to handle multiple connected components and we show that in the infinite system limit this modification has no significant impact on the measured quantities. After the network is relaxed, one proceeds with the next step in which the next node on the list is targeted. The procedure is pursued until the connectivity links cease to exist. The principal quantity of interest is the order parameter, which is the relative size m of the giant mutually connected component as compared to the original size of the network. For small healing probability w<w_c≈ 0.355 the model exhibits a first-order (hybrid) phase transition while for w>w_c the phase transition is of second-order <cit.>. § SIMULATION METHODPhase transitions are often accompanied by scaling laws but the numerical test of the relevant quantities for the CF model had been a challenging task. Recently, however, efficient algorithms have been developed, that allow for large scale simulation of the mutually connected components (MCC) in the CF model with computational time of O(N^1.2) <cit.>.The implementation of <cit.> is based on the idea that connectivity links only within MCCs are kept and the other ones are inactivated. Consider a node removal (and the removal of its links as a consequence) in the layer A that splits up a component. Then the new MCCs are to be found. The split is propagated to layer B by inactivating all the links that bridged the newly split components in layer B. Of course this might trigger further splitsthat must be propagated back to layer A and so on. This algorithm becomes very efficient using a proper graph data structure.The underlying fully dynamic graph algorithm (see <cit.>) can account for connectivity in a single layer and this accounting is efficient both for adding and removing edges. On each edge removal or inactivation it detects if the component in the layer is split. Then one can query the size of the new components and optionally the members of any of these or even whether two nodes belong to the same component, all of this very efficiently.The application of algorithm <cit.> to the healing problem needs some effort. Notably, the algorithm must integrate the step of adding healing links efficiently which is not obvious. In the following we generalize the algorithm to efficiently simulate interdependent networks with healing.In the cascading failure model only the largest giant mutually connected component (GMCC) is of interest. By deleting links the GMCC and the smaller mutually connected components get fragmented into smaller components. However, adding healing links might save a component from the fragmentation. Therefore the deletions are not to be propagated immediately.While the split of a component is easily propagated, the inverse, notably adding a link between two components in one layer can be computationally expensive. Adding a link requires a search to find which of the previously inactivated links need to be reactivated to find the GMCC. This search would possibly involve many small MCCs. The design challenge lies in rewriting the steps of the previous healing algorithm in such a way that healing links are added within components only. As a consequence it is assured that manipulation avoid reuniting components, i.e., components are always left intact or split. This choice is justified in the following. To fulfill the above constraints we propose an algorithm that buffers the links to be deleted while it adds the healing links before actually deleting or inactivating any links in the given layer, see the Algorithmin the . We find that the runtime of our algorithm scales as O(N^1.3) below w_c and at most as O(N^3) near or above the critical point, see <ref> in the . This runtime cannot be directly compared to that of the wave algorithm proposed in <cit.> because of the following two reasons. First, the runtime O(N⟨ h⟩)of the wave algorithm is given for a single external attack. Second, the number of waves (iterative steps) is known analytically to be ⟨ h ⟩∼ O(N^1/3) for ER-networks only <cit.>. Assuming that in most cases the waves need to be fully recalculated after an external failure, the wave algorithm has at least O(N^2) total runtime for w=0 (and exactly O(N^7/3) for ER-networks). In comparison, our algorithm is intended for fast updates at the cost of higher memory use but for the healing model the runtime is the bottleneck.With the generalized algorithm, we can investigate critical properties of the hybrid percolation transition of the CF model with healing thoroughly. We measure various critical exponents including susceptibility and correlation size that were missing in previous studies for the healing-enabled version of the two-dimensional (2D) lattice interdependent networks. § RESULTS ON THE CF MODEL WITH HEALING In the study of the healing we choose a 2D embedding topology with two N=L× L square lattices, both with periodic boundary conditions. Each node in one layer has a one-to-one dependent node in the other layer.The number of externally removed nodes is controlled. We define the control parameter p in the view of the one-by-one removal, i.e., in each time step a still functional random node is externally attacked and removed and time is measured in the number of successful attacks. That is, random nodes are targeted externally and they get disconnected by removing their links. The time is increased if the targeted node belonged to the GMCC. The link removal is eventually followed by a cascade of failures. At the end only the new giant mutually connected component is considered functional. This measurement of time is justified by the fact that smaller MCCs consist of a few nodes only and are often considered not viable <cit.>. The fraction of original nodes attacked externally is denoted by 1-p.We simulated system sizes N = 32^2, 64^2, 128^2, 256^2, 512^2, 1024^2, and 2048^2, with 384 network configurations for the largest system and increasingly more for smaller systems (# configurations ≈ 1.6· 10^9/N). §.§ Critical behavior of the GMCC The order parameter m(p) is defined as the size of the GMCC per node, which shows the typical behavior of the order parameter at a hybrid phase transition:m(p) = {[ 0, forp < p_c; m_0+u (p-p_c)^β_m, forp≥ p_c. ]. The critical values m_0 and q_c have been published <cit.> with great accuracy for the square lattice without healing (w=0). We used this case as a benchmark test. We measured p_c and q_c too and used this case as a benchmark test. We calculated the critical values for various w parameter settings using our algorithm. We check first whether the simulation results are in agreement with previous findings. The critical exponent _m is defined via finite size behavior of the interdependent network [This quantity is different from 1/α introduced in <cit.> as explained in the following. In the case the avalanche is triggered by an initial removal of a fraction of the nodes the exponent 1/α describes the finite size scaling of the avalanche duration (the number of back-and-forth propagation steps). However, in the healing model the avalanches are triggered by repeated single node removals therefore the avalanche dynamics is different. In this case a similar quantity, the exponent _a was introduced but they may eventually differ in value.]as σ∝ N^-1/_m where σ stands for the standard deviation of the critical point p_c. (For sake of simplicity, we incorporated the dimensionality d=2 into the definition _m ≡ν_m d = 2ν_m. Note that for all data points the smallest number of network realizations was 384 therefore the bias introduced due to the number of runs is negligible as demonstrated in the following. We used bootstrapping to measure the mean of the absolute difference between the standard deviation estimated from 384 realizations and the population standard deviation (estimated from all available data). This difference was about 3% of the population standard deviation. Over the interval of the seven system sizes this might introduce 0.01 absolute error in the slope measured on the log–log plots when the average errors are set methodically to get the largest effect. As known, the estimation of the variance from realizations is unbiased if the degrees of freedom is one less than the number of realizations. Therefore this error can be neglected compared to other fluctuations that appear in the numerical values of the exponents.) In case of w=0 the exponent was previously measured _m=2.2± 0.2 <cit.>.For the network with healing the above value for _m persists (see <ref> and <ref> in the ) until approaching w_c. Above w_c it gets stabilized at _m≈2 indicative of trivial scaling behavior in the sense that the standard deviation is inversely proportional to the square root of the number of nodes in accordance with the central limit theorem. Near w_c we could not disclose the true value of _m because of the following reason. Trying to simulate a system with a specific w near w_c results in an ensemble of systems mixing scaling behaviors below and above w_c. This mixing has an extra contribution, in addition to and dominating over the sample variance. The large _m indicates that the “unintentional” mixing part depends less on system size, at least for the system sizes that are accessible even with our efficient algorithm. The extra variance necessarily makes the distribution of the critical point wider as a consequence, it also makes it rather difficult and unreliable to extrapolate the critical point for the infinite system. In fact, the scaling breaks down and the determination of the critical exponents is hampered by the above effect in this regime.The dependence of the order parameter on p and w is shown in <ref>. We define the value of w_c as the smallest w for which we observe a continuous phase transition and we find w_c=0.355. This value agrees well with the w where _m has a sharp maximum. The scaling 1-m(p,w)=a(w)-a(w) m(1-1-p/a(w), 0) that is asymptotically satisfied in the w→ 0 limit <cit.> is confirmed by the new measurements. a(w)=(1-p_c(0)-Δ p_c(w))/(1-p_c(0)) where Δ p_c(w)≡ p_c(w)-p_c(0) ∝ w^δ and the exponent value is δ=1.006±0.009.The critical control parameter value p_c and the size of the breakdown the related jump m_0 of the order parameter (see <ref>) are extrapolated by finite size scaling, see <ref> in the . For w>w_c the size of the breakdown m_0=0 indicating that macroscopic cascades do not occur. During the process when eliminating nodes one by one small cascades may occur; the smallest cascades cease to exist only at w=1, see <ref>. The scaling exponent β_m quantitatively describes the order parameter in the scaling domain of the hybrid phase transition and is defined according to (<ref>) as (m(p)-m_0) ∝ (p-p_c)^β_m. Without healing, the previously obtained and analytically proved β_m = 0.5 <cit.> is reproduced and it holds up to very close to w_c, see <ref>. This value seems to be universal for 0≤ w<w_c.These results suggest that there exist two universality classes. One is characterized by the hybrid transition with β_m=0.5 and _m≈ 2.2. The other universality class has a vanishing giant component at breakdown and is characterized by exponents _m=2 and β_m=1. For the latter, see Figs. <ref> and <ref>, as well as, <ref>. The exponent γ_m is defined by the scaling of the susceptibility χ≡ N(⟨ m^2⟩-⟨ m⟩^2)∝ (p-p_c)^-γ_m. Unfortunately, we were able to calculate the γ_m values only with rather large error bars. They scatter between 1.41± 0.15 and 1.33±0.15 (see <ref>). We conclude that they do not contradict the assumption of universality. Due to lack of data we are unable to measure γ_m in the region w>w_c. Later we will present an argument that the exponent γ_m should be 0 in this region.§.§ Critical behavior of avalanchesThere is another set of critical exponents <cit.>, τ_a, σ_a, γ_a and _a describing the statistics of avalanches. They can be evaluated only for w<w_c where the number of avalanches is sufficient and avalanches are governed by scaling laws. The size s of the avalanche is the number of nodes failing due to a single external attack, in other words, this is the shrinking of the GMCC in one time step. The finite avalanches are those that happen before the breakdown. One can get a reliable statistics of the finite avalanches by simulating only a reasonable number of network realizations as we did. The idea is that in finite systems close enough to p_c all finite avalanches are in accordance with the critical scaling therefore one can use the avalanches in [p_c, p_c+Δ p] for Δ p < σ. We Δ p = 0.25σ because it yields many avalanches and critical behavior is assured.The exponent γ_a is defined with the average size of the finite cascades ⟨ s_finite⟩∝Δ p^-γ_a for which 1-β_m = γ_a holds <cit.>. This relationship is confirmed reasonably well for w≤0.1 where sufficient data is at our disposal, see <ref>. For w>w_c we would need even larger samples then studied to have sufficient statistics. Near w_c, however, the previously described mixing of realizations of network states from both continuous and discontinuous transitions makes our estimate for p_c less reliable. As a consequence the distance p-p_c from the critical point is less reliable too making it difficult to measure critical exponents.[Unfortunately one cannot gain insight by experimenting with p_c to get satisfactory scaling because m_0 is to be determined in parallel making the experimentation prone to errors.] The other exponents are defined as . p_s |_N=∞∝Δ p^-(τ_a-1)/σ_a and . p_s |_Δ p=0∝ N^(τ_a-1)/_aσ_a. The measurements support our hypothesis of a single universality class for avalanche-related exponents below w_c, see <ref>. The avalanche-related exponents are meaningless above w_c therefore we do not analyze them. §.§ Behavior near w=1 We have confirmed numerically that there is a critical value w_c for the healing above which macroscopic cascades disappear and the network tends to get more and more connected. Here we focus on the critical behavior close to w=1 and we prove that β=1 for this case.In a square latticebetween any two nodes U and V there exist initially at least two disjoint paths that only have U and V in common. Whenever we externally remove a node (different from U and V) at least one of those paths remains intact. As a consequence all the remaining nodes remain attached to the giant component. Thanks to the perfect healing (w=1) all possible bridges over the removed node are formed. This way the eventually cut paths between U and V are re-established and again there will be at least two disjoint paths between any two nodes. Correspondingly, in the case of interdependent layers, the removal of one node causes only the removal of its interdependent counterpart and no avalanches are induced: m_0=p_c=0, m≡ p, therefore β_m=1, χ_m≡0 and it does not make any sense to calculate γ_m or the exponents related to avalanches.When w⪅ 1 avalanches might occur but they are rare and small. For example, to initiate a cascade at least a small region of n nodes one needs to get separated from the connected component in one of the layers. To achieve this at least the perimeter of that region must be cut. In two dimensions the length of the perimeter is at least ∝√(n). Cutting means that healing links are not allowed to form bridges. This happens with probability at most (1-w)^√(n). When separated, the propagation of the damage from the initial n nodes may lead to a cascade of size s≥ n. As the dependency links have here unlimited range, the counterparts of the original nodes are far away from each other and it has small probability that their failure will lead to further separation of other components because such a separation must be prepared similarly. That is, for separating a single node in the 2D case at least three other connectivity links are needed to be cut previously without healing. This happens with probability smaller than (1-w)^3. So the typical cascades are of small size and one iteration. This has been confirmed by simulations. This means that, when approaching the critical point p_c≈0, the network is so densely connected that cascades are prevented almost surely so β_m=1 holds also in the vicinity of w=1. It is tempting to conclude that this observation points toward the existence of universality for w > w_c. Assuming that the small avalanches are mostly independent their number per unit cell is determined by the central limit theorem, that is the fluctuation of their number is inversely proportional to the square root of the system size, hence _m≈ 2.§ CONCLUSIONS We have generalized an efficient algorithm to simulate the CF model with healing. This allowed to measure the critical properties of the phase transition as a function of the healing probability. We revealed that below the critical healing probability w_c=0.355 the 2D interdependent network has a hybrid phase transition with both sets of exponents similar to the original model indicating universality. Above the critical healing however, as avalanches get eliminated the network is characterized by trivial scaling exponents in the sense that fluctuations follow the central limit theorem. The scaling relation 1-β_m=γ_a holds reasonably well for w<0.1. This also means that, despite the gradual shift in the critical point due to the change in the healing probability, the critical scaling of the original CF model dominates the effects of the healing in the w∈[0,w_c) range. Above the critical healing however, the healing takes over and avalanches get eliminated. In the w>w_c regime the network is characterized by trivial scaling exponents in the sense that fluctuations follow the central limit theorem. In summary, the healing can suppress the cascades in situations where the actors of the network cannot be repaired, e.g., economic crisis situations, but the critical behavior of the phase transition does change only when the healing is higher than a threshold value.This work was partially supported by H2020 FETPROACT-GSS CIMPLEX Grant No. 641191 § APPENDIX§.§ Algorithm *Let 𝒟_A be the set of edges to be deleted from layer A.*Let ℋ_A be the set of edges proposed as healing edges. This set is built as follows: take all the endpoints of the edges in 𝒟_A. For each node v among the endpoints list all possible pairs of the neighbors of v. Add the edge between each pair to ℋ_A with independent probability w if the edge connects two points whose dependents are in the same component in layer B and the edge is not already in ℋ_A nor does it exist in the network.*Create the edges ℋ_A to the layer A. During the previous step, these edges were not yet added to the layer A on purpose. Adding the edges in parallel with enumerating the nodes in 𝒟_A has unwanted side-effects that consist of nodes explored later encountering more healing links than nodes explored first. We want to avoid this and keep the algorithm independent of the order of enumeration.*Remove all edges in 𝒟_A. Whenever an edge removal splits up a connected component in a into two parts, the edges that run between the parts in layer B are scheduled for deletion, add them to 𝒟_B. (This step is the analogue to immediately inactivating edges in <cit.>.) *If 𝒟_B is not empty, repeat the above steps swapping the roles A↔ B until no more edges are removed.It is clear that the link creation Step <ref> is realized within the component before any deletion involving Step <ref> therefore the efficiency of the underlying dynamic graph algorithm is not degraded. 15 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Buldyrev et al.(2010)Buldyrev, Parshani, Paul, Stanley, and Havlin]Buldyrev2010 author author S. V. Buldyrev, author R. Parshani, author G. Paul, author H. E. Stanley,and author S. Havlin, 10.1038/nature08932 journal journal Naturevolume 464, pages 1025 (year 2010)NoStop [Li et al.(2012)Li, Bashan, Buldyrev, Stanley,and Havlin]Li2012 author author W. Li, author A. 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E volume 90, pages 012803 (year 2014), http://arxiv.org/abs/1211.2330 arXiv:1211.2330 NoStop [Grassberger(2015)]Grassberger2015 author author P. Grassberger, 10.1103/PhysRevE.91.062806 journal journal Phys. Rev. E volume 91, pages 062806 (year 2015)NoStop [Holm et al.(2001)Holm, de Lichtenberg, and Thorup]Holm2001 author author J. Holm, author K. de Lichtenberg,and author M. Thorup, 10.1145/502090.502095 journal journal J. ACM volume 48, pages 723 (year 2001)NoStop [Note1()]Note1 note This quantity is different from 1/α introduced in <cit.> as explained in the following. In the case the avalanche is triggered by an initial removal of a fraction of the nodes the exponent 1/α describes the finite size scaling of the avalanche duration (the number of back-and-forth propagation steps). However, in the healing model the avalanches are triggered by repeated single node removals therefore the avalanche dynamics is different. In this case a similar quantity, the exponent bar016ν_a was introduced but they may eventually differ in value.Stop [Note2()]Note2 note Unfortunately one cannot gain insight by experimenting with p_c to get satisfactory scaling because m_0 is to be determined in parallel making the experimentation prone to errors.Stop | http://arxiv.org/abs/1705.09829v4 | {
"authors": [
"Marcell Stippinger",
"János Kertész"
],
"categories": [
"physics.soc-ph",
"cond-mat.stat-mech",
"cs.SI"
],
"primary_category": "physics.soc-ph",
"published": "20170527152242",
"title": "Universality and scaling laws in the cascading failure model with healing"
} |
Term Models of Horn Clauses over Rational Pavelka Predicate Logic [================================================================= Generative adversarial networks (GANs) can implicitly learn rich distributions over images,audio, and data which are hard to model with an explicit likelihood.We present a practicalBayesian formulation for unsupervised and semi-supervised learning with GANs.Within this framework, we use stochastic gradient Hamiltonian Monte Carlo to marginalize theweights of the generator and discriminator networks.The resulting approach is straightforwardand obtains good performance without any standard interventions such asfeature matching or mini-batch discrimination.By exploring an expressive posterior over the parameters of the generator, the Bayesian GAN avoids mode-collapse, produces interpretable and diverse candidate samples, and provides state-of-the-artquantitative results for semi-supervised learning on benchmarks including SVHN, CelebA, and CIFAR-10, outperforming DCGAN, Wasserstein GANs, and DCGAN ensembles.§ INTRODUCTIONLearning a good generative model for high-dimensional natural signals, such as images, video and audio has long been one of the key milestones of machine learning. Powered by the learning capabilities of deep neural networks, generative adversarial networks (GANs) <cit.> and variational autoencoders <cit.>have brought the field closer to attaining this goal.GANs transform white noise through a deep neural network to generate candidate samplesfrom a data distribution.A discriminator learns, in a supervised manner, how to tune its parameters so as to correctly classify whether a given sample has come from the generator or the true data distribution. Meanwhile, the generator updates its parameters so as to fool the discriminator. As long as the generator has sufficient capacity, it can approximate the CDF inverse-CDF composition required to sample from a data distribution of interest.Since convolutional neural networks by design provide reasonable metrics over images (unlike, for instance, Gaussian likelihoods), GANs using convolutional neural networkscan in turn provide a compelling implicit distribution over images.Although GANs have been highly impactful, their learning objective can lead to mode collapse,where the generator simply memorizes a few training examples to fool the discriminator.This pathologyis reminiscent of maximum likelihood density estimation with Gaussian mixtures: by collapsing the varianceof each component we achieve infinite likelihood and memorize the dataset, which is not useful for a generalizable density estimate.Moreover, a large degree of intervention is required to stabilizeGAN training, including feature matching, label smoothing, and mini-batch discrimination <cit.>. To help alleviate these practical difficulties, recent work has focused on replacing the Jensen-Shannon divergenceimplicit in standard GAN training with alternative metrics, such as f-divergences <cit.> or Wasserstein divergences <cit.>. Much of this work is analogous to introducing various regularizers for maximum likelihood density estimation.But just as it can be difficult to choose the right regularizer, it can also be difficult to decide which divergence we wish to use for GAN training. It is our contention that GANs can be improved by fully probabilistic inference.Indeed, a posterior distribution over the parameters of the generator could be broad and highly multimodal.GAN training, which is based onmini-max optimization, always estimates this whole posterior distribution over the network weights as a pointmass centred on a single mode.Thus even if the generator does not memorize training examples, we would expect samples from the generator to be overly compact relative to samples from the data distribution. Moreover, each mode in the posterior over the network weights could correspond to wildly different generators,each with their own meaningful interpretations.By fully representing the posterior distribution over the parametersof both the generator and discriminator, we can more accurately model the true data distribution. The inferred data distribution can then be used for accurate and highly data-efficient semi-supervised learning.In this paper, we propose a simple Bayesian formulation for end-to-end unsupervised and semi-supervisedlearning with generative adversarial networks.Within this framework, we marginalize the posteriors over theweights of the generator and discriminator using stochastic gradient Hamiltonian Monte Carlo.Weinterpret data samples from the generator, showing exploration across several distinct modes in thegenerator weights.We also show data and iteration efficient learning of the true distribution. We also demonstrate state of the art semi-supervised learning performance on several benchmarks, includingSVHN, MNIST, CIFAR-10, and CelebA.The simplicity of the proposed approach is one of its greatest strengths: inference is straightforward, interpretable, and stable.Indeed all of the experimental resultswere obtained without feature matching or any ad-hoc techniques.We have made code and tutorials available at <https://github.com/andrewgordonwilson/bayesgan>.§ BAYESIAN GANSGiven a dataset 𝒟 = {x^(i)} of variables x^(i)∼ p_data(x^(i)), we wish to estimate p_data(x).We transform white noise z∼ p(z) through a generator G(z; θ_g), parametrized by θ_g, to produce candidate samples from the data distribution.We use a discriminatorD(x; θ_d), parametrized by θ_d, to output the probability that any x comes from the data distribution.Our considerations hold for general G and D, but in practice G and D are often neural networks with weight vectors θ_g and θ_d.By placing distributions over θ_g and θ_d,we induce distributions over an uncountably infinite space of generators and discriminators, corresponding to every possible setting of these weight vectors.The generator now represents a distribution over distributions of data.Sampling from the induced prior distribution over data instances proceeds as follows: (1) Sample θ_g ∼ p(θ_g);(2) Sample z^(1),…,z^(n)∼ p(z); (3) x̃^(j) = G(z^(j); θ_g) ∼ p_generator(x). For posterior inference, we propose unsupervised and semi-supervised formulations in Sec <ref> - <ref>.We note that in an exciting recent pre-print <cit.> briefly mention using a variational approach to marginalize weights in a generative model, as part of a general exposition on hierarchical implicit models (see also <cit.> for a nice theoretical exploration of related topics in graphical model message passing). While promising, our approach has several key differences:(1) our GAN representation is quite different, preserving a clear competition between generator and discriminator; (2) our representation for the posteriors isstraightforward, requires no interventions, provides novel formulations for unsupervised and semi-supervised learning, and has state of the art results on many benchmarks.Conversely, <cit.> is only pursued for fully supervised learning on a few small datasets; (3) we use sampling to explore a full posterior over the weights, whereas <cit.> perform a variational approximation centred on one of the modes of the posterior (and due to the properties of the KL divergence is prone to an overly compact representation of even that mode); (4) we marginalize z in addition to θ_g, θ_d; and (5) the ratio estimation approach in <cit.> limits the size of the neural networks they can use, whereas in our experiments we can use comparably deep networksto maximum likelihood approaches.In the experiments we illustrate the practical value of our formulation.Although the high level concept of a Bayesian GAN has been informally mentioned in various contexts, to the best of our knowledge we present the first detailed treatment of Bayesian GANs, including novel formulations, sampling based inference, and rigorous semi-supervised learning experiments. §.§ Unsupervised LearningTo infer posteriors over θ_g, θ_d, we can iteratively sample from the following conditional posteriors:p(θ_g | z, θ_d)∝(∏_i=1^n_g D( G(z^(i); θ_g); θ_d))p(θ_g | α_g) p(θ_d | z, X, θ_g)∝∏_i=1^n_d D(x^(i); θ_d) ×∏_i=1^n_g (1 - D( G(z^(i); θ_g); θ_d)) × p(θ_d | α_d).p(θ_g | α_g) and p(θ_d | α_d) are priors over the parameters of the generator and discriminator, with hyperparameters α_g and α_d, respectively.n_d and n_g are the numbers of mini-batch samples for the discriminator and generator, respectively.[For mini-batches, one must make sure the likelihood and prior are scaled appropriately.See Appendix <ref>.] We define X = {x^(i)}_i=1^n_d.We can intuitively understand this formulation starting from the generative process for data samples.Suppose we were to sample weights θ_g from the prior p(θ_g | α_g), and then condition on this sample of the weights to form a particular generative neural network. We then sample white noise z from p(z), and transform this noise through the network G(z; θ_g) to generate candidate data samples.The discriminator, conditioned on its weights θ_d, outputs a probability that these candidate samples came from the data distribution.Eq. (<ref>) says that if the discriminator outputs high probabilities, then the posterior p(θ_g | z, θ_d) will increase in a neighbourhood of the sampled setting of θ_g (and hence decrease for other settings).For the posterior over the discriminator weights θ_d, the first two terms of Eq. (<ref>) form a discriminative classification likelihood, labelling samples from the actual data versus the generator as belonging to separate classes.And the last term is the prior on θ_d.Marginalizing the noiseIn prior work, GAN updates are implicitly conditioned on a set of noise samples z.We can instead marginalize z from our posterior updates using simple Monte Carlo:p(θ_g | θ_d)= ∫ p(θ_g, z | θ_d) dz = ∫ p(θ_g | z, θ_d) p(z|θ_d)^=p(z) dz≈1/J_g∑_j=1^J_g p(θ_g | z^(j), θ_d), z^(j)∼ p(z)By following a similar derivation, p(θ_d | θ_g) ≈1/J_d∑_j^J_d p(θ_d | z^(j), X, θ_g), z^(j)∼ p(z).This specific setup has several nice features for Monte Carlo integration. First, p(z) is a white noise distribution from which we can take efficient and exact samples. Secondly, both p(θ_g | z, θ_d) and p(θ_d | z, X, θ_g), when viewed as a function of z, should be reasonably broad over z by construction, since z is used to produce candidate data samples in the generative procedure.Thus each term in the simple Monte Carlo sum typically makes a reasonable contribution to the total marginal posterior estimates. We do note, however, that the approximation will typically be worse for p(θ_d | θ_g) due to the conditioning on a minibatch of data in Equation <ref>. Classical GANs as maximum likelihood Our proposed probabilistic approach is a natural Bayesian generalization of the classical GAN: if one uses uniform priors for θ_g and θ_d, and performs iterative MAP optimization instead of posterior sampling over Eq. (<ref>) and (<ref>), then the local optima will be the same as for Algorithm 1 of <cit.>.We thus sometimes refer to the classical GAN as the ML-GAN.Moreover, even with a flat prior, there is a big difference between Bayesian marginalization over the whole posterior versus approximating this (often broad, multimodal) posterior with a point mass as in MAP optimization (see Figure <ref>, Appendix). Posterior samples By iteratively sampling from p(θ_g | θ_d) and p(θ_d | θ_g) at every step of an epoch one can, in the limit, obtain samples from the approximate posteriors over θ_g and θ_d. Having such samples can be very useful in practice. Indeed, one can use different samples for θ_g to alleviate GAN collapse and generate data samples with an appropriate level of entropy, as well as forming a committee of generators to strengthen the discriminator. The samples for θ_d in turn form a committee of discriminators which amplifies the overall adversarial signal, thereby further improving the unsupervised learning process. Arguably, the most rigorous method to assess the utility of these posterior samples is to examine their effect on semi-supervised learning, which is a focus of ourexperiments in Section <ref>. §.§ Semi-supervised LearningWe extend the proposed probabilistic GAN formalism to semi-supervised learning. In the semi-supervised setting for K-class classification, we have access to a set of n unlabelledobservations, {x^(i)}, as well as a (typically much smaller) set of n_s observations, {(x_s^(i), y_s^(i))}_i=1^N_s, with classlabels y_s^(i)∈{1, …, K}.Our goal is to jointly learn statistical structure from both the unlabelled and labelledexamples, in order to make much better predictions of class labels for new test examples x_* than if we only had access to the labelled training inputs.In this context, we redefine the discriminator such that D(x^(i) = y^(i); θ_d) gives the probability that sample x^(i) belongs to class y^(i).We reserve the class label 0 to indicate that a data sample is the output of the generator. We then infer the posterior over theweightsas follows:p(θ_g | z, θ_d)∝(∏_i=1^n_g∑_y=1^K D( G(z^(i); θ_g) = y; θ_d))p(θ_g | α_g) p(θ_d | z, X, y_s, θ_g)∝∏_i=1^n_d∑_y=1^KD(x^(i) = y; θ_d)∏_i=1^n_g D( G(z^(i); θ_g) = 0; θ_d)∏_i=1^N_s (D( x_s^(i) = y_s^(i); θ_d))p(θ_d | α_d)During every iteration we use n_g samples from the generator, n_d unlabeled samples, and all of the N_slabeled samples, where typically N_s≪ n.As in Section <ref>, we can approximatelymarginalize z using simple Monte Carlo sampling.Much like in the unsupervised learning case, we can marginalize the posteriors over θ_g and θ_d. To compute the predictive distribution for a class label y_* at a test input x_* we use a model average over all collected samples with respect to the posterior over θ_d:p(y_* | x_*, 𝒟) = ∫ p(y_* | x_*, θ_d)p(θ_d | 𝒟) dθ_d ≈1/T∑_k=1^T p(y_* | x_*, θ_d^(k)) , θ_d^(k)∼ p(θ_d | 𝒟).We will see that this model average is effective for boosting semi-supervised learning performance. In Section <ref> we present an approach to MCMC sampling from the posteriors over θ_g and θ_d. § POSTERIOR SAMPLING WITH STOCHASTIC GRADIENT HMC In the Bayesian GAN, we wish to marginalize the posterior distributions over the generator and discriminator weights, for unsupervised learning in <ref> and semi-supervised learning in <ref>.For this purpose, we use Stochastic GradientHamiltonian Monte Carlo (SGHMC) <cit.> for posterior sampling. The reason for this choice is three-fold: (1) SGHMC is very closely related to momentum-based SGD, which we know empirically works well for GAN training; (2) we can import parameter settings (such as learning rates and momentumterms) from SGD directly into SGHMC; and most importantly, (3) many of the practical benefits of a Bayesian approach to GAN inference come from exploring a rich multimodaldistribution over the weights θ_g of the generator, which is enabled by SGHMC. Alternatives, such as variational approximations, will typically centre their mass around a single mode, and thus providea unimodal and overly compact representation for the distribution, due to asymmetric biases of the KL-divergence. The posteriors in Equations <ref> and <ref> are both amenable to HMC techniques as we can compute the gradients of the loss with respect to the parameters we are sampling. SGHMC extends HMC to the case where we use noisy estimates of such gradients in a manner which guarantees mixing in the limit of a large number of minibatches.For a detailed review of SGHMC, please see <cit.>.Using the update rules from Eq. (15) in <cit.>,we propose to sample from the posteriors over the generator and discriminator weights as in Algorithm <ref>. Note that Algorithm <ref> outlines standard momentum-based SGHMC: in practice, we found it help to speed up the “burn-in” process by replacing the SGD part of this algorithm with Adam for the first few thousand iterations, after which we revert back to momentum-based SGHMC.As suggested in Appendix G of <cit.>, we employed a learning rate schedule which decayed according to γ / d where d is set to the number of unique “real” datapoints seen so far. Thus, our learning rate schedule converges to γ / N in the limit, where we have defined N = |𝒟|.§ EXPERIMENTSWe evaluate our proposed Bayesian GAN (henceforth titled BayesGAN) on six benchmarks (synthetic, MNIST, CIFAR-10, SVHN, and CelebA) each with four different numbers of labelled examples.We consider multiple alternatives, including: the DCGAN <cit.>, the recent Wasserstein GAN (W-DCGAN) <cit.>, an ensemble of ten DCGANs (DCGAN-10) which are formed by 10 random subsets 80% the size of the training set, and a fully supervised convolutional neural network. We also compare to the reported MNIST result for the LFVI-GAN, briefly mentioned in a recent pre-print <cit.>, where they use fully supervised modelling on the whole dataset with a variational approximation.We interpret many of the results from MNIST in detail in Section <ref>, and find that these observations carry forward to our CIFAR-10, SVHN, and CelebA experiments.For all real experiments we use a 5-layer Bayesian deconvolutional GAN (BayesGAN) for the generative model G(z | θ_g) (see <cit.> for further details about structure). The corresponding discriminator is a 5-layer 2-class DCGAN for the unsupervised GAN and a 5-layer, K+1 class DCGAN for a semi-supervised GAN performing classification over K classes. The connectivity structure of the unsupervised and semi-supervised DCGANs were the same as for the BayesGAN.Note that the structure of the networks in <cit.> are slightly different from <cit.> (e.g. there are 4 hidden layers and fewer filters per layer), so one cannot directly compare the results here with those in <cit.>. Our exact architecture specification is also given in our codebase.The performance of all methods could be improved through further calibrating architecture design for each individual benchmark.For the Bayesian GAN we place a 𝒩(0, 10I) prior on both the generator and discriminator weights and approximately integrate out z using simple Monte Carlo samples. We run Algorithm <ref> for 5000 iterations and then collect weight samples every 1000 iterations and record out-of-sample predictive accuracy using Bayesian model averaging (see Eq. <ref>).For Algorithm 1 we set J_g = 10, J_d = 1, M=2, and n_d = n_g = 64. All experiments were performed on a single TitanX GPU for consistency, but BayesGAN and DCGAN-10 could be sped up to approximately the same runtime as DCGAN through multi-GPU parallelization.In Table <ref> we summarize the semi-supervised results, where we see consistently improved performance over the alternatives. All runs are averaged over 10 random subsets of labeled examples for estimating error bars on performance (Table <ref> shows mean and 2 standard deviations).We also qualitatively illustrate the ability for the Bayesian GAN to produce complementary sets of data samples, corresponding to different representations of the generator produced by sampling from the posterior over the generator weights (Figures <ref>, <ref>, <ref>).The supplement also contains additional plots of accuracy per epoch and accuracy vs runtime for semi-supervised experiments. We emphasize that all of the alternatives required the special techniques described in <cit.> such as mini-batch discrimination, whereas the proposed Bayesian GAN needed none of these techniques. §.§ Synthetic DatasetWe present experiments on a multi-modal synthetic dataset to test the ability to infer a multi-modal posterior p(θ_g|𝒟). This ability not only helps avoid the collapse of the generator to a couple training examples, an instance of overfitting in regular GAN training, but also allows one to explore a set of generators with different complementary properties, harmonizing together to encapsulate a rich data distribution. We generate D-dimensional synthetic data as follows:z ∼𝒩(0, 10*I_d), A∼𝒩(0, I_D × d) , x= Az + ϵ, ϵ∼𝒩(0, 0.01*I_D),d ≪ D We then fit both a regular GAN and a Bayesian GAN to such a dataset with D = 100 and d = 2. The generator for both models is a two-layer neural network: 10-1000-100, fully connected, with ReLU activations. We set the dimensionality of z to be 10 in order for the DCGAN to converge (it does not converge when d=2, despite the inherent dimensionality being 2!). Consistently, the discriminator network has the following structure: 100-1000-1, fully-connected, ReLU activations.For this dataset we place an 𝒩(0, I) prior on the weights of the Bayesian GAN and approximately integrate out z using J=100 Monte-Carlo samples.Figure <ref> shows that the Bayesian GAN does a much better job qualitatively in generating samples (for which we show the first two principal components), and quantitatively in terms of Jensen-Shannon divergence (JSD) to the true distribution (determined through kernel density estimates). In fact, the DCGAN (labelled ML GAN as per Section <ref>) begins to eventually increase in testing JSD after a certain number of training iterations, which is reminiscent of over-fitting.When D=500, we still see good performance with the Bayesian GAN.We also see, with multidimensional scaling <cit.>, that samples from the posterior over Bayesian generator weights clearly form multiple distinct clusters, indicating that the SGHMC sampling is exploring multiple distinct modes, thus capturing multimodality in weight space as well as in data space. §.§ MNISTMNIST is a well-understood benchmark dataset consisting of 60k (50k train, 10k test) labeled images of hand-written digits. <cit.> showed excellent out-of-sample performance using only a small number of labeled inputs, convincingly demonstrating the importance of good generative modelling for semi-supervised learning. Here, we follow their experimental setup for MNIST. We evaluate the Bayesian DCGAN for semi-supervised learning using N_s = {20, 50, 100, 200} labelled training examples. We see in Table 1 that the Bayesian GAN has improved accuracy over the DCGAN, the Wasserstein GAN, and even an ensemble of 10 DCGANs! Moreover, it is remarkable that the Bayesian GAN with only 100 labelled training examples (0.2% of the training data) is able to achieve 99.3% testing accuracy, which is comparable with a state of the art fully supervised method using all 50,000 training examples!We show a fully supervised model using n_s samples to generally highlight the practical utility of semi-supervised learning.Moreover, <cit.> showed that a fully supervised LFVI-GAN, on the whole MNIST training set (50,000 labelled examples) produces 140 classification errors – almost twice the error of our proposed Bayesian GAN approach using only n_s = 100 (0.2%) labelled examples! We suspect this difference largely comes from (1) the simple practical formulation of the Bayesian GAN in Section <ref>, (2) marginalizing z via simple Monte Carlo, and (3) exploring a broad multimodal posterior distribution over the generator weights with SGHMC with our approach versus a variational approximation (prone to over-compact representations) centred on a single mode.We can also see qualitative differences in the unsupervised data samples from our Bayesian DCGAN and the standard DCGAN in Figure <ref>. The top row shows sample images produced from six generators produced from six samples over the posterior of the generator weights, and the bottom row shows sample data images from a DCGAN. We can see that each of the six panels in the top row have qualitative differences, almost as if a different person were writing the digits in each panel.Panel 1 (top left), for example, is quite crisp, while panel 3 is fairly thick, and panel 6 (top right) has thin and fainter strokes.In other words, the Bayesian GAN is learning different complementary generative hypotheses to explain the data.By contrast, all of the data samples on the bottom row from the DCGAN are homogenous.In effect, each posterior weight sample in the Bayesian GAN corresponds to a different style, while in the standard DCGAN the style is fixed.This difference is further illustrated for all datasets in Figure <ref> (supplement).Figure <ref> (supplement) also further emphasizes the utility of Bayesian marginalization versus optimization, even with vague priors.However, we do not necessarily expect high fidelity images from any arbitrary generator sampled from the posterior over generators; in fact, such a generator would probably have less posterior probability than the DCGAN, which we show in Section <ref> can be viewed as a maximum likelihood analogue of our approach.The advantage in the Bayesian approach comes from representing a whole space of generators alongside their posterior probabilities.Practically speaking, we also stress that for convergence of the maximum-likelihood DCGAN we had to resort to using tricks including minibatch discrimination, feature normalization and the addition of Gaussian noise to each layer of the discriminator. The Bayesian DCGAN needed none of these tricks. This robustness arises from a Gaussian prior over the weights which provides a useful inductive bias, and due to the MCMC sampling procedure which alleviates the risk of collapse and helps explore multiple modes (and uncertainty within each mode).To be balanced, we also stress that in practice the risk of collapse is not fully eliminated – indeed, some samples from p(θ_g | 𝒟) still produce generators that create data samples with too little entropy.In practice, sampling is not immune to becoming trapped in sharply peaked modes.We leave further analysis for future work.§.§ CIFAR-10CIFAR-10 is also a popular benchmark dataset <cit.>, with 50k training and 10k test images, which is harder to model than MNIST since the data are 32x32 RGB images of real objects.Figure <ref> shows datasets produced from four different generators corresponding to samples from the posterior over the generator weights.As with MNIST, we see meaningful qualitative variationbetween the panels.In Table <ref> we also see again (but on this more challenging dataset) that using Bayesian GANs as a generative model within the semi-supervised learning setup significantly decreases test set error over the alternatives, especially when n_s ≪ n.§.§ SVHNThe StreetView House Numbers (SVHN) dataset consists of RGB images of house numbers taken by StreetView vehicles. Unlike MNIST, the digits significantly differ in shape and appearance. The experimental procedure closely followed that for CIFAR-10. There are approximately 75k training and 25k test images. We see in Table <ref> a particularly pronounced difference in performance between BayesGAN and the alternatives. Data samples are shown in Figure <ref>. §.§ CelebAThe large CelebA dataset contains 120k celebrity faces amongst a variety of backgrounds (100k training, 20k test images).To reduce background variations we used a standard face detector <cit.> to crop the faces into a standard 50 × 50 size.Figure <ref> shows data samples from the trained Bayesian GAN. In order to assess performance for semi-supervised learning we createda 32-class classification task by predicting a 5-bit vector indicating whether or not the face: is blond, has glasses, is male, is pale and is young. Table <ref> shows the same pattern of promising performance for CelebA.§ DISCUSSIONBy exploring rich multimodal distributions over the weight parameters of the generator,the Bayesian GAN can capture a diverse set of complementary and interpretablerepresentations of data.We have shown that such representations can enablestate of the art performance on semi-supervised problems, using a simple inference procedure.Effective semi-supervised learning of natural high dimensional data is crucial for reducing the dependency ofdeep learning on large labelled datasets.Often labeling data is not an option, or it comes at a high cost –be it through human labour or through expensive instrumentation (such as LIDAR for autonomous driving). Moreover, semi-supervised learning provides a practical and quantifiable mechanism to benchmark the manyrecent advances in unsupervised learning.Although we use MCMC, in recent years variational approximations have been favoured for inference in Bayesianneural networks.However, the likelihood of a deep neural network can be broad withmany shallow local optima.This is exactly the type of density which is amenable to asampling based approach, which can explore a full posterior. Variational methods, by contrast, typically centre their approximation along a single mode and also provide an overly compact representation of that mode.Therefore in the future we may generally see advantages in following a sampling based approach in Bayesiandeep learning.Aside from sampling, one could try to betteraccommodate the likelihood functions common to deep learning using more general divergencemeasures (for example based on the family of α-divergences) instead of the KL divergencein variational methods, alongside more flexible proposal distributions.In the future, one could also estimate the marginal likelihood of a probabilistic GAN, having integratedaway distributions over the parameters.The marginal likelihood provides a natural utility function forautomatically learning hyperparameters, and for performing principled quantifiablemodel comparison between different GAN architectures.It would also be interesting to consider the Bayesian GANin conjunction with a non-parametric Bayesian deep learning framework, such as deep kernel learning<cit.>.We hope that our work will help inspire continued explorationinto Bayesian deep learning. Acknowledgements We thank Pavel Izmailov for helping to create a tutorial for the codebase and helpful comments, and Soumith Chintala for helpful advice, and NSF IIS-1563887 for support.apalike§ SUPPLEMENTARY MATERIAL In this supplementary material, we provide (1) futher details of the MCMC updates, (2) illustrate a tutorial figure, (3) show data samples from the Bayesian GAN for SVHN, CIFAR-10, and CelebA, and (4) give performance results as a function of iteration and runtime. §.§ Rescaling conditional posteriors to accommodate mini-batchesThe key updates in Algorithm <ref> involve iteratively computing log p(θ_g | z, θ_d) and log p(θ_d | z, X, θ_g), or log p(θ_d | z, X, 𝒟_s, θ_g) for the semi-supervised learning case (where we have defined the supervised dataset of size N_s as 𝒟_s). When Equations (<ref>) and (<ref>) are evaluated on a minibatch of data, it is necessary to scale the likelihood as follows:log p(θ_g | z, θ_d)=(∑_i=1^n_glog D( G(z^(i); θ_g); θ_d))N/n_g + log p(θ_g | α_g) + constantFor example, as the total number of training points N increases, the likelihood should dominate the prior.The re-scaling of the conditional posterior over θ_d, as well as the semi-supervised objectives, follow similarly. §.§ Additional Results | http://arxiv.org/abs/1705.09558v3 | {
"authors": [
"Yunus Saatchi",
"Andrew Gordon Wilson"
],
"categories": [
"stat.ML",
"cs.AI",
"cs.CV",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170526124756",
"title": "Bayesian GAN"
} |
1]Keith Levin 2]Avanti Athreya 2]Minh Tang 3] Vince Lyzinski 2]Youngser Park 2]Carey E. Priebe [1]University of Michigan, Ann Arbor, MI[2]Johns Hopkins University, Baltimore, MD[3]University of Massachusetts, Amherst, MAA Central Limit Theorem for an Omnibus Embedding of Multiple Random Graphs and Implications for Multiscale Network Inference [ December 30, 2023 ============================================================================================================================Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multi-sample graph inference are few. Here we describe an “omnibus" embedding in which multiple graphs on the same vertex set are jointly embedded into a single space with a distinct representation for each graph. We prove a central limit theorem for this embedding and demonstrate how it streamlines graph comparison, obviating the need for pairwise subspace alignments. The omnibus embedding achieves near-optimal inference accuracy when graphs arise from a common distribution and yet retains discriminatory power as a test procedure for the comparison of different graphs. Moreover, this joint embedding and the accompanying central limit theorem are important for answering multiscale graph inference questions, such as the identification of specific subgraphs or vertices responsible for similarity or difference across networks. We illustrate this with a pair of analyses of connectome data derived from dMRI and fMRI scans of human subjects. In particular, we show that this embedding allows the identification of specific brain regions associated with population-level differences. Finally, we sketch how the omnibus embedding can be used to address pressing open problems, both theoretical and practical, in multisample graph inference. Keywords: multiscale graph inference, graph embedding, multiple-graph hypothesis testing§ INTRODUCTIONStatistical inference across multiple graphs is of vital interdisciplinary interest in domains as varied as machine learning, neuroscience, and epidemiology. Inference on random graphs frequently depends on appropriate low-dimensional Euclidean representations of the vertices of these graphs, known as graph embeddings, typically given by spectral decompositions of adjacency or Laplacian matrices <cit.>. Nonetheless, while spectral methods for parametric inference in a single graph are well-studied, multi-sample graph inference is a nascent field. See, for example, the authors' work in <cit.> as among the only principled approaches to two-sample graph testing. What is more, for inference tasks involving multiple graphs—for instance, determining whether two or more graphs on the same vertex set are similar—discerning an optimal simultaneous embedding for all graphs is a challenge: how can such an embedding be structured to both provide estimation accuracy of common parameters when the graphs are similar, but retain discriminatory power when they are different?A flexible, robust embedding procedure to achieve both goals would be of considerable utility in a range of real data applications. For instance, consider the problem of community detection in large networks. While algorithms for community detection abound, relatively few approaches exist to address community classification; that is, to leverage graph structure to successfully establish which subcommunities appear statistically similar or different. In fact, the authors' work in <cit.> represents one of the earliest forays into statistically principled techniques for subgraph classification in hierarchical networks. Not surprisingly, the creation of a graph-statistical analogue of the classical analysis-of-variance F-test, in which a single test procedure would permit the comparison of graphs from multiple populations, is very much an open problem of immediate import. But even given a coherent framework for extracting graph-level differences across multiple populations of graphs, there remains the further complication of replicating this at multiple scales, by isolating—in the spirit of post-hoc tests such as Tukey's studentized range—precisely which subgraphs or vertices in a collection of vertex-matched graphs might be most similar or different. Our goal in this paper, then, is to provide a unified framework to answer the following questions: * Given a collection of random graphs, can we develop a single statistical procedure that accurately estimates common underlying graph parameters, in the case when these parameters are equal across graphs, but also delivers meaningful power for testing when these graph parameters are distinct? * Can we develop an inference procedure that identifies sources of graph similarity or difference at scales ranging from whole-graph to subgraph to vertex? For example, can we identify particular vertices that contribute significantly to statistical differences at the whole-graph level?* Can we develop an inference procedure that scales well to large graphs, addresses graphs that are weighted, directed, or whose edge information is corrupted, and which is amenable to downstream classical statistical methodology for Euclidean data?* Does such a statistical procedure compare favorably to existing state-of-the-art techniques for joint graph estimation and testing, and does it work well on real data?Here, we address each of these open problems with a single embedding procedure. Specifically, we describe an omnibus embedding, in which the adjacency matrices of multiple graphs on the same vertex-matched set are jointly embedded into a single space with a distinct representation for each graph and, indeed, each vertex of each graph. We then prove a central limit theorem for this embedding, a limit theorem similar in spirit to, but requiring a significantly more delicate probabilistic analysis than, the one proved in <cit.>. We show, in both simulated and real data, that the asymptotic normality of these embedded vertices has demonstrable utility. First, the omnibus embedding performs nearly optimally for the recovery of graph parameters when the graphs are from the same distribution, but compares favorably with state-of-the-art hypothesis testing procedures to discern whether graphs are different. Second, the simultaneous embedding into a shared space allows for the comparison of graphs without the need to perform pairwise alignments of the embeddings of different graphs. Third, the asymptotic normality of the omnibus embedding permits the application of a wide array of subsequent Euclidean inference techniques, most notably a multivariate analysis of variance (MANOVA) to isolate statistically significant vertices across several graphs. Thus, the omnibus embedding provides a statistically sound analogue of a post-hoc Tukey test for multisample graph inference. We demonstrate this with an analysis of real data, comparing a collection of magnetic resonance imaging (MRI) scans of human brains to identify dissimilar graphs and then to further pinpoint specific intra-graph features that account for global graph differences. The main theoretical results of this paper are a consistency theorem for the omnibus embedding, akin to <cit.>, and a central limit theorem, akin to <cit.>, for the distribution of any finite collection of rows of this omnibus embedding.We emphasize that distributional results for spectral decompositions of random graphs are few. The classic results of <cit.> describe the eigenvalues of the random graph and the work of <cit.> concerns distributions of eigenvectors of more general random matrices under moment restrictions, but <cit.> and <cit.> are among the only references for central limit theorems for spectral decompositions of adjacency and Laplacian matrices for a class of independent-edge random graphs broader than the model.Our consistency result shows that the omnibus embedding provides consistent estimates of certain unobserved vectors, called latent positions, that are associated to vertices of the graphs. At present, the best available spectral estimates of such latent positions involve averaging across graphs followed by an embedding, resulting in a single set of estimated latent positions, rather than in a distinct set for each graph. We find in simulations, that our omnibus-derived estimates perform competitively with these existing spectral estimates of the latent positions, while still retaining graph-specific information. In addition, we show that the omnibus embedding allows for a test statistic that improves on the state-of-the-art two-sample test procedure presented in <cit.> for determining whether two random dot product graphs <cit.> have the same latent positions.Specifically, <cit.> introduces a test statistic generated by performing a Euclidean, lower-dimensional embedding of the graph adjacency matrix <cit.> of each of the two networks, followed by a Procrustes alignment <cit.> of the two embeddings. Addressing the nonparametric analogue of this question—whether two graphs have the same latent position distribution—is the focus of <cit.>, which uses the embeddings of each graph to estimate associated density functions. The Procrustes alignment required by <cit.> both complicates the test statistic and, empirically, weakens the power of the test (see Section <ref>).Furthermore, it is unclear how to effectively adapt pairwise Procrustes alignments to tests involving more than two graphs. The omnibus embedding allows us to avoid these issues altogether by providing a multiple-graph representation that is well-suited to both latent position estimation and comparative graph inference methods. Our paper is organized as follows. In Sec. <ref>, we give an overview of our main results and present two real-data examples in which the omnibus embedding uncovers important multiscale information in collections of brain networks. In Sec. <ref>, we present background information and formal definitions. In Sec. <ref>, we provide detailed statements of our principal theoretical results. In Sec. <ref>, we present simulation data that illustrates the power of the omnibus embedding as a tool for both estimation and testing. In our Supplementary Material, we provide detailed proofs, including a sharpening of a vertex-exchangeability argument for bounding residual terms in the difference between omnibus estimates and true graph parameter values.We conclude with a discussion of extensions and open problems on multi-sample graph inference.§ SUMMARY OF MAIN RESULTS AND APPLICATIONS TO REAL DATA Recall that our goal is to develop a single spectral embedding technique for multiple-graph samples that (a) estimates common graph parameters, (b) retains discriminatory power for multisample graph hypothesis testing and (c) allows for a principled approach to identifying specific vertices that drive graph similarities or differences. In this section, we give an informal description of the omnibus embedding and a pared-down statement of our central limit theorem for this embedding, keeping notation to a minimum. We then demonstrate immediate payoffs in exploratory data analysis, leaving a more detailed technical descriptions of the method and our results for later sections.To provide a theoretically-principled paradigm for graph inference for stochastically-varying networks, we focus on a particular class of random graphs. We define a graph G to be an ordered pair of (V,ℰ) where V is the vertex or node set, and ℰ, the set of edges, is a subset of the Cartesian product of V × V. In a graph whose vertex set has cardinality n, we will usually represent V as V={1, 2, …, n}, and we say there is an edge between i and j if (i,j)∈ℰ.The adjacency matrixprovides arepresentation of such a graph: _ij=1if (i,j) ∈ℰ, and_ij=0otherwise. Where there is no danger of confusion, we will often refer to a graph G and its adjacency matrix interchangeably. Any model of a stochastic network must describe the probabilistic mechanism of connections between vertices. We focus on a class of latent position random graphs <cit.>,in which every vertex has associated to it a (typically unobserved) latent position, itself an object belonging to some (often Euclidean) space 𝒳.Probabilities of an edge between two vertices i and j, p_ij, are a function κ(·,·): 𝒳×𝒳→ [0,1] (known as the link function) of their associated latent positions (x_i, x_j). Thus p_ij=κ(x_i, x_j), and given these probabilities, the entries _ij of the adjacency matrixare independent Bernoulli random variables with success probabilities p_ij. We consolidate these probabilities into a matrix =(p_ij), and write ∼ to denote this relationship. The latent position graph model has tremendous utility in modeling natural phenomena. For example, individuals in a disease network may have a hidden vector of attributes (prior illness, high-risk occupation) that observers of disease dynamics do not see, but which nevertheless strongly influence the chance that such an individual may become ill or infect others. Because the link function is relatively unrestricted, latent position models can replicate a wide array of graph phenomena <cit.>. In a d-dimensional random dot product graph <cit.>, the latent space is an appropriately-constrained subspace of ℝ^d, and the link function is simply the dot product of the two latent d-dimensional vectors. The invariance of the inner product to orthogonal transformations is a nonidentifiability in the model, so we frequently specify accuracy up to a rotation matrix . Random dot product graphs are often divided into two types: those in which the latent positions are fixed, and those in which the latent positions are themselves random. We will address both cases here: in our theoretical results, the latent positions X_i ∈ℝ^d for vertex i are drawn independently from a common distribution F on ℝ^d; and our practical applications, we consider how to use an omnibus embedding to address the question of equality of potentially non-random latent positions.For the case in which the latent positions are drawn at random from some distribution F, an important graph inference task is the inference ofproperties of F from an observation of the graph alone. In the graph inference setting, there is both randomness in the latent positions, and given these latent positions, a subsequent conditional randomness in the existence of edges between vertices. A key to inference in such models is the initial step of consistently estimating the unobserved X_i's from a spectral decomposition of , and then using these estimates, denoted X̂_i, to infer properties of F. For an RDPG with n vertices, the n× d matrix of latent positionsis formed by taking vector _i associated to vertex i to be the i-th row of . Then =[p_ij], the matrix of probabilities of edges between vertices, is easily expressed as =^T. The aforementioned nonidentifiability is now transparent: ifis orthogonal, then ^T ^T= as well, so the rotated latent positionsgenerate the same the matrix of probabilities. Given such a model, a natural inference task is that of estimating the latent position matrixup to some orthogonal transformation.Because of the assumption that the matrixis of comparatively low rank, random dot product graphs can be analyzed with a number of tools from classical linear algebra, such as singular-value decompositions of their adjacency matrices. Nevertheless, this tractability does not compromise the utility of the model. Random dot product graphs are flexible enough to approximate a wide class of independent-edge random graphs <cit.>, including the stochastic block model <cit.>.Under mild assumptions, the adjacency matrixof a random dot product graph is a rough approximation of the matrix =[p_ij] of edge probabilities in the sense that the spectral norm of - can be controlled; see for example <cit.> and <cit.>. In <cit.>, <cit.> and <cit.>, it is established that, under eigengap assumptions on , a partial spectral decomposition of the adjacency matrix , known as the adjacency spectral embedding (ASE), allows for consistent estimation of the true, unobserved latent positions . That is, if we define =^1/2, whereis the diagonal matrix of the top d eigenvalues of A, sorted by magnitude, and ifare the associated unit eigenvectors, then the rows of this truncated eigendecomposition ofare consistent estimates {_i} of the latent positions {_i}. Of course, these latent positions are often the parameters we wish to estimate. In <cit.>, it is shown that embedding the adjacency matrix and then performing a novel angle-based clustering of the rows is key to decomposing large, hierarchical networks into structurally similar subcommunities. In <cit.>, it is shown that the suitably-scaled eigenvectors of the adjacency matrix converge in distribution to a Gaussian mixture. In this paper, we prove a similar result for an omnibus matrix generated from multiple independent graphs.The ASE provides a consistent estimate for the true latent positions in a random dot product graph up to orthogonal transformations. Hence a Procrustes distance between the adjacency spectral embedding of two graphs on the same vertex set serves as a test statistic for determining whether two random dot product graphs have the same latent positions <cit.>. Specifically, let ^(1) and ^(2) be the adjacency matrices of two random dot product graphs on the same vertex set (with known vertex correspondence), and letandbe their respective adjacency spectral embeddings. If the two graphs have the same generatingmatrices,it is reasonable to surmise that the Procrustes distancemin_∈^d × d -_F,where ·_F denotes the Frobenius norm of a matrix, will be relatively small. In <cit.>, the authors show that a scaled version of the Procrustes distance in (<ref>) provides a valid and consistent test for the equality of latent positions for a pair of random dot product graphs. Unfortunately, the fact that a Procrustes minimization must be performed both complicates the test statistic and compromises its power.Here, we instead consider an embedding of an omnibus matrix, defined as follows. Given two independent d-dimensional RDPG adjacency matrices ^(1) and ^(2), on the same vertex set with known vertex correspondence, the omnibus matrixis given by=[^(1) ^(1)+^(2)/2; ^(1) + ^(2)/2^(2) ],Note that this matrix easily extends to a sequence of graphs ^(1), ⋯, ^(m), where the block diagonal entries are the matrices ^(i) and the (l,k)-th off-diagonal block is the matrix ^(k)+ ^(l)/2.Analogously to our notation for the adjacency spectral embedding , letrepresent the d × d matrix of top d eigenvalues of , ordered again by magnitude, and letbe the mn × d-dimensional matrix of associated eigenvectors. Define the omnibus embedding, denoted OMNI(), by ^1/2.We stress that OMNI() produces m separate points in Euclidean for each graph vertex—effectively, one such point for each copy of the multiple graphs in our sample. This property renders the omnibus embedding useful for all manner of post-hoc inference. If we consider the rows of the omnibus embedding as potential estimates for the latent positions, two immediate questions are arise. Are these estimates consistent, and can we describe a scaled limiting distribution for them as graph size increases? We answer both of these in the affirmative.Key result 1: The rows of the omnibus embedding provide consistent estimates for graph latent positions. If the latent positions of the graphs ^(1), ⋯, ^m are equal, then under mild assumptions, the rows of OMNI() provide consistent estimates of their corresponding latent positions. Specifically, if h=n(s-1)+ i, where 1 ≤ i ≤ n and 1 ≤ s ≤ m, then there exists an orthogonal matrixsuch thatmax_1 ≤ i ≤ n(^1/2)_h-_i<C log mn/√(n)with high probability. This consistency result is especially useful because it bounds the error between true and estimated latent positions for all latent positions simultaneously. It further guarantees that the omnibus embedding competes well against the current best-performing estimator of the common latent positions , which is the adjacency spectral embedding of the sample mean matrix =∑_i=1^m ^(i)/m <cit.>. Thus, the omnibus embedding is not just consistent when the latent positions are equal; it is close to near-optimal, and we exhibit this clearly in simulations (see Sec. <ref>).Our second key result concerns the limiting distribution, as the graph size n increases, of the rows of the omnibus embedding in the case when the graphs are independent and have the same latent positions.Key result 2: For large graphs, the scaled rows of the omnibus embedding are asymptotically normal. Suppose that the latent positions for each graph are drawn i.i.d from a suitable distribution F, and that conditional on these latent positions, the adjacency matrices ^(1), ⋯, ^(m) are independent realizations of random dot product graphs with the given latent positions. Letrepresent that mn× d dimensional matrix of latent positions for the graphs. Let h=n(s-1) + i, where 1 ≤ i ≤ n and 1 ≤ s ≤ m-1.Then under mild assumptions, there exists a sequence of orthogonal matrices _n such that as n →∞, √(n)[(^1/2)_n-]_hconverges to a mean-zero d-dimensional Gaussian mixture. Hence the omnibus embedding allows for accurate estimation when the latent positions are equal; provides multiple points for each vertex; and under mild assumptions on the structure of the latent positions, these embeddings are approximately normal for large graph sizes. Even more remarkably,for testing whether two graphs have the same latent positions, we can build the omnibus embedding and consider only the Frobenius norm of the difference between the matrices defined by, respectively, the first n and the second n rows of this decomposition.This matrix difference, without any further Procrustes alignment, also serves as a test statistic for the equality of latent positions, which brings us to our third point. Key simulation evidence: The omnibus embedding has meaningful power for two- and multi-sample graph hypothesis testing. The omnibus embedding, without subsequent Procrustes alignments, yields an improvement in power over state-of-the-art methods in two-graph testing, as borne out by a comparison on simulated data. Combined with our earlier bounds on the 2 →∞-norm difference between true and estimated latent positions, this demonstrates that the omnibus embedding provides estimation accuracy when the graphs are drawn from the same latent positions and improved discriminatory power when they are different. What is more, the omnibus embedding produces multiple points for each vertex and our asymptotic normality guarantees that these points are approximately normal. As a consequence, the omnibus embedding not only permits the discovery of graph-wide differences, but also the isolation of vertices that contribute to these differences, via, for instance, a multivariate analysis of variance (MANOVA) applied to these embedded vectors. This leads us to our final point.Key real data analysis: the omnibus embedding isolates graph-wide differences and gives principled evidence for vertex significance in those differences, and works well on complicated real data. On weighted, directed, noisily-observed real graphs, slight modifications to the omnibus embedding procedure yield genuine exploratory insights into graph structure and vertex importance, with actionable import in application domains. To demonstrate this, we present in the next subsections detailed analyses of two neuroscientific data sets. The first is a connectomic data set, in which we analyze a collection of paired diffusion MRI (dMRI) brain scans across 57 patients <cit.>, a comparison of 114 graphs with 172 common vertices.Second, we consider the COBRE data set <cit.>, a collection of functional MRI scans of54 schizophrenic patients and 69 healthy controls, with each scan yielding a graph on 264 common vertices. In both data sets, we show how the omnibus embeddings can identify whole-graph differences as well as particular vertices involved in this difference. §.§ Discerning vertex-level difference in paired brain scans of human subjectsAs a case study in the utility of our techniques, we consider data from human brain scans collected on 57 subjects at Beijing Normal University. The data, labeled “BNU1", is available at .This diffusion MRI data comprises two scans on each of 57 different patients, for a total of 114 scans. The dMRI data was converted into weighted graphs via the Neuro-Data-MRI-to-Graphs (NDMG) pipeline of <cit.>, with vertices representing sub-regions defined via spatial proximity and edges by tensor-based fiber streamlines connecting these regions. As such, by condensing vertices in a given brain region still further, neuroscientists can represent this data at different scales. We focus on data at a resolution in which the m=114 graphs have n=172 common vertices. (We point out that in <cit.>, this same data, at slightly different scale, serves as a useful illustration of different structural properties uncovered by different spectral embeddings). Our inference goals are to determine which of these graphs appear statistically similar and to elucidate which vertices might be key contributors to such difference.We binarize the graphs via a simple thresholding operation, replacing all nonzero edge weights with 1, and leaving unchanged edges of weight zero. For reference, Figure <ref> shows a visualization of one pair of adjacency matrices. Alternative approaches to weighted graphs, such as a replacement of weights by a rank-ordering, are also possible and have proven useful <cit.>, but we do not pursue this here.With these m=114 binarized, undirected adjacency matrices, we generate the m-fold omnibus matrix M, the m-fold analogue of the matrix in Eq. (<ref>), which is of size (114 × 172)×(114 × 172). To select a dimension for the omnibus embedding, we apply the profile-likelihood method of <cit.> to M. This procedure performs model selection (i.e., estimates the rank of M) by locating an elbow in the screeplot of the eigenvalues of M (shown in Figure <ref>(a)) This yields an estimated embedding dimension of d̂=10. As a check, we perform the same estimation procedure on all of the 114 graphs as well. Reassuringly, we recover an estimated embedding dimension close to 10 for almost all of them, and proceed with d̂=10, as summarized by the boxplot in Fig. <ref>(b).We now construct a centered omnibus matrix, in which we subtract the sample mean =m^-1∑_l=1^m ^(l) from the omnibus matrix. That is, we construct an omnibus matrix from the centered graphs ^(l)=^(l)- instead of the observed graphs (l). Having constructed this centered omnibus matrix, we embed it into =10 dimensions, producing a matrix Ẑ∈^mn ×, with =10 columns and mn = 114×172 rows. Observe that Ẑ can be subdivided into 114 blocks each of size 172, one for each graph. For convenience, we denote these submatrices, each of size 172× 10, by X̂^(1), ⋯, X̂^(114). Under our model assumptions, each of these submatrices is an estimate of the latent position matrix of the corresponding brain graph.Because the omnibus embedding introduces an alignment between graphs by placing an average on the off-diagonal blocks of the omnibus matrix, we find that merely considering a Frobenius norm difference between blocks of the omnibus embedding, i.e.,X̂^(l)-X̂^(k)||_F,without any further Procrustes alignments, provides meaningful power in distinguishing between graphs with different latent positions (again, see Sec. <ref> for simulation evidence, and Sec. <ref> for theoretical discussion of why the omnibus embedding obviates the need for further subspace alignments). As a consequence, we can create a 114× 114 dissimilarity matrix =(_kl) ∈^114 × 114, defined as_kl:=X̂^(l)-X̂^(k)||_Fwhich records the Frobenius norm differences of the omnibus embeddings of the k-th and l-th graph in our collection. We illustrate this dissimilarity matrix in the first panel of Fig. <ref>, and we show how this matrix can be hierarchically clustered.Using classical multidimensional scaling <cit.>, we embed this dissimilarity matrix into 2-dimensional Euclidean space. This yields a collection of 114 points in ^2, each one of which represents one graph. We then cluster this collection of points using Gaussian mixture modeling, in which we select c=3 clusters according to the Bayesian Information Criterion (BIC). The three resulting clusters are depicted in Fig. <ref>(b). We remark that out of 57 subjects, only 10 subject scans are divided across clusters, suggesting that the clusters capture meaningful similarity across graphs. If it were truly the case that all of these graphs had the same latent positions, our main central limit theorem would ensure that for large graph sizes these embedded points would be asymptotically normal. Thus, if we consider these three clusters as identifying three distinct types of graphs, we can now compare embedded latent positions of individual vertices across graphs to determine which vertices play the most similar or different roles in their respective graphs. The (theoretical) asymptotic normality leads us to consider a multivariate analysis of variance (MANOVA). For each vertex, we consider the embedded points corresponding to that vertex that arise from the graphs in Cluster 1, Cluster 2, and Cluster 3, respectively. Because we have multiple embedded points for each vertex and these embedded points are asymptotically normal, MANOVA is, as an exploratory tool, principled. MANOVA produces p-value for each vertex, associated to the test of equality of the true mean vectors for the normal distributions governing the embedded points in each of the 3 classes Since there are 172 vertices, we obtain 172 corresponding p-values, and we correct for multiple comparisons using the Bonferroni correction.The p-values are ordered by significance in Fig. <ref>.We focus on one of the two most significant vertices, Vertex 98. Recall that by nature of the omnibus embedding, we have multiple points embedded in ^, each of which correspond to the 98-th vertex in one of the 114 graphs. Partitioning these 114 points according to the clusters associated to their respective graphs, we can perform nonparametric tests of difference across these collections of points to further illuminate how this vertex differs in its behavior across the three graph clusters. Fig. <ref> illustrateshow strikingly different are the first two principal dimensions of the embedded points for this vertex across the three different clusters. For contrast, we examine one of the least significant vertices, Vertex 124, and reproduce the analogous plots to those in Fig.<ref> for its first principal dimension. The results are displayed in Fig. <ref>.The contrast between Figs. <ref> and <ref> is striking. Because the omnibus embedding gives us multiple points for each vertex, we are able to isolate vertices that are responsible for between-graph differences and then interface with neuroscientists to discern what physical distinctions might be present at this vertex or brain location across the clusters of graphs.We recognize, of course, some immediate concerns with our procedure. First, the fact that our clusters are determined post-hoc implies that the embedded vectors are not independent samples from different populations. Second, the asymptotic normality of the embedded positions is a large-sample result, and applies to an arbitrary but finitely fixed collection of rows. Despite these limitations, we stress that our theoretical results supply a principled foundation on which to build a more refined analysis, and to date this is among the only approaches for the identification and comparison of individual vertices and their role in driving differences between (populations of) graphs. §.§ Identifying brain regions associated with schizophreniaWe next consider the COBRE data set <cit.>, a collection ofscans of both schizophrenic and healthy patients. Eachscan yields a graph on n = 264 vertices, corresponding to 264 brain regions of interest <cit.>, with edge weights given by correlations between BOLD signals measured in those regions. The data set containsscans for 54 schizophrenic patients and 69 healthy controls, for a total of m = 123 brain graphs.We follow the general framework of our BNU1 analysis above. Under the null hypothesis that all m graphs share the same underlying latent positions, the omnibus embedding yields for each vertex a collection of m points in ^d that are normally distributed about the true latent position of that vertex. By applying an omnibus embedding to the m=123 subjects in the COBRE dataset, we can therefore test, for each vertex i ∈ [264], whether or not the healthy and schizophrenic populations display a difference in that vertex, by comparing the latent positions of vertex i associated with the schizophrenic patients against those associated with the healthy patients. That is, let m_s = 54 denote the number of schizophrenic patients and m_h = 69 denote the number of healthy controls, with respective embeddings given by{ X^(j)_i : j =1,2,…,m_s },and { Y^(j)_i : j=1,2,…,m_h }We can test whether the samples{ X^(1)_i, X^(2)_i, …, X^(m_h)}⊆^dand { Y^(1)_i, Y^(2)_i, …, Y^(m_s)}⊆^dappear to come from the same distribution. By Theorem <ref>, if all m subjects' graphs are drawn from the same underlying RDPG, then it is natural to test the hypothesis that both the X_i^(j) and then Y_i^(j), 1 ≤ j ≤ m_k are drawn from the same normal distribution. We use Hotelling's t^2 test <cit.> (and we remark that experiments applying a permutation test for this same purpose yield broadly similar results). We note that while in the BNU1 data example in Section <ref>, we required a clustering to discover collections of similarly-behaving networks, the COBRE data set already has two populations of interest in the form of the healthy and schizophrenic patients.We begin by building the omnibus matrix of m=123brain graphs, each on n=264 vertices. In contrast to the BNU1 data presented above, here we work with the weighted graph obtained fromscans, rather than binarizing them. We apply a three-dimensional omnibus embedding to these m graphs, yielding 123 points in ^3 for each of the n=264 brain regions for a total of 32472 points. For each vertex i ∈{1,2,…,264}, there are m=123 points in ^3 each corresponding to vertex i in one of the brain graphs. 54 of these 123 points correspond to the estimated latent position of the i-th vertex in the schizophrenic patients, while the remaining 69 points correspond to the estimated latent position of the i-the vertex in the healthy patients. For each vertex i, we apply Hotelling's t^2 test to assess whether or not the healthy and schizophrenic estimated latent positions appear to come from different populations. Thus, for each of the 264 regions of interest, we obtain a p-value that captures the extent to which the estimated latent positions of the healthy and schizophrenic patients appear to differ in their distributions.Figure <ref> summarizes the result of the procedure just described. Using the Power parcellation <cit.>, we group the 264 brain regions into larger parcels, which capture what are believed by neuroscientists to correspond to functional subnetworks of the brain. For example, a parcel called the default mode network is associated with wakeful, undirected thought (i.e., mind wandering), and is implicated in schizophrenia <cit.>. We collect, for each of the 14 Power parcels, the p-values associated with all of the brain regions (i.e., vertices) in that parcel, and display in Figure <ref> a histogram of those p-values. Under this setup, parcels in which the populations are largely the same will have histograms that appear more or less flat, while parcels in which schizophrenic patients display different behavior from their healthy counterparts will result in left-skewed histograms. Observing Figure <ref>, we see strong visual evidence that the default mode, the sensory/somatomotor hand and the uncertain parcels are affected by schizophrenia. Here again we see the utility of the omnibus embedding. Thanks to the alignment of the embeddings across all 123 graphs in the sample, we obtain, after comparatively little processing, a concise summary of which vertices differ in their behavior across the two populations of interest. Further, this information can be summarized into an simple display of information—in this case, summarizing which Power parcels are likely involved in schizophrenia—that is interpretable by neuroscientists and other domain specialists.§ BACKGROUND, NOTATION, AND DEFINITIONSWe now turn toward a more thorough exploration of the theoretical results alluded to above. We begin by establishing notation and a few definitions that will prove useful in the sequel.§.§ Notation and DefinitionsFor a positive integer n, we let [n] = {1,2,…,n}, and denote the identity, zero and all-ones matrices by, respectively, ,and . For an n × n matrix , we let λ_i() denote the i-th largest eigenvalue ofandwe let σ_i() denote the i-th singular value of . We use ⊗ to denote the Kronecker product. For a vector , we letdenote the Euclidean norm of . For a matrix ∈^n_1 × n_2, we denote by _· j the column vector formed by the j-th column of , and let _i · denote the row vector formed by the i-th row of . For ease of notation, we let _i ∈^n_2 denote the column vector formed by transposing the i-th row of . That is, _i = (_i ·)^T. We letdenote the spectral norm of , _F denote the Frobenius norm ofand _ denote the maximum of the Euclidean norms of the rows of , i.e., _=max_i_i.Where there is no danger of confusion, we will often refer to a graph G and its adjacency matrixinterchangeably. Throughout, we will use C > 0 to denote a constant, not depending on n, whose value may vary from one line to another. For an event E, we denote its complement by E^c. Given a sequence of events { E_n }, we say that E_n occurs with high probability, and write E_nw.h.p., if [ E_n^c ] ≤ Cn^-2 for n sufficiently large. We note that E_n w.h.p. implies, by the Borel-Cantelli Lemma, that with probability 1 there exists an n_0 such that E_n holds for all n ≥ n_0.Our focus here is on d-dimensional random dot product graphs, for which the edge connection probabilities arise as inner products between vectors, called latent positions, that are associated to the vertices. Therefore, we define an an inner product distribution as a probability distribution over a suitable subset of ^d, as follows: (d-dimensional Inner Product Distribution) Let F be a probability distribution on ^d. We say that F is a d-dimensional inner product distribution on ^d if for all ,∈ F, we have ^T ∈ [0,1]. (Random Dot Product Graph) Let F be a d-dimensional inner product distribution with _1,_2,…,_nF, collected in the rows of the matrix =[_1, _2, …, _n]^T ∈^n × d. Supposeis a random adjacency matrix given by[|]= ∏_i<j(_i^T_j)^_ij(1-_i^T_j)^1-_ijWe then write (,) ∼(F,n) and say thatis the adjacency matrix of a random dot product graph with latent positions given by the rows of .We note that we restrict our attention here to hollow, undirected graphs.Given , the probability p_ij of observing an edge between vertex i and vertex j is simply _i^T_j, the dot product of the associated latent positions _i and _j. We define the matrix of such probabilities by =[p_ij]=^T, and write ∼() to denote that the existence of an edge between any two vertices 1 ≤ i < j ≤ n is a Bernoulli random variable with probability p_ij, with these edges independent. That is, if = ^T, then ∼() implies that conditioned on ,is distributed as in Eq. (<ref>). Note that if ∈^n × d is a matrix of latent positions and ∈^d × d is orthogonal,andgive rise to the same distribution over graphs in Equation (<ref>). Thus, the RDPG model has a nonidentifiability up to orthogonal transformation. The focus of this paper is on multi-graph inference. As such, we consider a collection of m random dot product graphs, all with the same latent positions, which motivates the following definition:(Joint Random Dot Product Graph)Let F be a d-dimensional inner product distribution on ^d. We say that random graphs ^(1),^(2),…,^(m) are distributed as a joint random dot product graph (JRDPG) and write (^(1),^(2),…,^(m),) ∼(F,n,m) if = [_1, _2,…,_n]^T ∈^n × d has its (transposed) rows distributed i.i.d. as _i ∼ F, and we havemarginal distributions (^(k),) ∼(F,n) for each k=1,2,…,m. That is, the ^(k) are conditionally independent given , with edges independently distributed as ^(k)_i,j∼( (^T)_ij ) for all 1 ≤ i < j ≤ n and all k ∈ [m]. Throughout, we let δ > 0 denote the eigengap of= _1 _1^T ∈^d × d,the second moment matrix of _1 ∼ F. That is, δ = λ_d() > 0 = λ_d+1(). We note thatcan be chosen diagonal without loss of generality after a suitable change of basis <cit.>. We assume further thatis such that its diagonal entries are in nonincreasing order, so that _1,1≥_2,2≥…≥_d,d = δ. We assume that the matrixis constant in n, so that d and δ are constants, while the number of graphs m is allowed to grow with n. We leave for future work the exploration of the case where the model parameters are allowed to vary with the number of vertices n.Since we rely on spectral decompositions, we begin with a straightforward one: the spectral decomposition of the positive semidefinite matrix =^T. (Spectral Decomposition of ) Sinceis symmetric and positive semidefinite, let = ^T denote its spectral decomposition, with ∈^n × d having orthonormal columns and ∈^d × d diagonal with nonincreasing entries ()_1,1≥ ()_2,2≥⋯≥ ()_d,d > 0. We note that while = ^T is not observed, existing spectral norm bounds <cit.> establish that if ∼(), the spectral norm of - is comparatively small. As a result, we regardas a noisy version of , and we begin our inference procedures with a spectral decomposition of . <cit.> Let ∈^n × n be the adjacency matrix of an undirected d-dimensional random dot product graph. The d-dimensional adjacency spectral embedding (ASE) ofis a spectral decomposition ofbased on its top d eigenvalues, obtained by (,d) = ^1/2, where ∈^d × d is a diagonal matrix whose entries are the top eigenvalues of(in nonincreasing order) and ∈^n × d is the matrix whose columns are the orthonormal eigenvectors corresponding to the eigenvalues in .We observe that without any additional assumptions, the top d eigenvalues ofare not guaranteed to be nonnegative. However, under our eigengap assumptions on , the i.i.d.-ness of the latent positions ensures that for large n, the eigenvalues ofwill be nonnegative with high probability (see Observation <ref> in the Supplementary Material). Given a set of m adjacency matrices distributed as(^(1),^(2),…,^(m),) ∼(F,n,m)for distribution F on ^d, a natural inference task is to recover the n latent positions _1,_2,…,_n ∈^d shared by the vertices of the m graphs. To estimate the underlying latent positions from these m graphs, <cit.> provides justification for the estimate = ( , d ), whereis the sample mean of the adjacency matrices ^(1),^(2),…,^(m). However,is ill-suited to any task that requires comparing latent positions across the m graphs, since theestimate collapses the m graphs into a single set of n latent positions. This motivates the omnibus embedding, which still yields a single spectral decomposition, but with a separate d-dimensional representation for each of the m graphs. This makes the omnibus embedding useful for simultaneous inference across all m observed graphs. (Omnibus embedding) Let ^(1),^(2),…,^(m)∈^n × n be (possibly weighted) adjacency matrices of a collection of m undirected graphs. We define the mn-by-mn omnibus matrix of ^(1), ^(2), …, ^(m) by= [ ^(1) 1/2(^(1) + ^(2))… 1/2(^(1) + ^(m)); 1/2(^(2) + ^(1)) ^(2)… 1/2(^(2) + ^(m));⋮⋮⋱⋮; 1/2(^(m) + ^(1)) 1/2(^(m) + ^(2))… ^(m) ],and the d-dimensional omnibus embedding of ^(1),^(2),…,^(m) is the adjacency spectral embedding of :(^(1),^(2),…,^(m),d)= ( , d ). If (^(1),^(2),…,^(m),) ∼(F,n,m), then the omnibus embedding provides a natural approach toestimatingwithout collapsing the m graphs into a single representation as with = (,d). Under the JRDPG, the omnibus matrix has expected value== _m ⊗ = ^Tfor ∈^mn × d having d orthonormal columns and ∈^d × d diagonal. Sinceis a reasonable estimate for = <cit.>, the matrix = (^(1),^(2),…,^(m),d) is a natural estimate of the mn latent positions collected in the matrix = [^T ^T …^T]^T ∈^mn × d. Here again, as in Remark <ref>,only recovers the true latent positionsup to an orthogonal rotation. The matrix= [;; ⋮; ] = ^1/2∈^mn × d,provides a reasonable canonical choice of latent positions, so that = for some suitably-chosen orthogonal matrix ∈^d × d, and our main theorem shows that we can recover(up to orthogonal rotation) by recovering . § MAIN RESULTS In this section, we give theoretical results on the consistency and asymptotic distribution of the estimated latent positions based on the omnibus matrix . In the next section, we demonstrate from simulations that the omnibus embedding can be successfully leveraged for subsequent inference, specifically two-sample testing.Lemma <ref> shows that the omnibus embedding provides uniformly consistent estimates of the true latent positions, up to an orthogonal transformation, roughly analogous to Lemma 5 in <cit.>. Lemma <ref> shows consistency of the omnibus embedding under thenorm, implying that all mn of the estimated latent positionsare near their corresponding true positions. We recall that the orthogonal transformationin the statement of the lemma is necessary since, as discussed in Remark <ref>, = ^T = ()()^T for any orthogonal ∈^d × d. With , , , anddefined as above, there exists an orthogonal matrix ∈^d × d such that with high probability, ^1/2-^1/2_≤Cm^1/2log mn /√(n) .This result is proved in the supplemental material. As noted earlier, our central limit theorem for the omnibus embedding is analogous to a similar result proved in <cit.>, but with the crucial difference that we no longer require that the second moment matrix have distinct eigenvalues. As in <cit.>, our proof here depends on writing the difference between a row of the omnibus embedding and itscorresponding latent position as a pair of summands: the first,to which a classical Central Limit Theorem can be applied, and the second, essentially a combination of residual terms, which converges to zero. The weakening of the assumption of distinct eigenvalues necessitates significant changes in how to bound the residual terms. In fact, <cit.> adapts a result of <cit.>—the latter of which depends on the assumption of distinct eigenvalues—to control these terms. Here, we resort to somewhat different methodology: we prove instead that analogous bounds to those in <cit.> hold for the estimated latent positions based on the omnibus matrix , and this enables us to establish that here, too, the rows of the omnibus embedding are also approximately normally distributed.Further, en route to this limiting result, we compute the explicit variance of the omnibus matrix, and show that as m, the number of graphs embedded, increases, this contributes to a reduction in the variance of the estimated latent positions.Let (^(1),^(2),…,^(m),) ∼(F,n,m) for some d-dimensional inner product distribution F and letdenote the omnibus matrix as in (<ref>). Let = withas defined in Equation (<ref>), with estimate = (^(1),^(2),…,^(m),d). Let h = m(s-1) + i for i ∈ [n],s ∈ [m], so that _h denotes the estimated latent position of the i-th vertex in the s-th graph ^(s). That is, _h is the column vector formed by transposing the h-th row of the matrix = ^1/2 = (^(1),^(2),…,^(m),d). Let Φ(,) denote the cdf of a (multivariate) Gaussian with mean zero and covariance matrix , evaluated at ∈^d. There exists a sequence of orthogonal d-by-d matrices ()_n=1^∞ such that for all ∈^d,lim_n →∞[ n^1/2(- )_h ≤] = ∫_ FΦ(, () ) dF(),where () = (m+3)^-1() ^-1/(4m),is as defined in (<ref>) and() = [ (^T _1 - ( ^T _1)^2 ) _1 _1^T ].This result is proved in the supplemental material.§ EXPERIMENTAL RESULTSIn this section, we present experiments on synthetic data exploring the efficacy of the omnibus embedding described above. We consider both estimation of latent positions and two-sample graph testing.§.§ Recovery of Latent PositionsPerhaps the most ubiquitous estimation problem for RDPG data is that of estimating the latent positions (i.e., the rows of the matrix ); consequently, we begin by exploring how well the omnibus embedding recovers the latent positions of a given random dot product graph. If one wishes merely to estimate the latent positionsof a set of m graphs (^(1),^(2),…,^(m),) ∼(F,n,m), the estimate = ( ∑_i=1^m ^(i)/m, d ), the embedding of the sample mean of the adjacency matrices performs well asymptotically <cit.>. Indeed, all else equal, the embeddingis preferable to the omnibus embedding if only because it requires an eigendecomposition of an n-by-n matrix rather than the much larger mn-by-mn omnibus matrix.Of course, the omnibus embedding can still be used to to estimate the latent positions, potentially at the cost of increased variance. Figure <ref> compares the mean-squared error of various techniques for estimating the latent positions for a random dot product graph. The figure plots the (empirical) mean squared error in recovering the latent positions of a 3-dimensional JRDPG as a function of the number of vertices n. Each point in the plot is the empirical mean of 50 independent trials. In each trial, the vertex latent positions are drawn i.i.d. from a Dirichlet with parameter [1, 1, 1]^T ∈^3. Having generated a random set of latent positions, we generate two graphs, ^(1),^(2)∈^n × n independently, based on this set of latent positions. Thus, we have (^(1),^(2),) ∼(F,n,2), where F = ([1, 1, 1]^T) is a Dirichlet with parameter [1, 1, 1]^T ∈^3, and n varies. The lines correspond to * ASE1 (red): we embed only one of the two observed graphs, and use only the ASE of that graph to estimate the latent positions in . That is, we consider (^(1)) as our estimate of , ignoring entirely the information present in ^(2). This condition serves as a baseline for how much additional information is provided by the second graph ^(2).* Abar (gold): we embed the average of the two graphs, = (^(1) + ^(2))/2 as = ( , 3 ). As discussed in, for example, <cit.>, this is the lowest-variance estimate of the latent positions .* OMNI (green): We apply the omnibus embedding to obtain = (,3), whereis as in Equation (<ref>). We then use only the first n rows of ∈^2n × d as our estimate of . Thus, this embedding takes advantage of the information available in both graphs ^(1) and ^(2), but does not use both graphs equally, since the first rows ofare based primarily on the information contained in ^(1).* OMNIbar (purple): We again apply the omnibus embedding to obtain estimated latent positions = (,3), but this time we use all available information by averaging the first n rows and the second n rows of .* PROCbar (blue): We separately embed the graphs ^(1) and ^(2), obtaining two separate estimates of the latent positions in ^3. We then align these two sets of estimated latent positions via Procrustes alignment, and average the aligned embeddings to obtain our final estimate of the latent positions.First, let us note that ASE applied to a single graph (red) lags all other methods. This is expected, since all other methods assessed in Figure <ref> use information from both observed graphs ^(1) and ^(2) rather than only ^(1). We see that all other methods perform essentially equally well on graphs of 50 vertices or fewer. Given the dearth of signal in these smaller graphs, we do not expect any method to recover the latent positions accurately.Crucially, however, we see that the OMNIbar estimate (purple) performs nearly identically to the Abar estimate (gold), the natural choice among spectral methods for the estimation latent positions <cit.>. The Procrustes estimate (in blue) provides a two-graph analogue of ASE (red): it combines two ASE estimates via Procrustes alignment, but does not enforce an a priori alignment of the estimated latent positions in the manner of the omnibus embedding does (we discuss this enforced alignment in <ref> as well.) As predicted by the results in <cit.> and <cit.>, the Procrustes estimate is competitive with the Abar (gold) estimate for suitably large graphs. The OMNI estimate (in green) serves, in a sense, as an in-between method, in that it uses information available from both graphs, but in contrast to Procrustes (blue), OMNIbar (purple) and Abar (gold), it does not make complete use of the information available in the second graph. For this reason, it is noteworthy that the OMNI estimate outperforms the Procrustes estimate for graphs of 80-100 vertices. That is, for certain graph sizes, the omnibus estimate appears to more optimally leverage the information in both graphs than the Procrustes estimate does, despite the fact that the information in the second graph has comparatively little influence on the OMNI embedding. §.§ Two-graph Hypothesis Testing We now turn to the matter of using the omnibus embedding for testing the semiparametric hypothesis that two observed graphs are drawn from the same underlying latent positions. Suppose we have a set of points _1,_2,…,_n,_1,_2,…,_n ∈^d. Let the graph G_1 with adjacency matrix ^(1) have edges distributed independently as ^(1)_ij∼( _i^T _j ). Similarly, let G_2 have adjacency matrix ^(2) with edges distributed independently as ^(2)_ij∼( _i^T _j ). As discussed previously, while = (^(1)+^(2))/2 may be optimal for estimation of latent positions, it is not clear how to use the embedding (,d) to test the following hypothesis:H_0 : _i = _i ∀ i∈ [n].On the other hand, the omnibus embedding provides a natural test of the null hypothesis (<ref>) by comparing the first n and last n embeddings of the omnibus matrix= [^(1) (^(1) + ^(2))/2; (^(1) + ^(2))/2^(2) ].Intuitively, when H_0 holds, the distributional result in Theorem <ref> holds, and the i-th and (n+i)-th rows of (^(1),^(2),d) are equidistributed (though they are not independent). On the other hand, when H_0 fails to hold, there exists at least one i ∈ [n] for which the i-th and (n+i)-th rows ofare not identically distributed, and thus the corresponding embeddings are also distributionally distinct. This suggests a test that compares the first n rows of (^(1),^(2),d) against the last n rows (see below for details). Here, we empirically explore the power this test against its Procrustes-based alternative from <cit.>.Our setup is as follows. We draw _1,_2,…,_n ∈^3 i.i.d. according to a Dirichlet distribution F with parameter = [1, 1, 1]^T. Assembling these n points into a matrix = [_1 _2 …_n]^T ∈^n × 3, we can generate a graph G_1 with adjacency matrix ^(1) with entries ^(1)_ij∼( (^T)_ij ). Thus, (^(1),) ∼(F,n). We generate a second graph G_2 by first drawing random points _1,_2,…,_nF. Selecting a set of indices I ⊂ [n] of size k < n uniformly at random from among all such nk sets, we let G_2 have latent positions_i = _i i ∈ I_iAssembling these points into a matrix = [_1, _2, …, _n]^T ∈^n × 3, we generate graph G_2 with adjacency matrix ^(2) with edges generated independently according to ^(2)_ij∼( (^T)_ij ). The task is then to test the hypothesisH_0 := .To test this hypothesis, we consider two different tests, one based on a Procrustes alignment of the adjacency spectral embeddings of G_1 and G_2 <cit.> and the other based on the omnibus embedding. Both approaches are based on estimates of the latent positions of the two graphs. In both cases we use a test statistic of the form T = ∑_i=1^n _i - _i _F^2, and accept or reject based on a Monte Carlo estimate of the critical value of T under the null hypothesis, in which _i = _i for all i ∈ [n]. In each trial, we use 500 Monte Carlo iterates to estimate the distribution of T.We note that in the experiments presented here, we assume that the latent positions _1,_2,…,_n of graph G_1 are known for sampling purposes, so that the matrix = ^(1) is known exactly, rather than estimated from the observed adjacency matrix ^(1). This allows us to sample from the true null distribution. As proved in <cit.>, the estimated latent positions _1 = (^(1)) and _2 = ( ^(2) ) recover the true latent positions _1 and _2 (up to rotation) to arbitrary accuracy in (2,∞)-norm for suitably large n <cit.>. Without using this known matrix , we would require that our matrices have tens of thousands of vertices before the variance associated with estimating the latent positions would no longer overwhelm the signal present in the few altered latent positions.Three major factors influence the complexity of testing the null hypothesis in Equation (<ref>): the number of vertices n, the number of changed latent positions k = |I|, and the distances _i - _i_F between the latent positions. The three plots in Figure <ref> illustrate the first two of these three factors. These three plots show the power of two different approaches to testing the null hypothesis (<ref>) for different sized graphs and for different values of k, the number of altered latent positions. In all three conditions, both methods improve as the number of vertices increases, as expected, especially since we do not require estimation of the underlying expected matrixfor Monte Carlo estimation of the null distribution of the test statistic. We see that when only one vertex is changed, neither method has power much above 0.25. However, in the case of k = 5 and k = 10, is it clear that the omnibus-based test achieves higher power than the Procrustes-based test, especially in the range of 30 to 250 vertices.Figure <ref> shows the effect of the difference between the latent position matrices under null and alternative. We consider a 3-dimensional RDPG on n vertices, in which one latent position, i ∈ [n], is fixed to be equal to _i = (0.8, 0.1, 0.1)^T and the remaining latent positions are drawn i.i.d. from a Dirichlet with parameter = (1,1,1)^T. We collect these latent positions in the rows of the matrix ∈^n × 3. To produce the latent positions ∈^n × 3 of the second graph, we use the same latent positions in , but we alter the i-th position to be _i = (1-λ)_i + λ (0.1,0.1,0.8)^T for λ∈ [0,1] a “drift” parameter, controlling how much the latent position changes between the two graphs. Intuitively, correctly rejecting H_0 := is easier for larger values of λ; the greater the gap between latent position matrices under null and alternative, the more easily our test procedure should discriminate between them. Figure <ref> shows how the size of the drift parameter influencesthe power. We see that for n=30 vertices (top left), neither the omnibus nor Procrustes test has power appreciably better than approximately 0.05, largely in agreement with the what we observed in Figure <ref>. Similarly, when n=200 vertices (bottom right), both methods perform approximately equally (though omnibus does appear to consistently outperform Procrustes testing). The case of n=50 and n=100 vertices (upper right and bottom left, respectively), though, offers a fascinating instance in which the omnibus test consistently outperforms the Procrustes test. Particularly interesting to note is the n=50 case (top right), in which we see that performance of the Procrustes test is more or less flat as a function of drift parameter λ, while the omnibus embedding clearly improves as λ increases, with performance climbing well above that of Procrustes for λ > 0.8.§ DISCUSSION AND CONCLUSION The omnibus embedding is a simple, scalable procedure for the simultaneous embedding of multiple graphs on the same vertex set, the output of which are multiple points in Euclidean space for each graph vertex. For a wide class of latent position random graphs, this embedding generates accurate estimates of latent positions and supplies empirical power for distinguishing when graphs are statistically different. Our consistency results in the 2 →∞ norm for the omnibus-derived estimates are competitive with state-of-the-art spectral approaches to latent position estimation, and our distributional results for the asymptotic normality of the rows of the omnibus embedding render principled the application of classical Euclidean inference techniques, such as analyses of variance, for the comparison of multiple population of graphs and the identification of drivers of graph similarity or difference at multiple scales, from whole graphs to subcommunities to vertices. We illustrate the utility of the omnibus embedding in data analyses of two different collections of noisy, weighted brain scans, and we uncover new insights into brain regions and vertices that are responsible for graph-level differences in two distinct data sets. Further, we quantifythe impact of multiple graphs on the variance of the rows of the embedding, specifically in relation to the variance given in <cit.>. This result shows that as the number of graphs, m, grows, a significant reduction in the variance is achievable. Experimental data suggest that the omnibus embedding is competitive with state-of-the-art, multiple-graph spectral estimation of latent positions, and we surmise that the variance of the rows in the omnibus embedding is close to optimal for latent position estimators derived from the adjacency spectral embedding. That is, the variance of the omnibus embedding is asymptotically equal to the variance obtained by first averaging the m graphs to get(which corresponds, in essence, to the maximum likelihood estimate for ), and then performing an adjacency spectral embedding of . Let _i correspond to the i-th row of = (^(1),^(2),…,^(m),d), and let _i denote the i-th row of the adjacency spectral embedding of . Let _i denote the average value of the m rows ofcorresponding to the i-th vertex, namely, the average of the m vectors corresponding to the i-th vertex in the omnibus embedding. We conjecture that averaging the rows of the omnibus embedding accounts for all of the reduction in variance when one compares a single row of the omnibus embedding and a single row of . Figure <ref> provides weak evidence in favor of this conjecture, since it illustrates that the MSE of both the omnibus- and Procrustes-based estimates of the latent position estimates are very close to that of the estimate based on the mean adjacency matrix. (Decomposition of Variance) With notation as above, for large n,( √(n)_i ) ≈( √(n)_i ) <( √(n)_i ).We have also demonstrated that the omnibus embedding can be profitably deployed for two-sample semiparametric hypothesis testing of graph-valued data. Our omnibus embedding provides a natural mechanism for the simultaneous embedding of multiple graphs into a single vector space. This eliminates the need for multiple Procrustes alignments, which were required in previously-explored approaches to multiple-graph testing <cit.>. In the two-graph hypothesis testing framework of <cit.>, each graph is embedded separately. Under the assumption of equality of latent positions (i.e., under H_0 in Equation (<ref>)), we note that embedding the first graph estimates the true latent positionsup to a unitary transformation in ^d × d. Call this estimate _1. Similarly, _2, the estimates based on the second graph, estimatesonly up to some potentially different unitary rotation, i.e., _2 ≈^* for some unitary ^*. Procrustes alignment is thus required to discover the rotation aligning _1 with _2. In <cit.>, it was shown that this Procrustes alignment, given bymin_∈_d_1 - _2 _F,converges under the null hypothesis. The effect of this Procrustes alignment on subsequent inference is ill-understood. At the very least, it has the potential to introduce variance, and our simulations in Section <ref> suggest that it negatively impacts performance in both estimation and testing settings. Furthermore, when the matrix = ^T does not have distinct eigenvalues (i.e., is not uniquely diagonalizable), this Procrustes step is unavoidable, since the difference _1 - _2_F need not converge at all.In contrast, our omnibus embedding builds an alignment of the graphs into its very structure. To see this, consider, for simplicity, the m=2 case. Let ∈^n × d be the matrix whose rows are the latent positions of both graphs G_1 and G_2, and let ∈^2n × 2n be their omnibus matrix. Then == [; ] = [ ;][ ;]^T.Suppose now that we wish to factorizeas= [ ; ^* ][ ; ^* ]^T = [ (^*)^T ^T; ^* ^T ].That is, we want to consider graphs G_1 and G_2 as being generated from the same latent positions, but in one case, say, under adifferent rotation. This possibility necessitates the Procrustes alignment in the case of separately-embedded graphs. In the case of the omnibus matrix, the structure of thematrix implies that ^*=_d. Thus, in contrast to the Procrustes alignment, the omnibus matrix incorporates an alignment a priori. Simulations show that the omnibus embedding outperforms the Procrustes-based test for equality of latent positions, especially in the case of moderately-sized graphs.To further illustrate the utility of this omnibus embedding, consider the case of testing whether three different random dot product graphs have the same generating latent positions. The omnibus embedding gives us a single canonical representation of all three graphs: Let ^O_1, ^O_2, and ^O_3 be the estimates for the three latent position matrices generated from the omnibus embedding. To test whether any two of these random graphs have the same generating latent positions, we merely have to compare the Frobenius norms of their differences, as opposed to computing three separate Procrustes alignments. In the latter case, in effect, we do not have a canonical choice of coordinates in which to compare our graphs simultaneously.In our analysis of BNU1 data, we “center" our omnibus matrix, by first considering ^(i)=^(i)- and then performing an omnibus embedding on the ^(i) matrices. While our theorems are written for the uncentered case, the analysis of the centered version proceeds along similiar lines. We findthat in many practical settings, centering meaningfully improves our ability to detect differences across graphs, and we offer the following conjectures as to why.First, we surmise that centering allows us to better assess covariance structure between estimated latent positions, and thereby improve clustering in a dissimilarity matrix. Second, centering can mitigate the effect of degree heterogeneity across graphs. Third, centering can dampen the potentially noisy impact of common subgraphs, if they exist, to more clearly address graph difference.Investigating the impact of centering, both for theory and practice, is ongoing, and it is a prominent open problem in the analysis of the omnibus embedding. Of course, other open problems abound, such as an analysis of the omnibus embedding when the m graphs are correlated, are weighted, or are corrupted by occlusion or noise; a closer examination of the impact of the Procrustes alignment on power; the development of an analogue to a Tukey test for determining which graphs differ when we test equality of multiple graphs; the comparative efficiency of the omnibus embedding relative to other spectral estimates; and finally, results for the omnibus embedding under the alternative, when the graph distributions are unequal. The elegance of the omnibus embedding, especially its anchoring in a long and robust history of spectral inference procedures, makes it an ideal point of departure for multiple graph inference, and the richness of the open problems it inspires suggests that the omnibus embedding will remain a key part of the graph statistician's arsenal. § ACKNOWLEDGMENTSThis research is partly sponsored by the Air Force Research Laboratory and DARPA, under agreement number FA8750-18-2-0035; as well as DARPA, under agreement numbers FA8750-12-2-0303, N66001-14-1-4028 and N66001-15-C-4041. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory and DARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The authors also gratefully acknowledge the support of NSF grant DMS-1646108 and NIH grant BRAIN U01-NS108637.plainnatSUPPLEMENTARY MATERIALWe collect here the technical proofs supporting our main result, Theorem <ref>.We consider the (transposed) h-th row of the matrix√(n)( ^1/2^T- ^1/2),where ,∈^d × d are orthogonal transformations. We follow the reasoning of Theorem 18 in <cit.>, decomposing this matrix as √(n)( ^1/2^T- ^1/2) = √(n)( + ), where ,∈^mn × d. To prove our central limit theorem, we show that the (transposed) h-th row of √(n) converges in probability toand that the (transposed) h-th row of √(n) converges in distribution to a mixture of normals.We note that Lemma <ref>, Observation <ref>, Lemma <ref>, Propositions <ref> and<ref>, Lemma <ref>, and Lemma <ref> provide the groundwork for establishing our consistency result, Lemma <ref>, and, thereafter, for showing that the h-th row of √(n) converges in probability to . Next, Lemma <ref> establishes that the h-th row of √(n) converges in distribution to a mixture of normals; the proof of Theorem <ref> then follows from Slutsky's Theorem.We begin with a standard matrix concentration inequality, reproduced from <cit.>.<cit.>Consider independent random Hermitian matrices ^(1),^(2),…,^(k)∈^n × n with ^(i) = and ^(i)≤ L with probability 1 for all i for some fixed L > 0. Define = ∑_i=1^k ^(i), and let v() = ^2. Then for all t ≥ 0,[ ≥ t ] ≤ 2nexp{ - t^2/2 / v() + Lt/3 }.We will apply this matrix Bernstein inequality to the omnibus matrix to obtain a bound on -, from which it will follow by Weyl's inequality <cit.> that the eigenvalues ofare close to those of .Let ∈^mn × mn be the omnibus matrix of ^(1),^(2),…,^(m), where(^(1),^(2),…,^(m),) ∼(F,n,m).Then - ≤ C m n^1/2log^1/2 mnw.h.p. Condition on some = ^T, so that== [ …; … ⋮; ⋮ ⋮ ⋱ ⋮; … … ]∈^mn × mn.We will apply Theorem <ref> to -. For all q ∈ [m] and i,j ∈ [n], let ^ij = _i_j^T + _j_i^T and define block matrix _q,i,j∈^mn × mn with blocks of size n-by-n by_q,i,j = [ … ^ij … ; … ^ij … ;⋮⋱ ⋮⋮; … ^ij … ;^ij…^ij 2^ij^ij…^ij; … ^ij …0_n;⋮⋱ ⋮⋮; … ^ij …],where the ^ij terms appear in the q-th row and q-th column. Using this definition, we have-= ∑_q=1^m ∑_1 ≤ i < j ≤ n^(q)_ij - _ij/2_q,i,j,which is a sum of mn2 independent zero-mean matrices, with (^(q)_ij - _ij ) _q,i,j/2 ≤√(m+1) for all q ∈ [m] and i,j ∈ [n].To apply Theorem <ref>, it remains to consider the variance term v(- ). We note first that, letting _ij = _i_i^T + _j_j^T ∈^n × n, we have_q,i,j_q,i,j = [_ij…_ij_ij_ij…_ij;⋮⋱⋮⋮⋮⋱⋮;_ij…_ij_ij_ij…_ij;_ij…_ij (m+3)_ij_ij…_ij;_ij…_ij_ij_ij…_ij;⋮⋱⋮⋮⋮⋱⋮;_ij…_ij_ij_ij…_ij ],where the (m+3)_ij term appears in the q-th entry on the diagonal. Using the fact that the maximum row sum is an upper bound on the spectral norm <cit.>,v(-) = ∑_q=1^m ∑_1 ≤ i < j ≤ n (^(q)_ij - _ij)^2 / 4 _q,i,j_q,i,j≤ (m+1)^2(n-1) / 4 .Applying this upper bound on v(-) in Theorem <ref>, with t = 12(m+1)√( (n-1) log mn ), we obtain[- ≥ 12(m+1)√( (n-1) log mn )] ≤ 2m^-3n^-2.Integrating over allyields the result. Let λ_1 ≥λ_2 ≥…≥λ_d > 0 denote the top d eigenvalues of = ^T, and letbe as in Equation (<ref>). Then σ() = {mλ_1,mλ_2,…,mλ_d,0,…,0}. This is immediate from the structure of , as defined in Equation (<ref>). Let F be an inner product distribution on ^d with random vectors _1,_2,…,_n, F. With probability at least 1 - d^2/n^2, it holds for all i ∈ [d] that | λ_i() - nλ_i(^T) | ≤ 2d√( n log n ). Further, we have for all i ∈ [d], λ_i() ≤ C nm δ with high probability. A slightly looser version of this bound appeared in <cit.>. We include a proof of this improved result for the sake of completeness.Note that for 1 ≤ i ≤ d, we have λ_i() = λ_i(^T) = λ_i( ^T ). Hoeffding's inequality applied to (^T- n^T)_ij = ∑_t=1^n (_ti_tj - _i _j) yields, for all i,j ∈ [d],[ |(^T ) - n^T|_ij≥ 2√( n log n )] ≤2/n^2.A union bound over all i,j ∈ [d] implies that ^T- n^T _F^2 ≤ 4d^2 n log n with probability at least 1 - 2d^2/n^2. Upper bounding the spectral norm by the Frobenius norm, we have ^T- n^T ≤ 2d √( n log n ) with probability at least 1 - 2d^2/n^2, and Weyl's inequality <cit.> thus implies | λ_i() - nλ_i( ^T ) | ≤ 2d √( n log n ) for all 1 ≤ i ≤ d, from which Observation <ref> and the reverse triangle inequality yieldλ_i() = mλ_i() ≥ m λ_d() ≥ m| nλ_d( ^T ) - 2d √(n log n) | ≥ C mnfor suitably large n. The next several lemmas follow the reasoning in <cit.>, in particular Proposition 16, Lemma 17 and Theorem 18, and thus they are stated here without proof. <cit.>Let = ^T be the eigendecomposition of , where ∈^mn × d has orthonormal columns and ∈^d × d is diagonal and invertible. Let ∈^d × d be the diagonal matrix of the top d eigenvalues ofand ∈^mn × d be the matrix with orthonormal columns containing the top d corresponding eigenvectors, so that ^T is our estimate of , as described above. Let ^T be the SVD of ^T. Then^T- ^T _F ≤ C log mn / nw.h.p. It will be helpful to have the following two propositions, both of which follow from standard applications of Hoeffding's inequality. With notation as above,^T ( - ) _F≤ C √( mn(m + log mn) ) w.h.p.With notation as above,^T (-) _F≤ C √( m log mn ) w.h.p.In what follows, we let = ^T, whereandare as defined in Lemma <ref>, i.e., ^T is the SVD of ^T. The following lemma shows that the matrix“approximately commutes” with several diagonal matrices that will be of import in later computations.<cit.>Let = ^T be as defined above. Then- _F ≤ C m log mnw.h.p. , ^1/2 - ^1/2_F ≤ C m^1/2log mn / n^1/2 w.h.p.and^-1/2 - ^-1/2_F ≤ C (mn)^-3/2 w.h.p. To prove our central limit theorem, we require somewhat more precise control on certain residual terms, which we establish in the following key lemma. Define _1= ^T-_2= ^1/2 - ^1/2 _3=- ^T+ _1 =- . Then the following convergences in probability hold: √(n)[ (-)(^-1/2 - ^-1/2) ]_h , √(n)[ ^T (-) ^-1/2]_h , √(n)[ ( - ^T)(-) _3 ^-1/2]_h , and with high probability, _1^1/2+_2_F ≤C m^1/2log mn/n^1/2. We begin by observing that _1 ^1/2 + _2 _F ≤_1 _F ^1/2 + _2 _F. Lemma <ref> and the trivial upper bound on the eigenvalues ofensures that _1 _F ^1/2≤ C m^1/2log mn / n^1/2 w.h.p. , Combining this with Equation (<ref>), we conclude that _1 ^1/2 + _2 _F ≤ C m^1/2log mn / n^1/2 w.h.p. We will establish (<ref>), (<ref>) and (<ref>) order. To see (<ref>), observe that √(n) (-)(^-1/2 - ^-1/2) _F ≤√(n) (-) ^-1/2 - ^-1/2_F, and application of Proposition <ref> and Lemma <ref> imply that with high probability √(n) (-)(^-1/2 - ^-1/2) _F ≤ C √(log mn / mn^3 ), which goes to 0 as n →∞. To show the convergence in (<ref>), we recall that ^1/2=^T, and observe that since the rows of the latent position matrixare necessarily bounded in Euclidean norm by 1, and since the top d eigenvalues ofare of order mn, it follows that _≤ C (mn)^-1/2 w.h.p. Next, Proposition <ref> and Observation <ref> imply that (^T (-) ^-1/2)_h≤_^T (-) ^-1/2 ≤ C log^1/2 mn / m^1/2 nw.h.p.,which implies (<ref>). Finally, to establish (<ref>), we must bound the Euclidean norm of the vector [ ( - ^T)(-) _3 ^-1/2]_h, where, as defined above, _3 =-. Let _1 and _2 be defined as follows: _1 = ( - ^T)(-)(-^T) ^-1/2 _2 =( - ^T)(-)(^T- ) ^-1/2 Recalling that _3=-, we have ( - ^T)(-) _3 ^-1/2 = ( - ^T)(-)(-^T ) ^-1/2 + ( - ^T)(-)(^T-) ^-1/2= _1 + _2. We will bound the Euclidean norm of the h-th row of each of these two matrices on the right-hand side, from which a triangle inequality will yield our desired bound on the quantity in Equation (<ref>). Recall that we use C to denote a positive constant, independent of n and m, which may change from line to line. Let us first consider _2 = ( - ^T)(-)(^T- ) ^-1/2. We have _2 _F ≤ ( - ^T)(-)^T- _F ^-1/2. By submultiplicativity of the spectral norm and Lemma <ref>, ( - ^T)(-)≤ C m n^1/2log^1/2 mn with high probability. From Lemma <ref> and Observation <ref>, respectively, we have with high probability ^T- _F ≤ C n^-1log mn and^-1/2≤ C (mn)^-1/2 Thus, we deduce that with high probability, _2 _F ≤ C m^1/2log^3/2 mn / n from which it follows that √(n)_2 _F0, and hence √(n) (_2)_h0. Turning our attention to _1, and recalling that ^T =I, we note that (_1)_h= [ ( - ^T)(-)(-^T) ^-1/2]_h = [ ( - ^T)(-)(I-^T) ^T ^-1/2]_h ≤^-1/2[ ( - ^T)(-)(-^T)^T ]_h . Let ϵ > 0 be a constant. We will show that lim_n →∞[ √(n)(_1)_h> ϵ] = 0. For ease of notation, define _1 = ( - ^T)(-)(-^T)^T. We will show that lim_n →∞[ √(n)[ _1 ]_h > n^1/4]= 0, which will imply (<ref>) since, by Observation <ref>, ^-1/2≤ C(mn)^-1/2 w.h.p. We verify the bound in (<ref>) by showing that the Frobenius norms of the rows of _1 are exchangeable, and thus all of these Frobenius norms have the same expectation. We can then invoke Markov's inequality to bound the probability that the Frobenius norm of any fixed row exceeds a specified threshold. To prove exchangeability of the Frobenius norms of the rows of _1,note that if ∈^mn × mn is any permutation matrix, right multiplication of an mn × mn matrix G by ^T merely permutes the columns of G; hence the Frobenius norm of the i-th row of G^T is the same as the Frobenius norm of the i-th row of G. For any n× n matrix real symmetric matrix G, let 𝒫_d( G) denote the projection onto the eigenspace defined by the top d eigenvalues (in magnitude) of G. Similarly, let 𝒫_d^⊥( G) denote the projection onto the orthogonal complement of that eigenspace. For the matrix , for example, the columns ofare a basis for the eigenspace associated to the top d eigenvalues, and the matrix ^T is the unique projection operator into this eigenspace. Observe that ^T^T ^T=^T^T, and thus ^T^T is the projection matrix onto the eigenspace of top d eigenvalues of ^T if and only if ^T is the corresponding projection matrix for . Similarly,^T is the unique projection operator onto the eigenspace defined by the top d eigenvalues (in magnitude) of , and ^T^T is the corresponding projection matrix for ^T. Now, for any pair of n × n matrices ( G,H), let ℒ( G, H) represent the following operator: ℒ( G,H)=𝒫_d^⊥( G)( G- H)𝒫_d^⊥( G)𝒫_d( H) We see that ℒ(, )=_1, and by uniqueness of projections, we note that ℒ(^T, ^T)=( (-^T)^T(-)^T (-^T) ^T (^T)^T=_1 ^T Since we assume that the latent positions for our graphs are i.i.d, the entries of the matrix pair (, ) have the same joint distribution as the entries of the pair (^T, ^T). Therefore, the entries of the matrix ℒ(, ) have the same distribution as those of ℒ(^T, ^T).By Eq. (<ref>), this implies that _1 has the same distribution as _1 ^T. Since the Frobenius norm of any row of _1 ^T is exactly equal to the Frobenius norm of the corresponding to row of _1, we conclude that the Frobenius norms of rows of _1 have the same distribution as the Frobenius norms of the rows of _1, thereby establishing that the Frobenius norms of the rows of _1 are exchangeable. This row-exchangeability for the Frobenius norms of _1 implies that each row has the same expectation, and hence mn(_1)_h ^2 = _1 _F^2. Applying Markov's inequality, [ √(n)[ _1 ]_h> t ] ≤ n [ ( - ^T)(-)(-^T)^T ]_h ^2 / t^2 =( - ^T)(-)(-^T)^T _F^2 / m t^2 . We will proceed by showing that with high probability, ( - ^T)(-)(-^T)^T _F ≤ C m log mn, whence choosing t = n^1/4 in (<ref>) yields that lim_n→∞[ √(n)[ ( - ^T)(-) (-^T)^T ]_h> n^1/4] = 0, and (<ref>) will follow. We have ( - ^T)(-)(-^T)^T _F ≤- - ^T _F Theorem <ref> implies that the first term in this product is at most C mn^1/2log^1/2 mn with high probability, and the final term in this product is, trivially, at most 1. To bound the second term, we will follow reasoning similar to that in Lemma <ref>, combined with the Davis-Kahan theorem. The Davis-Kahan Theorem <cit.> implies that for a suitable constant C > 0, ^T - ^T ≤ C- /λ_d() . By Theorem 2 in <cit.>, there exists orthonormal ∈^d × d such that - _F ≤ C ^T - ^T _F. We observe further that the multivariate linear least squares problem min_∈^d × d - _F^2 is solved by = ^T. Thus, combining all of the above, - ^T _F^2 ≤ - _F^2 ≤ C ^T - ^T _F^2 ≤ C ^T - ^T ^2 ≤ C- /λ_d() ≤ C log^1/2 mn / n^1/2 w.h.p. Thus, we have ( - ^T)(-)(-^T)^T _F ≤- - ^T _F ≤ C m log mnw.h.p. , which implies (<ref>), as required, and thus the convergence in (<ref>) is established, completing the proof. We are now ready to prove Lemma <ref> on the consistency of the omnibus embedding; that is, we can now prove that there exists an orthogonal matrix ∈^d × d such that with high probability,^1/2-^1/2_≤Cm^1/2log mn /√(n) . Observe that ^1/2 - ^1/2 = (-)^-1/2 + (-)(^-1/2 - ^-1/2) - ^T (-) ^-1/2 + ( - ^T)(-) _3 ^-1/2 + _1 ^1/2 + _2. With _1 and _2 defined in Equation (<ref>), the arguments in the proof of Lemma <ref> imply that with high probability(-)(^-1/2 - ^-1/2) ≤ Cm^1/2 n^-1log^1/2 mn^T (-) ^-1/2_F ≤ C n^-1/2log^1/2mn( - ^T)(-) _3 ^-1/2_F≤_1_F+_2_F ≤ C n^-1/2 m^1/2log mn + C m^1/2 n^-1log^3/2mnAs a consequence, there exists an orthogonal matrixsuch that ^1/2-_F ≤(-)^-1/2_F + C m^1/2log mn /√(n) w.h.p. From this, we deduce that max_i(^1/2-^1/2)_i≤1/λ_d()max_i((-))_i + C m^1/2log mn/√(n) w.h.p.Standard application of Hoeffding's inequality as in Proposition <ref> shows that with high probability,max_i((-))_i≤ C(m^1/2 +log^1/2 mn).The desired bound follows from Observation <ref> applied to λ_d(). We are now ready to consider the asymptotic distributional behavior of our estimates of the latent positions.By the definition of the JRDPG (Definition <ref>), the latent positions of the expected omnibus matrix == ^T are given by= [;; ⋮; ] = ^1/2∈^mn × d.Recall that we denote the matrix of these “true” latent positions by = [ ^T, ^T, …, ^T ]^T ∈^mn × d, so that = for some suitably-chosen orthogonal matrix .Fix some i ∈ [n] and some s ∈ [m] and let h = m(s-1) + i. Conditional on _i = _i ∈^d, there exists a sequence of d-by-d orthogonal matrices {} such that n^1/2^T [ ( - ) ^-1/2]_h ( 0, (_i) ),where (_i) ∈^d × d is a covariance matrix that depends on _i. For each n=1,2,…, choose orthogonal ∈^d × d so that = (and hence =, as well). At least one suchexists for each value of n, since, as discussed previously, the true latent positionsare specified only up to some rotation = ^1/2 =. We haven^1/2^T [ ( - ) ^-1/2]_h= n^1/2^T [ ^-1 - ^-1]_h = n^1/2^T ^-1[- ]_h =n^1/2^T ^-1/ m [- ]_h,where we have used the fact that = m.Recalling the structure of = (see Equation (<ref>)) and recalling that _j = (_j ·)^T, we haven^1/2 ^T [ ( - ) ^-1/2]_h =n^1/2^T ^-1/ m ( ∑_q=1^m ∑_j=1^n ( ^(q)_ij + ^(s)_ij/2 - _ij) _j ) =n^1/2^T ^-1/ m ( ∑_j ≠ i( m+1/2(^(s)_ij - _ij) + ∑_q ≠ s^(q)_ij - _ij/2) _j ) - n^1/2^T ^-1_ii_i = ( n ^T ^-1) [ n^-1/2∑_j ≠ i((m+1) /2m(^(s)_ij - _ij) + 1/ m ∑_q ≠ s^(q)_ij - _ij/2) _j ] - n ^T ^-1_ii_i / n^1/2. Conditioning on _i = _i ∈^d, we first observe that_ii/ n^1/2_i = _i^T _i / n^1/2 x_i → 0a.s.further, the scaled sumn^-1/2∑_j ≠ i (m+1 /2m(^(s)_ij - _ij) + 1/ m ∑_q ≠ s^(q)_ij - _ij/2) _j = n^-1/2∑_j ≠ i((m+1) /2m(^(s)_ij - _j^T _i) + 1/ m ∑_q ≠ s^(q)_ij - _j^T _i /2) _jis a sum of n-1 independent 0-mean random variables, each with covariance matrix given by(_i) =m+3 / 4m [( _i^T _j - (_i^T _j)^2 ) _j _j^T ].The multivariate central limit theorem thus implies thatn^-1/2∑_j ≠ i((m+1) / 2m (^(s)_ij - _j _i^T) + 1/ m ∑_q ≠ s^(q)_ij - _j _i^T /2) _j ( , (_i) ) . By the strong law of large numbers,1/n^T- → a.s.However, we also have1/n ()^T- ^T = ( 1/n^T - ) ^T → a.s. ,and = ^1/2^T ^1/2 = ()^T . Thus,1/n - ^T → a.s.Since all matrices involved are order d, which is fixed in n, the convergences in the preceding three equations can be thought of either as element-wise or under any matrix norm. In particular, we have 1/n - ^T → 0, whence Weyl's inequality <cit.> implies that the eigenvalues of /n converge to those of . Since both /n andare diagonal, this implies that /n →. We note that in the case wherehas distinct diagonal entries, this implies that → I as in <cit.>, though in the case wherehas repeated eigenvalues, no such convergence is guaranteed. Thus we have shown that n ^T ^-1→^-1 almost surely. Combining this fact with (<ref>), the multivariate version of Slutsky's theorem yields n^1/2^T [ ( - ) ^-1/2]_h (, (_i) )where (_i) = ^-1(_i) ^-1. Integrating over the possible values of _i with respect to distribution F completes the proof. We are now ready to prove our main result, Theorem <ref>. Let h ∈ [mn], with h = (m-1)s + i for s,i ∈ [n]. We wish to consider the (transposed) h-th row of the matrix √(n)( ^1/2 - ^1/2), where we recall from Lemma <ref> that = ^T, where ^T is the SVD of ^T. We follow the reasoning of Theorem 18 in <cit.>, decomposing this matrix as √(n)( ^1/2 - ^1/2) = √(n)( + ), where ,∈^mn × d. We will show that the (transposed) h-th row of √(n) converges in probability toand, using Lemma <ref>, that the (transposed) h-th row of √(n) converges in distribution to a mixture of normals. An application of Slutsky's Theorem yields the desired result.Recall our earlier definitions of _1, _2, and _3:_1= ^T-_2= ^1/2 - ^1/2 _3=-= - ^T + _1As we noted in the proof of Lemma <ref>, adding and subtracting appropriate quantities, we deduce as in Eq. (<ref>) that^1/2 - ^1/2 = (-)^-1/2 + (-)(^-1/2 - ^-1/2) - ^T (-) ^-1/2 + ( - ^T)(-) _3 ^-1/2 + _1 ^1/2 + _2.Applying Lemma <ref> and integrating over _i, we have that there exists a sequence of orthogonal matrices {}_n=1^∞ such thatlim_n →∞[ √(n)^T [(-)^-1/2 ]_h ≤] = ∫_ FΦ(, () ) dF().Now consider ^1/2^T - ^1/2. From Equation (<ref>), we have( ^1/2^T - ^1/2)= ( - )^-1/2 + ^T ,where= (-)(^-1/2 - ^-1/2) - ^T (-) ^-1/2 + ( - ^T)(-) _3 ^-1/2 + _1 ^1/2 + _2.Since ^T is unitary, it suffices to show that the h-th row of , as defined in (<ref>), when multiplied by √(n), goes toin probability, from which Slutsky's Theorem will yield our desired result. This is precisely the content of Lemma <ref>, except that we need to establish the following convergence in probability:√(n)[ ( _1 ^1/2 + _2 ) ]_h . We recall that by Lemma <ref> and Equation (<ref>),_1_≤_^T-≤C log mn/n^3/2 w.h.p.Combining this with Observation <ref> and Lemma <ref> along with Equation (<ref>) again,(_1 ^1/2 + _2)_h ≤_1_^1/2 + __2≤C log mn/n w.h.p. ,from which the convergence in (<ref>) follows, completing the proof. | http://arxiv.org/abs/1705.09355v5 | {
"authors": [
"Keith Levin",
"Avanti Athreya",
"Minh Tang",
"Vince Lyzinski",
"Youngser Park",
"Carey E. Priebe"
],
"categories": [
"stat.ME",
"62H12, 62H15, 05C80"
],
"primary_category": "stat.ME",
"published": "20170525204815",
"title": "A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference"
} |
University of Münster and CERN Recent results on jet production in heavy ion collisions at RHIC and the LHC are discussed, with emphasis on inclusive jet yields and semi-inclusive hadron-triggered and vector boson-triggered recoil jet yields as well as their azimuthal angular correlations. I will also discuss the constraints that these observables impose on the opacity of the medium, the flavour dependence of energy loss, the interplay of perturbative and non perturbative effects and the change of the degrees of freedom of the medium with the resolution of the probe. § INTRODUCTION The scope of the heavy ion jet physics program at RHIC and LHCis to understand the behaviour of QCD matter at the limit of high density and temperature via the study of the dynamics of the jet-medium interactions. Jet physics in heavy ion collisions is a multiscale problem. Hard scales govern the perturbative production of the elementary scattering and subsequent branching in vacuum and in medium. While jet constituents may interact strongly with the medium at scales corresponding to thetemperature of the plasma.The characterization of medium modifications of jet distributions benefits from observables that are well-defined, that preserve the infrared and collinear safety of the measurement and thus allow fora direct connection to the theory. It also requires the control ofthe large combinatorial background present in heavy ion collisions.The first generation of jet measurements are the jet production cross sections and their suppression relative to the vacuum proton-proton reference. These “disappeareance” measurements indicate that a significant amount of energy is radiated out of the jet area but do not impose severe constrains on the dynamics of jet-medium interactions.The second generation of observables are semi-inclusive jet rates, using hadrons, jets or vector-bosons as triggers.Coincidence measurements allow exploration of interjet broadening and inspection of low jet momenta and high resolution R. Vector-boson triggers allow for a quantification of the energy lost by the recoiling jet. The third generation of observables are jet shapes. Possible modifications of the intrajet distributions are currently explored via the jet mass <cit.>, dispersion p_TD, angularities <cit.>, fragmentation functions <cit.> and jet-track correlations <cit.>. The role of color coherence in medium <cit.> is explored by new observables like 2-subjetiness <cit.>, andsoft drop subjet momentum balance <cit.>.Those two measurements were designed to understand whether the subjet structure is resolvedby the medium, depending on the angular scale, and consequently, determine whether subjets interact in the medium independently or coherently. The second and third generation observables can be combined in the measurement of substructure of recoiling jets, as it is done in <cit.>.Note as well that this third generation of jet observables won't be discussed here since they were the subject of another talk <cit.>. Open questions include the flavour dependence of energy loss, the dependence of energy loss on jet substructure, the role of color-coherence effects, the interplay between weak and strong coupling effects, the role of the medium response or correlated background and the change with the probe resolution scale of the medium degrees of freedom. § INCLUSIVE JET SUPPRESSIONInclusive jet yields have been measured in Pb-Pb collisions at the LHC over a wide kinematic range from a few tens of GeV to the TeV scale.The suppression of such yields relative to thebinary collision-scaled proton-proton reference, is flat up to jet transverse momentum p_T of 1 TeV for jet resolution R=0.4.The strong suppression of 1 TeV jets is a striking observation, given the short size of the medium compared to the hadronization length of the shower of such energetic probes. Fig. <ref> (top left), shows ATLAS results for inclusive jet and hadron nuclear modification factor in Pb-Pb collisions at 5.02 TeV <cit.>. Data for prompt J/ψ and Z bosons are also shown. In Fig. <ref> (bottom), the jet suppression is magnified and compared to 2.76 TeV results. There is no evidence of collision energy dependence. It should be noted that the magnitude of the suppression R_AA is not a direct measure of the opacity of the medium to high p_T probes, but rather depends on other elements such the the slope of the underlying hard parton spectrum and the quark/gluon fractions. A different medium opacity at the two colliding energies could manifest itself in a comparable R_AA. The measurement of jet yields at forward rapidities is complementary to studying the √(s) dependence, since it also offers the possibility to vary the medium density, the quark/gluon content and the spectral slope. It was predicted <cit.>that at forward rapidities and close to the kinematic limit where the spectrum becomes steep, the suppression of forward jets will increase relative to midrapidity. The ATLAS measurement, see Fig. <ref> (top right), confirms this prediction qualitatively. At forward rapidity and high jet energy, the jet yields are suppressed relative to midrapidity by 30%. Another interesting question that rises when exploring jet production at the TeV scale is the role of nuclear effects. At midrapidity, for E=1 TeV and √(s_NN)= 5 TeV, the parton momentum fraction is x_t=2E/√(s_NN) =0.4 and EMC effects may emerge. § SEMI-INCLUSIVE RECOIL JET YIELDS AND MOMENTUM BALANCE Semi-inclusive trigger-jet coincidence measurements provide several advantages:* Analysis based on semi-inclusive coincidence measurementsallow for a precise subtraction of the background uncorrelated to thehard scattering, without imposing fragmentation bias on the jetpopulation. When such background is removed, full correction of the yields can be achieved down to low jet p_T and large R.* When the trigger is a vector boson that does not interact strongly with the medium, the momentum imbalance between the trigger and the recoil jet provides a direct measurement of the energy lost out of the jet area. * Different choices of trigger bias towards different recoil jet flavour, allowing for the exploration of the flavour dependence of energy loss. The different trigger choices also vary the geometric bias. * Semi-inclusive measurements are self-normalized and as a consequence they don't need an interpretation of event activityin terms of geometry. This is an advantage when studying energy loss in small systems as was shown <cit.>.The STAR collaboration has reported the semi-inclusive yields of track-based jets recoiling from high p_T hadrons (9<p_T^trig<30 GeV) <cit.>. A key aspect of the analysis is that uncorrelated background is removed via event mixing and the fully corrected jet yields are measured down to nearly zero p_T for R up to 0.5. The central and peripheral corrected yields are shown inFig. <ref> (top left) together with a PYTHIA and a NLO calculation, and their ratio, I_CP is shown in the lower panel of the same figure. ALICE has also measured the distribution of track-based jets recoiling from a high p_T trigger track <cit.>. To remove combinatorial background, the difference of two exclusive trigger track classes is taken and a new observable is defined Δ_Recoil. The trigger track class subtraction is proven to be effectively equivalent to the event mixing except for the treatment of multiple parton interactions at low p_T,MPIs, that are not subtracted in the latter<cit.>. The ratio of central Pb-Pb Δ_Recoil to PYTHIA Δ_Recoil is shown in Fig. <ref> (top right) down to p_T=20 GeV and R=0.5. In Fig. <ref> (bottom), the ratio of the recoil jet distributions measured with R=0.2 relative to those measured with R=0.5 is shown both by STAR (left) and ALICE (right). ALICE data reveals no in-medium redistribution of energy within R=0.5 compared to the vacuum reference. The finite suppression of the recoil jet distributions in Fig. <ref> together with the low infrared cut-off of these measurements, indicates that the medium-induced energy loss arises predominantly from radiation at angles larger than 0.5 relative to the jet axis. The lost energy is reflected in the magnitude of the spectrum shift under the assumption of negligible trigger track energy loss and it is estimated to be 8 ± 2 GeV at ALICE, in the jet momentum range of 60 to 100 GeV while STAR results indicate approximately 4 GeV in the jet momentum range of 10 to 20 GeV. CMS and ATLAS have measured jet production in association with isolated γ and Z bosons in proton-proton and Pb-Pb collisions <cit.>.The γ and Z triggers have 40<p_T^γ<60 and p_T^Z>60 respectively, while the recoiling jets have p_T^jet> 30 GeV with R=0.3 and R=0.4, respectively.The CMS measurement of R_Jγ , the number of jet coincidences per γ trigger, is shown in Fig. <ref> (top left). The central Pb-Pb data is below the vacuum reference (proton-proton results smeared by background fluctuations) indicating that a significant fraction of the recoil jets lose energy and their momentum is shifted below the 30 GeV threshold. The hardening of the recoil jet yield with increasing photon momentum is reflectedin the increasing trend ofR_Jγ.The momentum asymmetry x_JB, the ratio of the jet and boson momentum, would be a δ function only at LO. Higher-order corrections broaden its distribution. A shift in the mean of this distribution is used to quantify the average energy lost by the recoil jets. Fig. <ref> (top right) shows the average fraction of the photon momentum carried by the recoiling jets as a function of centrality. One can see that in the most central bin, the average shift between Pb-Pb data and the smeared pp reference is compatible with ≈ 10 GeV of energy radiated out of the jet area.In Fig.<ref> (bottom left) the asymmetry distribution (in this case for Z-triggered recoiled jets) is shown for central Pb-Pb collisions compared to several theory models. It should be noted that the data points are not fully corrected (not unfolded for background fluctuations and detector effects) so theory predictions are consistently smeared with parametrizations of the response provided by the experiment. In the absense of a consistent theoretical model incorporating weakly and strongly coupled energy loss, both limits are compared independently to data. JEWEL <cit.> is a Monte Carlo model that considers elastic and radiative energy loss in the medium in a weakly coupled framework. In the Hybrid model <cit.> parton showers aregenerated by Pythia and the interactions with the medium are implemented by changing the momenta of the partons using an analytical calculation of the lost energy according to strong coupling expectations from string calculations in the gauge/gravity duality. Its prediction is the green curve in plot. To explore the sensitivity of the observable to the details of energy loss, two parametric perturbative limits for the energy loss are also considered, one proportional to the third power of the temperature, as expected from radiative losses, one proportional to the squared of the temperature, as expected from collisional losses. One can see that the differences between the different approaches are not significant enough to discriminate between the details of the energy loss process as implemented in the models.Other purely perturbative calculations<cit.> reproduce boson-tagged jet data farily well. Z and γ triggers enhance the sample of quark recoiled jets compared to jet triggers, so these data may illuminate the parton-flavour dependence of energy loss. At high jet p_T where quark mass effects are negligible, a b-jet is essentially a quark jet.Triggering on b-dijets is interesting because the requirement of a large azimuthal gap between the dijet system suppresses the contribution to b-jet production of gluon splitting. Such a contribution confuses the flavour and color charge of the object propagating in the medium. The beauty dijet sample is consequently a cleaner measurement of prompt heavy quark flavour than inclusive beauty jet R_AA. In Fig. <ref> (bottom right) one can see that the momentum imbalance of the b dijet system is consistent with that of the inclusive dijet system across all centrality bins for trigger and recoil jets of p_T > 100 and p_T > 40 GeV respectively.Since the jet-triggered recoil jet population is expected to be gluon-dominated, the previous plot points to no significant differences in quark and gluon energy loss in the given kinematic regime.§ MEDIUM-INDUCED ACOPLANARITYThe combined analysis of jet energy loss and momentum broadening can constrain the underlying mechanism of energy loss. In the perturbative BDMPS formalism <cit.>, for instance, energy loss and momentum broadening are linearly coupled by a single parameter, the transport coefficient q̂. On the contrary, in the limit of strong coupling <cit.>, energy loss and broadening are uncorrelated. Momentum broadening can be studied experimentally via intrajet shapes sensitive to the redistribution of jet momentum and constituents to wider angles <cit.> and it can be studied as a change in the jet direction as a whole <cit.> Here we focus on the second approach and we inspect momentum broadening via the azimuthal angular correlation of dijet systems. §.§ Broadening of the primary peak of the azimuthal angular correlation In a purely perturbative framework, there are two ingredients that contribute to the azimuthal decorrelation of the recoil jet: one is the vacuum soft and collinear radiation and other is the medium-induced effects. The latter are described as random kicks of momentum at each scattering of the partonic projectile with the medium. The multiple kicks result in a total accumulated momentumQs=q̂· L where L is the medium length. The relative contributions of vacuum and in-medium effects to the decorrelation measured at RHIC and LHChas being studied theoretically using resummation techniques <cit.>. Fig. <ref> shows the vacuum contribution to momentum broadening for the different kinematic choices of trigger hadrons and recoil hadrons/jets in hadron-hadron and hadron-jet correlations at RHIC and LHC. One can see that at the LHC kinematics the vacuum distributions are broader than at RHIC and thus medium-induced broadening, which would be combined in quadrature with these distributions,would be more difficult to measure. Fig. <ref> shows the hadron-jet angular correlations measured at RHIC <cit.> and at LHC <cit.> for different kinematic choices for the trigger and recoil jet momentum, compared to the calculations that also incorporate medium-broadening, parameterized as ⟨q̂L ⟩.One can see that as kinematic cuts select harder scatterings, the sensitivity to different choices of broadening parameter is decreased. In the strongly coupled limit <cit.> there is no notion of scattering centers or of multiple discrete scatterings. However, under some limits, coloured excitations acquire transverse momentum according to a gaussian with width Q^2=q̂L where q̂ = K T^3 and K is a free parameter of the theory.In Fig. <ref> (left), one can see that very different choices for the broadening parameter lead to negligible changes in the azimuthal correlation for the kinematics selected by the CMS cuts. Fig. <ref> (right) shows the azimuthal angular correlation for Z-jet pairs compared to purely perturbative JEWEL model and to the hybrid model with different assumptions for the energy lossrate. The four curves can describe the data and the conclusion is thatthis observable, with the given kinematic cuts, is not very sensitive to thedetails of the microscopic dynamics of the interaction with themedium. §.§ Tails of the azimuthal correlationThe main peak of the azimuthal angular correlation helps to constrain the average momentum broadening while the tails of such distribution might encode fundamental information on the dynamics of the degrees of freedom of the medium. Recent calculations <cit.> show that the probability of large angle and semi-hard parton deflections in medium, the so-called Moliere regime, is parametrically larger in the case of quasi-particle than in the case of strongly-coupled degrees of freedom. This is shown in Fig. <ref> (left).The calculations are done in terms of transverse momentum k_T and in the limit of infinite parton energy. More realistic calculations will open a very interesting possibility of measuring an excess of large angle deflections in heavy ion collisions relative to the vacuum reference as a signature of quasi-particle degrees of freedom. The inspection of the tails of the azimuthal correlation requires large statistics and good control of the background and of the higher order contributions needed to describe that range already in vacuum. ALICE <cit.> has measured the integrated yield above azimuthal angle threshold in hadron-jet correlations in Pb-Pb collisions. Fig. <ref> (right) shows the integrated yield above azimuthal angle threshold Δφ_thresh in the hadron-jet correlation both fordata and the smeared reference for pp collisions. No significant change of trend of data relative to the smearedreference with angular threshold is observed. However, note that the recoil jet p_T has momentum above 40 GeV and that low p_T is preferable, since the deflection angle resulting from the given momentum from the medium decreases with jet energy.Unfolding in 2Dwill help to correct simultaneously the p_T of the jet and the azimuthal angle between the jet and the trigger hadron or boson, allowing for fully corrected measurements of Δ_ϕ at significantly smaller jet p_T.§ CONCLUSIONSA large sample of differential jet measurements is currently available for systematic comparisons to the theory models.Most of the jet results from LHC Run 2 can be described simultaneusly by weakly coupled and strongly coupled theory models, implying lack of sensitivity to discriminate among different microscopic pictures for jet energy loss. However, several of the experimental measurements have strong cuts on jet p_T that might be biased to a kinematic regime dominated by vacuum effects, whereby medium effects become a small perturbation that is difficult to disentangle.The access to low jet p_Tand to more differential intrajet measurements will open new possibilities in the data-theory comparison.§ ACKNOWLEDGEMENT I thank the organisers for the interesting conference and the opportunity to give this talk, and Peter Jacobs, Christian Klein-Boesing and Marco van Leeuwen for discussions and critical reading of this manuscript.elsarticle-num 05Acharya:2017goa S. Acharya et al. [ALICE Collaboration],arXiv:1702.00804 [nucl-ex]. Cunqueiro:2015dmx L. Cunqueiro [ALICE Collaboration],Nucl. Phys. A 956 (2016) 593 ATLAS:2017iya The ATLAS collaboration [ATLAS Collaboration],ATLAS-CONF-2017-005. CMS:2015tla CMS Collaboration [CMS Collaboration],CMS-PAS-HIN-14-016. CasalderreySolana:2012ef J. Casalderrey-Solana, Y. Mehtar-Tani, C. A. 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"authors": [
"Leticia Cunqueiro"
],
"categories": [
"nucl-ex"
],
"primary_category": "nucl-ex",
"published": "20170527163943",
"title": "Jet Production at RHIC and LHC"
} |
1,2]Oleg SofryginOleg Sofrygin is the corresponding author, [email protected] 1]Zheng Zhu 1]Julie A Schmittdiel 1]Alyce S. Adams 1]Richard W. Grant 2]Mark J. van der Laan 1]Romain Neugebauer[1]Division of Research,Kaiser Permanente, Northern California, Oakland, CA, U.S.A. [2]Division of Biostatistics, School of Public Health, UC Berkeley, USATargeted Learning with Daily EHR Data [ December 30, 2023 ===================================== Electronic health records (EHR) data provide a cost and time-effective opportunity to conduct cohort studies of the effects of multiple time-point interventions in the diverse patient population found in real-world clinical settings. Because the computational cost of analyzing EHR data at daily (or more granular) scale can be quite high, a pragmatic approach has been to partition the follow-up into coarser intervals of pre-specified length. Current guidelines suggest employing a 'small' interval, but the feasibility and practical impact of this recommendation has not been evaluated and no formal methodology to inform this choice has been developed. We start filling these gaps by leveraging large-scale EHR data from a diabetes study to develop and illustrate a fast and scalable targeted learning approach that allows to follow the current recommendation and study its practical impact on inference. More specifically, we map daily EHR data into four analytic datasets using 90, 30, 15 and 5-day intervals. We apply a semi-parametric and doubly robust estimation approach, the longitudinal TMLE, to estimate the causal effects of four dynamic treatment rules with each dataset, and compare the resulting inferences. To overcome the computational challenges presented by the size of these data, we propose a novel TMLE implementation, the 'long-format TMLE', and rely on the latest advances in scalable data-adaptive machine-learning software, xgboost and h2o, for estimation of the TMLE nuisance parameters.keywords: Big data; Causal inference; Dynamic treatment regimes; EHR; Machine learning; Targeted minimum loss-based estimation.§ INTRODUCTION The availability of linked databases and compilations of electronic health records (EHR) has enabled the conduct of observational studies using large representative population cohorts. This data typically provides information on the nature of clinical visits (e.g, ambulatory, emergency department, email, telephone, acute inpatient hospital stay), medication dispensed, diagnoses, procedures, laboratory test results and any other information that is continuously generated from patients' encounters with their healthcare providers. For instance, EHR-based cohort studies have been used to estimate the relative effectiveness of time-varying interventions in real-life clinical settings.The advances in causal inference have provided a sound methodological basis for designing observational studies and assessing the validity of their findings. For example, the “new user design” <cit.> advocates for applying the same rigor and selection criteria used in RCT design to EHR-based observational studies <cit.>. Moreover, advances in semi-parametric and empirical process theory have allowed for flexible data-adaptive estimation methods that can incorporate machine learning into analyses of comparative effectiveness. By lowering the risk of model misspecification, these data-adaptive approaches can further strengthen the validity of evidence based on observational studies. Finally, some of these semi-parametric approaches also allow drawing valid inference based on formal asymptotic results. For example, the recently proposed Targeted Minimum Loss-Based Estimation (TMLE) for longitudinal data <cit.> – a doubly robust and locally-efficient substitution estimator.While recent methodological advances have significantly improved the potential strength of evidence from observational studies, the practical tools for conducting such analyses have not kept up with the growing size of EHR data. In particular, implementation and application of machine learning to large scale EHR data has proved to be challenging <cit.>. EHR data typically includes almost continuous event dates (e.g., data is updated daily), rather than the discrete event dates from interval assessments more common in epidemiologic cohort studies and many RCTs (e.g., data is updated every 3 months). To mitigate the high computing cost of analyzing EHR data at the daily (or more granular) scale, an analyst typically discretizes study follow-up by choosing a small number of cutoff time points. These cutoffs determine the duration of each follow-up time interval and the total number of analysis time points. The granular EHR data on each subject is then aggregated into interval-specific measurements for downstream analysis.Current literature suggests choosing a small time interval <cit.> to define evenly spaced cutoff time points, however there are no clear guidelines for deciding on the optimal duration of this interval (referred to as the 'time unit' from hereon). Moreover, in practice, the effect of selecting different time unit on causal inferences has not been previously examined within the same EHR cohort. Notably, the choice of a time unit is often driven by the computational complexity of the estimation procedure, as much as the subject-specific domain knowledge <cit.>. For example, in <cit.> the authors applied longitudinal TMLE for estimating the comparative effectiveness of four dynamic treatment regimes by coarsening the daily EHR data into the 90-day time unit. However, such coarsening introduces measurement error, which can in turn lead to bias in the resulting effect estimates. For example, the treatment level assigned to a patient for one 90-day time-interval might misrepresent the actual treatment experienced. Intuitively, analyzing data as it is observed (using the original event dates) should improve causal inferences by avoiding the reliance on arbitrary coarsening algorithms.In this paper, we propose a fast and scalable targeted learning implementation for estimating the effects of complex treatment regimes using EHR data coarsened with a time unit that can more closely (compared to current practice) approximate the original EHR event dates. We demonstrate the feasibility of the proposed approach and evaluate how the choice of progressively larger time units may effect inference by re-analyzing EHR data from a large diabetes comparative effectiveness study described in <cit.>. We used the granular EHR data generated from the patient's encounters with the healthcare system and discretized the patient-specific daily follow-up by mapping it into equally-sized time bins. In separate analyses, the time unit was varied from 90 days, down to 30, 15 and 5 days. These four time-units yielded four analytic datasets, each based on the same pool of subjects, but with a different level of follow-up coarsening as defined by selected time-unit. Notably, the 5-day time-unit produced a dataset that was nearly a replica of the original granular EHR dataset. We then applied an analogue of the double robust estimating equation method first proposed by <cit.>, similar to the TMLE described in <cit.>, to each of these four datasets. We also compared our results to those obtained from the previous TMLE analysis based on 90-day time-unit in <cit.>.The 90, 30, 15 and 5-day time-unit resulted in datasets with roughly 0.62, 1.81, 3.59 and 8.23 million person-time observations for the entire duration of the follow-up, respectively. The large number of person-time observations and the high computational complexity of our chosen estimation procedures required developing novel statistical software. We carried out our analysis by implementing a new R package, 𝚜𝚝𝚛𝚎𝚖𝚛 <cit.>, which streamlines the analysis of comparative effectiveness of static, dynamic and stochastic interventions in large-scale longitudinal data. As part of the 𝚜𝚝𝚛𝚎𝚖𝚛 R package, we have implemented a computationally efficient version of the longitudinal TMLE, to which we refer as the long-format TMLE. Furthermore, for estimation of the nuisance parameters, we relied on the latest machine learning tools available in R language <cit.>, such as the Extreme Gradient Boosting with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 <cit.> and fast and scalable machine learning with 𝚑2𝚘 <cit.>. Both of these packages implement a number of distributed and highly data-adaptive algorithms designed to work well in large data.The contributions of this article can be summarized as follows. First, to the best of our knowledge, this is the first time the performance of longitudinal TMLE has been evaluated on the same EHR data under varying discretizations of the follow-up time. Furthermore, we present a novel and computationally efficient version of the longitudinal TMLE. We also present a possible application of the new 𝚜𝚝𝚛𝚎𝚖𝚛 software which allowed us to analyze such large scale EHR data. Finally, we hope that our new software will help advance future reproducible research with EHR data and will contribute to research on time-unit selection and its effects on inference.The remainder of this article is organized as follows. In Section <ref>, we describe our motivating research question. In Section <ref> we formally describe the observed data, our statistical parameter and introduce a novel implementation of the longitudinal TMLE with data-adaptive estimation of its nuisance parameters. In Section <ref>, we describe our analyses, present the benchmarks for computing times with the 𝚜𝚝𝚛𝚎𝚖𝚛 R package and present our analyses results. Finally, we conclude with a discussion in Section <ref>. Additional materials and results are provided in our Web Supplement. § MOTIVATING STUDY: COMPARATIVE EFFECTIVENESS OF DYNAMIC REGIMES IN DIABETES CAREThe diabetes study and context that motivated this work was previously described in <cit.>. Briefly, it has long been hypothesized that aggressive glycemic control is an effective strategy to reduce the occurrence of common and devastating microvascular and macrovascular complications of type 2 diabetes (T2DM). A major goal of clinical care of T2DM is minimization of such complications through a variety of pharmacological treatments and interventions to achieve recommended levels of glucose control. The progressive nature of T2DM results in frequent revisiting of treatment decisions for many patients as glycemic control deteriorates. Widely accepted stepwise guidelines start treatment with metformin, then add a secretagogue if control is not reached or deteriorates. Insulin or (less frequently) a third oral agent is the next step. Thus, it is common for T2DM patients to be on multiple glucose-lowering medications.Current recommendations specify target hemoglobin A1c of <7% for most patients <cit.>. However, evidence supporting the effectiveness of a blanket recommendation is inconsistent across several outcomes <cit.>, especially when intensive anti-diabetic therapy is required. The effects of intensive treatment remain uncertain, and the optimal target levels of A1c for balancing benefits and risks of therapy are not clearly defined. Furthermore, no additional major trials addressing these questions are underway.For these reasons, using the electronic health records (EHR) from patients of seven sites of the HMO Research Network <cit.>, a large retrospective cohort study of adults with T2DM was conducted to evaluate the impact of various glucose-lowering strategies on several clinical outcomes. More specifically, the original analyses were based on TMLE and Inverse Probability Weighting estimation approaches using EHR data coarsened with the 90-day time unit to contrast cumulative risks under the following four treatment intensification (TI) strategies denoted by d_θ: 'patient initiates TI at the first time her A1c level reaches or drifts above θ% and patient remains on the intensified therapy thereafter' with θ=7, 7.5, 8, or 8.5. Here, we report on secondary analyses to evaluate the impact of the same glucose-lowering strategies on the development or progression of albuminuria, a microvascular complication in T2DM using a novel TMLE implementation and smaller time units. § DATA AND MODELING APPROACHES Below, we first describe the structure of the analytic dataset that results from coarsening EHR data based on a particular choice of time unit.§.§ Data structure and causal parameter The observed data on each patient in the cohort consist of measurements on exposure, outcome, and confounding variables updated at regular time intervals between study entry and until each patient's end of follow-up. The time (expressed in units of 90, 30, 15 or 5 days) when the patient's follow-up ends is denoted by T̃ and is defined as the earliest of the time to failure, i.e., albuminuria development or progression, denoted by T or the time to a right-censoring event denoted by C. The following three types of right-censoring events experienced by patients in the study were distinguished: the end of follow-up by administrative end of study, disenrollment from the health plan and death. For patients with normoalbuminuria at study entry, i.e., microalbumin-to-creatinine ratio (ACR) <30, we defined failure as an ACR measurement indicating either microalbuminuria (ACR 30 to 300) or macroalbuminuria (ACR>300). For patients with microalbuminuria at study entry, we defined failure as an ACR measurement indicating macroalbuminuria. Addition inclusion and exclusion criteria described in <cit.> yielded the final sample size n=51,179.At each time point t=0,…,T̃, the patient's exposure to an intensified diabetes treatment is represented by the binary variable A^T(t), and the indicator of the patient's right-censored status at time t is denoted by A^C(t). The combination A(t)=(A^T(t),A^C(t)) is referred to as the action at time t. At each time point t=0,…,T̃, covariates, such as A1c measurements (others are listed in Table I of <cit.>), are denoted by the multi-dimensional variable L(t) and defined from EHR measurements that occur before the action at time t, A(t), or are otherwise assumed not to be affected by the actions at time t or thereafter, (A(t),A(t+1),…). In addition, data collected at each time t includes an outcome process denoted by Y(t) - an indicator of failure prior to or at t, formally defined as Y(t)=I(T≤ t). By definition, the outcome is thus missing at t=T̃ if the person was right-censored at t.To simplify notation, we use over-bars to denote covariate and exposure histories, e.g., a patient's exposure history through time t is denoted by A̅(t)=(A(0),…,A(t)). We assume the analytic dataset is composed of n independent and identically distributed (iid) realizations (O_i:i=1,…,n) of the following random variable O=(L̅(T̃),A̅(T̃),Y̅(T̃-1),(1-A^C(T̃))Y(T̃))∼ P. We also assume that each O_i is drawn from distribution P belonging to some model ℳ. By convention, we extend the observed data structure using first Y_i(T̃_i)=0 if A_i^C(T̃_i)=1 and then O_i(t)=O_i(T̃_i) for t>T̃_i. Note that these added degenerate random variables will not be used in the practical implementation of our estimation procedure, yet they will allow us to simplify the presentation in the following section. In particular, this convention implies that whenever Y(T̃_i)=1, the outcomes Y_i(t) are deterministically set to 1 for all t>T̃_i.One common way to store (O_1,…,O_n) in a computer is with the so-called “long-format” dataset, where each row contains a record of a single person-time observation O_i(t)=(L_i(t),A_i(t),Y_i(t)), for some i∈{1,…,n} and t∈{0,…,T̃_i}. For time-to-event data, the long-format can be especially convenient, since only the relevant (non-degenerate) information is kept, while all degenerate observation-rows such that t>T̃_i are typically discarded. An alternative way to store the same analytic dataset is by using the so-called “wide-format”, which includes values for the degenerate part of the observed data structure O_i(t) for T̃_i<t≤ K, where K=max(T̃_i:i=1,...,n). For the remainder of this paper we assume that data are stored in long-format.In this study, we aim to evaluate the effect of dynamic treatment interventions on the cumulative risk of failure at a pre-specified time point t_0. The dynamic treatment interventions of interest correspond to treatment decisions made according to given clinical policies for initiation of an intensified therapy based on the patient's evolving A1c level. These policies denoted by d_θ were described above. Formally, these policies are individualized action rules <cit.> defined as a vector function d̅_θ=(d_θ,0,…,d_θ,t_0) where each function, d_θ,t for t=0,…,t_0, is a decision rule for determining the action regimen (i.e., a treatment and right-censoring intervention) to be experienced by a patient at time t, given the action and covariate history measured up to a given time t. More specifically here, we consider the action rule (L(t),A(t-1))↦ d_θ,t(L(t),A(t-1))∈{0,1}×{0} as a function for assigning the treatment action A(t). Note that these rules are restricted to set the censoring indicators A^C(t)=0. Furthermore, at each time t, we define the dynamic rule d_θ,t by setting A^T(t)=0 if and only if the patient was not previously treated with an intensified therapy (i.e., A(t-1)=0) and the A1c level at time t (an element of L(t)) was lower than or equal to θ, and, otherwise, setting A^T(t)=1. Finally, for each observation O_i, we define the treatment process A̅_i^θ(t)=(A_i^θ(0),…,A_i^θ(t)) as the treatment sequence that would result from sequentially applying the previous action rules, i.e., (d_θ,0(L_i(0)),…,d_θ,t(L_i(t),A_i^θ(t-1))), starting at time 0 through t. Note that the rules defined by d̅_θ are deterministic for a given O_i and thus A̅_i^θ is fixed conditional on the event (O_i=o_i). For notation convenience, we also set A̅_i^θ(t)=A̅_i^θ(T̃_i), for t>T̃_i, noting that, in practice, A̅_i^θ(t) will be evaluated only when t≤T̃_i and can be set as arbitrary for all other t.Suppose Y_d̅_θ(t_0) for θ=θ_1,θ_2 denotes a patient's potential outcome at time t_0 had she been treated between study entry and time t_0 according to the decision rule d̅_θ. Note that the corresponding sequence of treatment interventions is not necessarily equal to A̅_i^θ(t_0). The parameter of interest denoted by ψ^θ_1,θ_2(t_0) is then defined as the causal risk difference between the cumulative risks of two distinct dynamic treatment strategies d̅_θ_1 and d̅_θ_2 at t_0: ψ^θ_1,θ_2(t_0)=ψ^θ_1(t_0)-ψ^θ_2(t_0), where ψ^θ(t_0)=P(Y_d̅_θ(t_0)=1), i.e., the cumulative risk associated with rule d̅_θ at t_0. The above definition of the causal parameter of interest relies on the counterfactual statistical framework, which is omitted here for brevity. We refer the reader to earlier work in <cit.> and <cit.> for a detailed description of the relevant concepts.§.§ Identifiability and the statistical parameter of interest As discussed in the next paragraph, identifiability of the causal parameter with the observational data relies on at least two assumptions: no unmeasured confounding and positivity <cit.>. If the counterfactual outcomes are not explicitly defined based on the more general structural framework through additional explicit assumptions encoded by a causal diagram <cit.>, then an additional consistency assumption is made <cit.>.Without loss of generality, suppose that we are interested in estimating the cumulative risk ψ^θ(t_0) at t_0=1 under fixed dynamic regimen d̅_θ=(d_θ,0,d_θ,1). Define L'(k)=(L(k),O̅(k-1)), for k=0,…,t_0, where by convention (A(-1),L(-1)) is an empty set and Y(-1)=0. Let o=(l̅(t̃),a̅(t̃),y̅(t̃)) denote a particular fixed realization of O. We can now define the following recursive sequence of expectations:Q_1(a,l)=E_P(Y(1)|A(1)=a,L'(1)=l), Q_0(a,l) = E_P(Q_1(A^θ(1),L'(1))|A(0)=a,L'(0)=l) Q_-1=E_P(Q_0(A^θ(0),L'(0)),where we remind that (A^θ(0),A^θ(1)) was previously defined as the sequence of treatments A^θ(0)=d_θ,0(L(0)) and A^θ(1)=d_θ,1(L̅(1),A^θ(0)). Note that, by definition Q_1 is 1 whenever Y(0)=1, since in this case Y(1) is degenerate.Each Q_k, for k=-1,0, is defined by taking the previous conditional expectation, Q_k+1, evaluating it at A^θ(k+1) and L^'(k+1) and then marginalizing over the intermediate covariates L^'(k+1). Under the identifiability assumptions mentioned above, we show in Web Supplement A that the statistical parameter Ψ^θ(t_0)(P)=Q_-1 is equal to the causal cumulative risk ψ^θ(t_0) for dynamic rule d̅_θ. We note that the above representation of Q_-1 is analogous to the iterative conditional expectation representation used in <cit.>, with one notable difference: our parameter evaluates Q_1 and Q_0 with respect to the latest values of the counterfactual treatment, A^θ(1) and A^θ(0), respectively. This is in contrast to the iterative conditional expectations in <cit.>, where conditioning in Q_1 would be evaluated with respect to the entire counterfactual history of exposures A̅^θ(1). As we show in Web Supplement A, these two parameter representations happen to be equivalent. However, our particular target parameter representation above will allow us to develop a TMLE that is computationally faster and more scalable to a much larger number of time-points.We introduce the following notation, which will be useful for the description of the TMLE in next section: let Q=(Q_1,Q_0) and, for k=0,1, define the treatment mechanism g_A(k)(a(k)|l'(k))=P(A(k)=a(k)|L^'(k)=l'(k)), i.e., the conditional probability that A(k) is equal to a(k), conditional on events L'(k) being set to some fixed history l'(k). Finally, let g=(g_A(0),g_A(1)).§.§ Long-format TMLE for time-to-event outcomes Doubly robust approaches allow for consistent estimation of Ψ^θ(t_0)(P) even when either the outcome model for Q or the exposure model for g is misspecified. Among the class of doubly robust estimators, those that are based on the substitution principle, such as the longitudinal TMLE in <cit.>, might be preferable, since the substitution principle may offer improvements in finite sample behavior of an estimator. The TMLE described in <cit.> is an analogue of the double robust estimating equation method presented in <cit.>. While there are several possible ways to implement the longitudinal TMLE procedure (e.g., <cit.>), our implemented version, referred to as “long-format TMLE”, has been adapted to work efficiently with large scale time-to-event EHR datasets.As a reminder, for each subject i, we defined L'_i(k) to include i's entire covariate history up to time-point k, in addition to L_i(k) itself. However, due to the curse of dimensionality, estimating Q_k based on all L'(k), when k is sufficiently large and L(k) is high-dimensional, will generally result in a poor finite-sample performance. Thus, to control the dimensionality , we replace L'(k) with a user-defined summary f_k(L'(k)), where f_k(·) is an arbitrary mapping (L̅(k),A̅(k-1))↦ R^d such that d is fixed for all k=0,…,max(T̃). For example, in our data analyses presented in the following sections we defined the mapping f_k(L'(k)) as (L(0),L(k),A(k-1)). This approach allows the practitioner to control the dimensionality of the regression problem when fitting each Q_k model. Furthermore, by forcing all relevant confounders for time-point k to be defined in a single person-time row via the mapping f_k(·) (i.e., (f_k(·),A_i(k),Y_i(k))), we can also simplify the implementation of the iterative part of TMLE algorithm that fits the initial model for Q_k, as we describe next.Applying the mappings f_k(·) to our observed data on n subjects, (O_1,…,O_n), results in a new, reduced, long-format representation of the data, where each row of the reduced dataset is defined by the following person-time observation (f_k(L'_i(k)),A_i(k),Y_i(k)), for some i∈{1,…,n} and k∈{0,…,T̃_i}. For example, for subject i with T̃_i=0, the entire reduced representation of O_i consists of just a single person-time row (f_0(L_i(0)),A_i(0),Y_i(0)). Our proposed TMLE algorithm, outlined below, will work directly with this reduced long-format representation of (O_1,...,O_n). The algorithm relies on the representation of Ψ^θ(t_0)(P) from the previous section, where one makes predictions based on only the last treatment value for each time-point. This in turn allows us to keep the input data in the reduced long-format at all times, substantially lowering the memory footprint of the procedure.We now describe the long-format TMLE algorithm for estimating parameter Ψ^θ(t_0)(P) indexed by the fixed dynamic regimen d̅_θ=(d_θ,0,...,d_θ,t_0), where for simplicity, we let t_0=1. A more detailed description of this TMLE is also provided in the Web Supplement B. Briefly, the algorithm proceeds recursively by estimating each Q_k in Q, for k=1,0. Prior to that, we instantiate a new variable Q̃_(k+1)=Y(k), for k=0,…,T̃ and we make one final modification to our reduced long-format dataset by adding a new column of subject-specific cumulative weight estimates, defined for each row k as ŵt̂(k)=∏_j=0^kI(A̅(j)=A̅^θ(j))/ĝ_A(j)(O), where k=0,…,T̃ and ĝ_A(j) is the estimator of g_A(j) at j. For iteration k=1, one starts by obtaining an initial estimate Q̂_k of Q_k by regressing Q̃_(k+1) against (A(k),f_k(L'(k))) based on some parametric (e.g., logistic) model, for all subjects such that T̃≥ k and A^C(k)=0 (i.e., this fit is performed among subjects who were at risk for the event at time k). Alternatively, the regression fit can be obtained by using a subset of subjects such that A(k)=A^θ(k) or based on a data-adaptive estimation procedure as discussed later. Next, one estimates the intercept ε^k with an intercept-only logistic regression for the outcome Q̃_(k+1) using the offset Q̂_k(A^θ(k),f_k(L'(k))), and the weights ŵt̂(k), where (x)=log(x/1-x). The last regression defines the TMLE update Q̂_k^* of Q̂_k as (Q̂_k(a^θ(k),f_k(l'(k)))+ε̂^k), for any realization (a^θ(k),l'(k)) and (x)=1/1+e^-x. Finally, if k>0, for all subjects such that T̃≥ k, we compute the TMLE update, defined as Q̂_k^*(A^θ(k),f_k(L'(k))), and use this update to over-write the previously defined instance of Q̃_k. Note that Q̃_k remains set to Y(k-1) for all subjects with T̃<k. The illustration of this over-writing scheme for Q̃_k is also presented in Figure <ref> for iteration k=1, using a toy example for three hypothetical subjects. The same procedure is now repeated for iteration k=0, using the outcome Q̃_(k+1), resulting in TMLE update Q̂_k^*, for all subjects i=1,…,n. Finally, the TMLE of Ψ^θ(t_0)(P)=Q^-1 is defined as Ψ̂^θ(t_0)=1/n∑_i=1^nQ̂_0^*(A_i^θ(0),f_0(L'_i(0))). TMLE estimate of the causal RD ψ^θ_1,θ_2(t_0) can be now evaluated as Ψ̂^θ_1(t_0)(P)-Ψ̂^θ_2(t_0)(P), where the above described procedure is carried out separately to estimate Ψ^θ_1(t_0)(P) and Ψ^θ_2(t_0)(P), for rules d̅_θ_1 and d̅_θ_2, respectively.Note that in above description, each initial estimate Q̂_k can be obtained by either stratifying the subjects based on A(k)=A^θ(k) (referred to as “stratified TMLE”) or by pooling the estimation among all subjects at risk of event at time k (referred to as “pooled TMLE”). Furthermore, in our data analyses described in Section <ref>, we use the stratified TMLE procedure. Finally, the inference can be obtained using the approach described in our Web Supplement C, based on the asymptotic results from prior papers. §.§ Data-adaptive estimation via cross-validation The double-robustness property of the TMLE means that its consistency hinges on the crucial assumption that at least one of the two nuisance parameters (g, Q) is estimated consistently. Current guidelines suggest that the nuisance parameters, such as propensity scores, should be estimated in a flexible and data-adaptive manner <cit.>. However, traditionally in observational studies, these nuisance parameters have been estimated based on logistic regressions, with main terms and interaction terms often chosen based on the input from subject matter experts <cit.>. In contrast, a data-adaptive estimation procedure provides an opportunity to learn complex patterns in the data which could have been overlooked when relying on a single parametric model.For instance, improved finite-sample performance from data-adaptive estimation of the nuisance parameters has been previously noted with Inverse Probability Weighting (IPW) estimation, a propensity score-based alternative to TMLE <cit.>.It has been suggested that problems with parametric modeling approaches can arise even in studies with no violation of the positivity assumption. For example, the predicted propensity scores from the misspecified logistic models might be close to 0 or 1, resulting in unwarranted extreme weights which could be avoided with data-adaptive estimation. Similarly, these extreme weights may also lead to finite-sample instability for doubly-robust estimation approaches, such as the long-format TMLE. These considerations provide further motivation for the use of the data-adaptive estimation procedures.Many machine learning (ML) algorithms have been developed and applied for data-adaptive estimation of nuisance parameters in causal inference problems <cit.>. However, the choice of a single ML algorithm over others is unlikely to be based on real subject-matter knowledge, since: “in practice it is generally impossible to know a priori which [ML procedure] will perform best for a given prediction problem and data set” <cit.>. To hedge against erroneous inference due to arbitrary selection of a single algorithm, an ensemble learning approach known as discrete Super Learning (dSL) <cit.> can be utilized. This approach selects the optimal ML procedure among a library of candidate estimators. The optimal estimator is selected by minimizing the estimated expectation of a user-specified loss function (e.g., the negative log-likelihood loss) <cit.>. Cross-validation is used to assess an expected loss associated with each candidate estimator, which protects against overfitting and ensures that the final selected estimator (called the 'discrete super learner') performs asymptotically as well (in terms of the expected loss) as any of the candidate estimators considered <cit.>. Because of the general asymptotic and finite-sample formal dSL results, in this work we favor the approach of super learning over other existing ensemble learning approaches.Prior applications of super learning in R <cit.> have noted the high computational cost of aggressive super learning, especially for large datasets (e.g., dSL library contains a large number of computationally costly ML algorithms) <cit.>. Because of these limitations, which are also compounded by the choice of a smaller time unit in our study, we implemented a new version of the discrete super learner, the 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 R package <cit.>. This R package is utilized for estimation of the TMLE nuisance parameters by the R package 𝚜𝚝𝚛𝚎𝚖𝚛. Below, we describe how 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 builds on the latest advances in scalable machine learning software, 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 <cit.> and 𝚑2𝚘 <cit.>, which makes it feasible to conduct more aggressive super learning in EHR-based cohort studies with small time units.In our implementation of the discrete super learner, we focus on the negative log-likelihood loss function. The candidate machine learning algorithms that can be included in our ensembles are distributed high-performance 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 and 𝚑2𝚘 implementations of the following algorithms: random forests (RFs), gradient boosting machines (GBMs) <cit.>, logistic regression (GLM), regularized logistic regression, such as, LASSO, ridge and elastic net <cit.>. GBM is an automated and data-adaptive algorithm that can be used with large number of covariates to fit a flexible non-parametric model. For overview of GBMs we refer to <cit.>. The advantage of procedures like classification trees and GBM is that they allow us to search through a large space of model parameters, accounting for the effects of many covariates and their interactions, thereby reducing bias in the resulting estimator regardless of the distribution of the data that defined the true values of the nuisance parameters. The large number of possible tuning parameters available for the estimation procedures in our ensemble required conducting a grid search over the space of such parameters. We note that 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 leverages the internal cross-validation implemented in 𝚑2𝚘 and 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 R packages for additional computational efficiency. § ANALYSIS In this section, we demonstrate a possible application of our proposed targeted learning software. We show that the 𝚜𝚝𝚛𝚎𝚖𝚛 and 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 R packages provide fast and scalable software for the analyses of high-dimensional longitudinal data. We estimate the effects of four dynamic treatment regimes on a time-to-event outcome using EHR data from the diabetes cohort study described in Sections <ref> and <ref>. These analyses are based on large EHR cohort study, using four progressively smaller time units (i.e., four nested discretizations of the same follow-up data). For instance, the choice of the smaller time unit might more closely approximate the original EHR daily event dates, as it is the case in our application. We also evaluate the practical impact of these four progressively smaller time units on inferences. Finally, we compare the long-format TMLE results to those obtained from IPW estimator. All results are also compared to prior published findings from alternate IPW and TMLE analyses.The choice of time-unit of 90, 30, 15, and 5 days results in four analytic datasets, each constructed by applying the SAS macro %_𝙼𝚂𝙼𝚜𝚝𝚛𝚞𝚌𝚝𝚞𝚛𝚎 <cit.> to coarsen the original EHR data. The maximum follow-up in each dataset is subsequently truncated to the first two years, i.e., 8 quarters, 24 months, 48 15-day intervals, and 144 5-day intervals, respectively. For each analytic dataset, we evaluate the counterfactual cumulative risks at t_0 associated with four treatment intensification strategies, with t_0 fixed to 8 distinct time points. That is, for the 90-day (resp. 30-day) analytic dataset, Ψ^θ(t_0)(P) is estimated for t_0=0,1,…,7 (resp. t_0=2,5,…,23) and θ=7,7.5,8,8.5. Similarly, for the 15-day (resp. 5-day) analytic dataset, Ψ^θ(t_0)(P) is estimated for t_0=5,10,…,47 (resp. t_0=17,35,…..,143) and θ=7,7.5,8,8.5.With each of the four analytic datasets and for each of the 32 target cumulative risks, we evaluate the following two estimators: the stratified long-format TMLE (Section <ref>) and a bounded IPW estimator (based on a saturated MSM for counterfactual hazards <cit.>). The TMLE and IPW estimators are implemented based on unstabilized and stabilized IP weights truncated at 200 and 40, respectively <cit.>. Each of the two estimators above uses two alternative strategies for nuisance parameter estimation: a-priori specified logistic regression models (parametric approach) and discrete super learning with 10-fold cross-validation (data-adaptive approach).§.§ Nuisance parameter estimation approaches For the parametric approach, the estimators of each Q_k and g_A(k), for k=0,…,t_0, are based on separate logistic regression models that include main terms for all baseline covariates L(0), the exposures (A(k),A(k-1)) and the most recent measurement of time-varying covariates L(k), i.e., the reduced dataset is defined with f_k(L'(k))=(L(0),L(k),A(k-1)). This covariate selection approach results in approximately 150 predictors for each regression model for Q_k and g_A(k), at each time point k. In addition, estimation of g relies on fitting separate logistic models for treatment initiation and continuation. Furthermore, logistic models for two types of right-censoring events (health plan disenrollment and death) are fit separately, while it is assumed that right-censoring due to end of the study is completely at random (i.e., we use an intercept-only logistic regression). Each of the logistic models for estimating g are fit by pooling data over all time-points k=0,… t_0. Furthermore, the same sets of predictors that are used for estimation of Q_k are also included in estimation of g, in addition to a main term for the value of time k. Finally, for 30, 15 and 5 day time units, these logistic models also include the indicators that the follow-up time k belongs to a particular two-month interval. The %_𝙼𝚂𝙼𝚜𝚝𝚛𝚞𝚌𝚝𝚞𝚛𝚎 SAS macro <cit.> implements automatic imputation of the missing covariates and creates indicators of imputed values. The indicators of imputation are also included in each L(k). The documentation for the SAS macro provides a detailed description of the implemented schemes for imputation. For the data-adaptive estimation approach, each of the above-described logistic model-based estimators is replaced with a distinct discrete super learner. Each discrete super learner uses the same set of predictors that are included in the corresponding model from the parametric approach. The replication R code for the specification of each dSL is available from the following github repository: http://www.github.com/osofr/stremr.paperwww.github.com/osofr/stremr.paper. In short, for estimation of each component of g, for 90, 30 and 15 day time-unit, the dSL uses the following ensemble of 89 distributed (i.e., parallelized) estimators that are each indexed by a particular tuning parameter choice: Random Forests (RFs) with 𝚑2𝚘 (8); Gradient Boosting Machines (GBMs) with 𝚑2𝚘 (1); GBMs with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 (18); Generalized Linear Models (GLMs) with 𝚑2𝚘 (2); and regularized GLMs with 𝚑2𝚘 (60). For 5 day time-unit, the dSL for each component of g uses the following smaller ensemble of 20 estimators: RFs with 𝚑2𝚘 (1); GBMs with 𝚑2𝚘 (1); GBMs with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 (1); GLMs with 𝚑2𝚘 (2); and regularized GLMs with 𝚑2𝚘 (15). An abbreviated discussion of some of the tuning parameters that index the ML algorithms considered is provided in Remark <ref>. To obtain discrete super learning estimates for each component of Q=(Q_k:k=0,…,t_0), we rely solely on the candidate estimators available in the 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 R package because of computational constraints. Specifically, the dSL ensemble for each Q_k, for all four analytic datasets, is restricted to the following 8 estimators: GBMs with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 (3); GLMs with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 (1); and regularized GLMs with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 (4). To achieve maximum computational efficiency with 𝚜𝚝𝚛𝚎𝚖𝚛, it is essential to be able to parallelize the estimation of Ψ_t_0 over multiple time points t_0. However, the estimators implemented in the most recent 𝚑2𝚘 version 3.10.4.7 do not allow outside parallelization for different values of t_0 (i.e., 𝚑2𝚘 does not allow fitting two distinct discrete super learners in parallel). For this reason, the data-adaptive estimation of each component of Q was performed solely with 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 R package.We use the following parameters to fine-tune the performance of GBMs and RFs in 𝚑2𝚘 and 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 R packages: 𝚗𝚝𝚛𝚎𝚎𝚜 (𝚑2𝚘) and 𝚗𝚛𝚘𝚞𝚗𝚍𝚜 (𝚡𝚐𝚋𝚘𝚘𝚜𝚝), 𝚖𝚊𝚡_𝚍𝚎𝚙𝚝𝚑 (𝚑2𝚘 and 𝚡𝚐𝚋𝚘𝚘𝚜𝚝), 𝚜𝚊𝚖𝚙𝚕𝚎_𝚛𝚊𝚝𝚎 (𝚑2𝚘) and 𝚜𝚞𝚋𝚜𝚊𝚖𝚙𝚕𝚎 (𝚡𝚐𝚋𝚘𝚘𝚜𝚝), 𝚌𝚘𝚕_𝚜𝚊𝚖𝚙𝚕𝚎_𝚛𝚊𝚝𝚎_𝚙𝚎𝚛_𝚝𝚛𝚎𝚎 (𝚑2𝚘) and 𝚌𝚘𝚕𝚜𝚊𝚖𝚙𝚕𝚎_𝚋𝚢𝚝𝚛𝚎𝚎 (𝚡𝚐𝚋𝚘𝚘𝚜𝚝), 𝚕𝚎𝚊𝚛𝚗_𝚛𝚊𝚝𝚎 (𝚑2𝚘) and 𝚕𝚎𝚊𝚛𝚗𝚒𝚗𝚐_𝚛𝚊𝚝𝚎 (𝚡𝚐𝚋𝚘𝚘𝚜𝚝). Furthermore, 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 provides additional “shrinkage” tuning parameters, 𝚖𝚊𝚡_𝚍𝚎𝚕𝚝𝚊_𝚜𝚝𝚎𝚙 and 𝚕𝚊𝚖𝚋𝚍𝚊. These parameters allow controlling the smoothness of the resulting fit, with higher values generally resulting in a more conservative estimator fit that might be less prone to overfitting. Furthermore, these two tuning parameters can be useful to reduces the risk of spurious predicted probabilities that are near 0 and 1 and thus might help obtain a more stable propensity score fit of g. Finally, for regularized logistic regression with 𝚑2𝚘 R package, we use 𝚕𝚊𝚖𝚋𝚍𝚊_𝚜𝚎𝚊𝚛𝚌𝚑 option for computing the regularization path and finding the optimal regularization value λ <cit.>. Similarly, for regularized logistic regressions in 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 we use a grid of candidate λ values and select the optimal value by minimizing the cross validation mean-squared error.§.§ Benchmarks Our benchmarks provide the running times for conducting the above described analyses with long-format TMLE, using the four EHR datasets. We report separate running times for estimation of the nuisance parameter g and the TMLE Q̂^*. All analyses were implemented on Linux server with 32 cores and 250GB of RAM. Whenever possible, the computation was parallelized over the available cores. For instance, the data-adaptive estimation of g and Q was parallelized by using the distributed machine learning procedures implemented in 𝚑2𝚘 and 𝚡𝚐𝚋𝚘𝚘𝚜𝚝 R packages. Similarly, the estimation of TMLE survival at 8 different time-points t_0 was parallelized by the 𝚜𝚝𝚛𝚎𝚖𝚛 R package. The compute times (in hours) are presented in Table <ref>. These results are based on the two estimation strategies for the nuisance parameters, as described above. For example, the sum of dSL ĝ and dSL Q̂^* for 5 day time-unit is the total time it takes to obtain the results for all 4 survival curves at the bottom of the right-panel in Figure <ref>. These results show that dSL is computationally costly, but the running times do not preclude routine application of the 𝚜𝚝𝚛𝚎𝚖𝚛 and 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 R packages in EHR-based datasets, even for studies that use a small time-unit.§.§ Results The long-format TMLE survival estimates for data-adaptive estimation of (g,Q) for all four time-units are presented in Figure <ref>. Additional results using 90 and 30-day time-unit are presented in Figure <ref>, the corresponding results for 15 and 5-day time-unit are presented in Figure <ref>. All plots in Figures <ref> and <ref> present the TMLE point estimates of risk differences (RDs) for any two of the four dynamic regimens. The estimates are plotted for 8 different values of t_0, along with their corresponding 95% confidence intervals (CIs). Results from the long-format TMLE analyses with parametric estimation of (Q,g) are presented in Figures 1-3 of the Web Supplement D. Results from the IPW for parametric and data-adaptive estimation strategies are presented in Figures 4-9 of the Web Supplement D. Finally, the distributions of the untruncated and unstabilized IP weight estimates ŵt̂(t) obtained with the two estimation strategies for g are presented in Tables 1 and 2 of the Web Supplement D.The top-left panel in Figure <ref> and top panel in Figure <ref> demonstrates that we have replicated the prior TMLE results for the 90-day time-unit from <cit.>. The point estimates from all four analyses provide consistent evidence, suggesting that earlier treatment intensification provides benefits in lowering the long term cumulative risk of onset or progression of albuminuria. However, the new results are inconclusive for the earliest treatment intensification with dynamic regime d_7.0 due to increasing variability of the estimates with progressively smaller time-unit. Moreover, this trend is also observed for the other three dynamic rules, as the variance estimates increase substantially with the smaller time-unit. Finally, the data-adaptive approaches clearly produce tighter confidence intervals, than with the logistic regression alone.The distribution of the propensity score based weights for each time unit of analyses is also reported as part of the same supplementary materials. Finally, we conduct an alternative set of analyses by including a large number of two-way interactions for estimation of each Q_k, for k=0,…,t_0. However, these analyses did not materially change our findings and the results are thus omitted. § DISCUSSION In this work we've studied the impact of choosing a different time-unit on inference for comparative effectiveness research in EHR data. Current guidelines suggest choosing the time-unit of analysis by dividing the granular (e.g., daily) subject-level follow-up into small (and equal) time interval. Relying on overly discretized EHR data might invalidate the validity of the analytic findings in a number of ways. For example, naïve discretization might introduce measurement error in the true observed exposure, accidentally reverse the actual time ordering of the events, and may result in failure to adjust for all measured time-varying confounding. Thus, choosing a small time-unit is a natural way to reduce the reliance on ad-hoc data-coarsening decisions. As we've shown in our data analysis, the choice of time-unit may indeed impact the inference.The size and dimensionality of the currently available granular EHR data presents novel computational challenges for application of semi-parametric estimation approaches, such as longitudinal TMLE and data-adaptive estimation of nuisance parameters. In our example, the actual EHR data is generated daily from patient's encounters with the healthcare system. To address these challenges we have developed and applied the “long-format TMLE” – a new algorithmic solution for the existing targeted learning methodology. We also apply a data-adaptive approach of discrete super learning (dSL) to estimation of the corresponding nuisance parameters, based on its novel implementation in the 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 R package. Our benchmarks show that 𝚜𝚝𝚛𝚎𝚖𝚛 and 𝚐𝚛𝚒𝚍𝚒𝚜𝚕 R packages can be routinely applied to EHR-based datasets, even for studies that use a very small time-unit.Our analyses demonstrate a substantial increase in the variance of the estimates associated with the selection of the smaller time-unit. As a possible explanation, it should be pointed out that the choice of the smaller time-unit increases the total number of considered time-points and will typically result in a larger set of time-varying covariates. As a result, one would expect that the estimation problem becomes harder for the smaller time-unit (in part, due to the growing dimensionality of the time-varying covariate sets, and, in part, due to larger number of nuisance parameters that need to be estimated). This can potentially lead to a larger variance of the underlying estimates, as was observed in our applied study. It remains to be seen if a single estimation procedure could leverage different levels of discretization and the choice of the different time-unit within the same dataset to provide a more precise estimate. However, we leave the formal methodological analysis of this subject for future research.Finally, we point out that the 𝚜𝚝𝚛𝚎𝚖𝚛 R package implements additional estimation procedures that are outside the scope of this paper, e.g., long-format TMLE for stochastic interventions, handling problems with multivariate and categorical exposures at each time-point, no-direct-effect-based estimators <cit.> of joint dynamic treatment and monitoring interventions, iterative longitudinal TMLE <cit.>, sequentially double robust procedures, such as the infinite-dimensional TMLE <cit.>, and other procedures described in the package documentation and the following github page: http://www.github.com/osofr/stremrwww.github.com/osofr/stremr. § ACKNOWLEDGEMENTS The authors thank the following investigators from the HMO research network for making data from their sites available to this study: Denise M. Boudreau (Group Health), Connie Trinacty (Kaiser Permanente Hawaii), Gregory A. Nichols (Kaiser Permanente Northwest), Marsha A. Raebel (Kaiser Permanente Colorado), Kristi Reynolds (Kaiser Permanente Southern California), and Patrick J. O’Connor (HealthPartners). This study was supported through a Patient-Centered Outcomes Research Institute (PCORI) Award (ME-1403-12506). All statements in this report, including its findings and conclusions, are solely those of the authors and do not necessarily represent the views of the Patient-Centered Outcomes Research Institute (PCORI), its Board of Governors or Methodology Committee. This work was also supported through an NIH grant (R01 AI074345-07).§ WEB APPENDIX A. REPRESENTING THE TARGETED PARAMETER AS A SEQUENCE OF RECURSIVELY DEFINED ITERATED CONDITIONAL EXPECTATIONS In this section we show that our representation of the statistical target parameter from the main text is indeed a valid mapping for the desired statistical quantity. We will first generalize the framework presented in main text to the case of arbitrary stochastic interventions. We will then present the mapping of the statistical estimand for an arbitrary stochastic intervention in terms of iterated conditional expectations. We will finish the proof by showing that the dynamic intervention considered in main text of the paper is just a special case of a stochastic intervention, making the presented proof also valid for the statistical estimand considered in the main text of the paper. §.§ Observed data, likelihood and the statistical model Suppose we observe n i.i.d. copies of a longitudinal data structureO=(L(0),A(0),…,L(τ),A(τ),L(τ+1)=Y)∼ P,where A(t) denotes binary valued intervention node, L(0) baseline covariates, L(t) is a time-dependent confounder realized after A(t-1) and before A(t), for t=0,…,τ and L(τ+1)=Y is the final (binary) outcome of interest. There are no restrictions on the dimension and support of L(t), t=0,…,τ and we assume that the outcome of interest Y at τ+1 is always observed (i.e., no right-censoring censoring, the data structure in O is never degenerate for any t=τ,…,0). The density p(o) of O∼ P can be factorized according to the time-ordering as p(o) =∏_t=0^τ+1p(l(t)|l̅(t-1),a̅(t-1))∏_t=0^τp(a(t)|l̅(t),a̅(t-1)) =∏_t=0^τ+1q_t(l(t)|l̅(t-1),a̅(t-1))∏_t=0^τg_t(a(t)|l̅(t),a̅(t-1)),for some realization o of O. Recall that l̅(t)=(l(0),...,l(t)) and a̅(t)=(a(0),...,a(t)). Also recall that ((l(-1),a(-1)) is defined as an empty set. Note also that q_t denotes the conditional density of L(t), given L̅(t-1),A̅(t-1), while Q_t denotes its conditional distribution. Similarly, g_t denotes the conditional density of A(t), given (L̅(t),A̅(t-1)), while G_t denotes its conditional distribution. We assume the densities q_t are well-defined with respect to some dominating measures μ_L(t), for t=0,…τ+1 and g_t are well-defined with respect to some dominating measures μ_A(t), for t=0,…,τ. Similarly, assume the density p of O is a well-defined density with respect to the product measure μ. Let q=(q_t:t=1,…,τ+1) and g=(g_t:t=1,…,τ), so that the distribution of O is parameterized by (q,g). Consider a statistical model ℳ for P that possibly assumes knowledge on g. If 𝒬 is the parameter set of all values for q and 𝒢 the parameter set of possible values of g, then this statistical model can be represented as ℳ={P_q,g=QG : q∈𝒬,g∈𝒢}. In this statistical model q puts no restrictions on the conditional distributions Q_t of L(t) given (L̅(t-1),A̅(t-1)), for t=0,…,τ+1. §.§ Stochastic interventions Define the intervention of interest by replacing the conditional distribution G with a new user-supplied intervention G^*={G_t^*:t=0,…,τ} that has a density g^*={g_t^*:t=0,…,τ}, which we assume is well-defined. Namely, G^* is a multivariate conditional distribution that encodes how each intervened exposure A^*(t) is generated, conditional on (L̅(t),A̅(t-1)), for t=0,…,τ. When g_t^* is non-degenerate it is often referred to as a “stochastic intervention” <cit.>. Furthermore, any static or dynamic intervention <cit.> on A(t) can be formulated in terms of a degenerate choice of g_t^*. Therefore, the stochastic interventions are a natural generalization of static and dynamic interventions. We make no further restrictions on G_t^* beyond assuming that A(t) and A^*(t) belong to the same common space 𝒜={0,1} for all t=0,…,τ. §.§ G-computation formula and statistical parameter Define the post-intervention distribution P_q,g^* by replacing the factors G in P_q,g with a new user-supplied stochastic intervention G^*, with its corresponding post-intervention density p_q,g^* given by p_q,g^*(o)=∏_t=0^τ+1q_t(l(t)|l̅(t-1),a̅(t-1))∏_t=0^τg_t^*(a(t)|l̅(t),a̅(t-1)).The distribution P_q,g^* of p_q,g^* is referred to as the G-computation formula for the post-intervention distribution of O, under the stochastic intervention G^* <cit.>. It can be used for identifying the counterfactual post-intervention distribution defined by a non-parametric structural equation model <cit.>.Let O^* denote a random variable with density p_q,g^* (<ref>) and distribution P_q,g^* , defined as a function of the data distribution P of O:O^*=(L(0),A^*(0),…,L^*(τ),A^*(τ),L^*(τ+1)=Y^*).Consider the statistical mapping Ψ(P_q,g) defined as Ψ(P_q,g)=E_P_q,g^*Y^*=∫_o∈𝒪ydP_q,g^*(o).Note that the above mapping is defined with respect to the post-intervention distribution P_q,g^*, and hence, E_P_q,g^*Y^* is entirely a function of the observed data generating distribution P and the known stochastic intervention G^*. In other words, E_P_q,g^*Y^* is identified by the observed data distribution P and it depends on P through Q=(Q_t:t=0,…,τ+1).Under additional causal identifying assumptions of sequential randomization and positivity, as stated in following section, the statistical parameter E_P_q,g^*Y^* can be interpreted as the mean causal effect of the longitudinal stochastic intervention G^* on Y. However, formal demonstration of these identifiability results requires postulating a causal non-parametric structural equation model (NPSEM), with counterfactual outcomes explicitly defined and additional explicit assumptions encoded by a causal diagram <cit.>. The G-computation formula and the post-intervention distribution P_q,g^* play the key role in establishing these identifiability results. We refer the interested reader to <cit.> for the example application of the G-computation formula towards identifiability of the average causal effect of static regimens in longitudinal data with time-varying confounding. Note that the latter causal parameter corresponds with the choice of a degenerate G^* that puts mass one on a single vector a̅(τ). We also refer to <cit.> for the application of the post-intervention distribution P_q,g^* towards identifying the average causal effects of arbitrary stochastic interventions in network-dependent longitudinal data. §.§ Target parameter as a function of iterated conditional means By applying the Fubini's theorem and re-arranging the order of integration, we can re-write the target parameter E_P_q,g^*Y^* in terms of the iterated integrals as follows, E_P_q,g^*Y^*=∫_o∈𝒪ydP_q,g^*(o) =∫_l̅(τ),a̅(τ)[∫_yydQ_τ+1(y|l̅(τ),a̅(τ))]∏_t=0^τdG_t^*(o)∏_t=0^τdQ_t(o) =∫_l̅(τ),a̅(τ)[∫_yydQ_τ+1(y|l̅(τ),a̅(τ))]∏_t=0^τ{ dG_t^*(o)dQ_t(o)} .For conciseness and with some abuse of notation, we used G_t^*(o) and Q_t(o) for denoting G_t(a(t)|l̅(t),a̅(t-1)) and Q_t(l(t)|l̅(t-1),a̅(t-1)), respectively, for t=0,…τ. We also assumed that 𝒪 represents the set of all values of the observed data O. Note that the inner-most integral in the last line is the conditional expectation of Y with respect to the conditional distribution Q_τ+1 of L(τ+1) given (A̅(τ),L̅(τ)), namely,Q̅_τ+1,1(a̅(τ),l̅(τ))≡ E_Q_τ+1[Y|A̅(τ)=a̅(τ),L̅(τ)=l̅(τ)] =∫_y∈𝒴ydQ_τ+1(y|l̅(τ),a̅(τ)).By applying Fubini's theorem one more time, we integrate out a(τ) with respect to the conditional stochastic intervention G_τ^*(o) as follows, E_P_q,g^*Y^*=∫_l̅(τ),a̅(τ)[Q̅_τ+1,1(a̅(τ),l̅(τ))]∏_t=0^τ{ dG_t^*(o)dQ_t(o)} =∫_l̅(τ),a̅(τ-1){∫_a(τ)[Q̅_τ+1,1(a̅(τ),l̅(τ))]dG_τ^*(o)} dQ_τ(o)∏_t=0^τ-1{ dG_t^*(o)dQ_t(o)} ,where we also note that the inner-most integral defines the following conditional expectation,Q̅_τ+1(a̅(τ-1),l̅(τ))≡ E_G_τ^*[Q̅_τ+1,1(A̅(τ),L̅(τ))|A̅(τ-1)=a̅(τ-1),L̅(τ)=l̅(τ).]. We have now demonstrated the first two steps of the algorithm that represents the integral E_p_q,g^*Y^* in terms of the iterated conditional expectations. The rest of the integration process proceed in a similar manner, with the integration order iterated with respect to (q_t,g_t-1^*), for t=τ,…,0, moving backwards in time until we reach the final expectation over the marginal distribution Q_0 of L(0).For notation convenience, let Q̅_τ+1,1≡Q̅_τ+1,1(A̅(τ),L̅(τ)) and Q̅_τ+1≡Q̅_τ+1(A̅(τ-1),L̅(τ)). The full steps that represent E_p_q,g^*Y^* as iterated conditional expectations are as follows[ ; Q̅_t+1,1=E_q_t+1(Q̅_t+1|A̅(t),L̅(t)); Q̅_t+1=E_g_t^*(Q̅_t+1,1|A̅(t-1),L̅(t));…;Q̅_t=0=E_L(0)Q̅_1; =E_p_q,g^*Y̅^* ] Note that this representation allows the effective evaluation of E_p_q,g^*Y^* by first evaluating a conditional expectation with respect to the conditional distribution of L(τ+1), then the conditional mean of the previous conditional expectation with respect to the conditional distribution of A^*(τ), and iterating this process of taking a conditional expectation with respect to L(t) and A^*(t-1) until we end up with a conditional expectation over A^*(0), given L(0), and finally we take the marginal expectation with respect to the distribution of L(0). §.§ Applying the iterative representation of the estimand for stochastic intervention to fixed dynamic intervention Define g_t^* for t=0,…,τ so that they define our dynamic intervention of interest. In particular, let g_t^* be a degenerate distribution that puts mass one on a single value of a(t) and that is equal to g_t^*(a(t)|l̅(t),a̅(t-1))=I(a(t)=a^θ(t)),for t=0,…τ. Then the above representation of E_P_q,g^*Y̅^* in terms of the iterated conditional expectations is equivalent to the mapping Ψ^θ(τ) as presented in the main text of the paper. To see this, note that the above iterated integration steps with respect to the stochastic intervention g_t^* simplify to Q̅_t+1(A̅(t-1),L̅(t)) =E_G_t^*[Q̅_t+1,1(A̅(t),L̅(t))|A̅(t-1),L̅(t).] =Q̅_t+1,1(A̅(t),L̅(t))I(A(t)=A^*(t)) =E_Q_t+1(Q̅_t+1| A(t)=A^*(t),A̅(t-1),L̅(t)),for t=0,…,τ. By plugging this result back into the above described iterated means mapping of E_P_q,g^*Y̅^*, we obtain exactly the same sequential G-computation mapping as the one presented in the main text of this paper for the parameter Ψ^θ(τ). This finishes the proof, since it shows that indeed our statistical parameter representation in the main text is valid, i.e., it produces the desired statistical estimand identified by the post-intervention G-computation formula.§ WEB APPENDIX B. CAUSAL PARAMETER AND CAUSAL IDENTIFYING ASSUMPTIONS Let g^*={g_t^*:t=0,…,τ} denote the stochastic intervention of interest which determines the random assignment of the observed treatment nodes A̅=(A(0),…,A(τ)). Let Y_g^* denote the patient's potential outcome at time τ+1 had the patient been treated according to the randomly drawn treatment strategy g^*. Similarly, let L_g^*(t) denote the counterfactual values of the patient's time-varying or baseline covariates at time t=0,…,τ, under intervention g^*. Finally, let EY_g^* denote the causal parameter of interest, defined as the mean causal effect of intervention g^* on the outcome.The causal validity of the statistical estimand presented above rests on the following two untestable identifying assumptions:[Sequential Randomization Assumption (SRA)] For each t=0,…,τ, assume that A(t) is conditionally independent of (L_g^*(t+1),…,L_g^*(τ),Y_g^*,) given the observed past (L̅(t),A̅(t-1)). [Positivity Assumption (PA)] For k=0,…,τ, assumesup_o∈𝒪∏_t=0^kg_t^*(a(t)|l̅(t),a̅(t-1))∏_k=0^kg_t^*(a(t)|l̅(t),a̅(t-1))<∞P_0-a.e.,where 𝒪 is the support of O=(L(0),A(0),…,L(τ),A(τ),L(τ+1)=Y)∼ P_0.One can obtain the causal identifiability results for dynamic intervention (A_i^θ(0),…,A_i^θ(τ)) by simply letting g_t^* be a degenerate distribution that puts mass one on a single value of a(t) and setting it equal to g_t^*(a(t)|l̅(t),a̅(t-1))=I(a(t)=a^θ(t)),for t=0,…τ. § WEB APPENDIX C. SUMMARY OF PRACTICAL IMPLEMENTATION OF TMLE FOR FIXED DYNAMIC RULE We describe the long-format TMLE algorithm for estimating parameter Ψ^θ(t_0)(P) indexed by the fixed dynamic regimen d̅_θ=(d_θ,0,...,d_θ,t_0). For notational convenience we let t_0=1. To define TMLE we need to use the efficient influence curve (EIC) of the statistical target parameter Ψ^θ(t_0) at P, which is given by: D^*(θ,t_0)(P)=D^0,*(θ,t_0)(P)+D^1,*(θ,t_0)(P)+D^2,*(θ,t_0)(P), where D^0,*(θ,t_0)(P) =(Q_0-Ψ^θ(t_0))D^1,*(θ,t_0)(P) =I(A(0)=A^θ(0))/g_A(0)(O)(Q_1-Q_0)D^2,*(θ,t_0)(P) =I(A̅(1)=A̅^θ(1))/g_A(0)(O)g_A(1)(O)(Y(t_0)-Q_1). The TMLE algorithm described below involves application of the sequential G-computation formula from <cit.>. In this implementation of TMLE one carries out the TMLE update step by fitting a separate ε^j for updating each Q_j, for j=1,0, and sequentially carrying out these updates starting with Q_1 and going backwards. In addition, it involves first targeting the regression before defining it as outcome for the next regression backwards in time. For our particular example with two time-points, this entails obtaining an estimate Q̂_1 of Q_1, running the first TMLE update on Q̂_1 to obtain a targeted estimate Q̂_1^* . This is followed by using the estimate Q̂_1^* to obtain an initial estimate Q̂_0 of Q_0 and running the second TMLE update on Q̂_0 to obtain a targeted estimate Q̂_0^*. Finally, TMLE estimate Ψ̂^θ(t_0) of Ψ^θ(t_0)=E_P(Q_0(A^θ(0),L'(0))) can be obtained from Q̂_0^* as Ψ̂^θ(t_0)=1/n∑_i=1^nQ̂_0^*(A_i^θ(0),f_0(L'_i(0))).In practice, the initial estimate Q̂_1 is obtained by first regressing the outcomes Y_i(1) against (A_i(1),L'_i(1)), among the subjects i∈{1,…,n} that at t=1: a) were at risk of experiencing the event of interest (i.e., Y_i(t-1)=0); b) were uncensored at t (i.e., A_i^C(t)=0); and c) (optionally) had their observed exposure indicator at time t match the values allocated by their dynamic treatment rule d̅_θ (i.e., A_i^T(t)=A_i^θ,T). More generally, fitting Q_1 can rely on data-adaptive techniques based on the following log-likelihood loss function:L_1(Q_1)=-{ Y(1)log Q_1+(1-Y(1))log(1-Q_1)} .The resulting model fit produces a mapping (a(1),l'(1))→ E_n(Y(1)|a(1),l'(1)) that can be now used to obtain an estimate Q̂_1 of Q_1. That is, the prediction Q̂_1 is obtained as E_n(Y_i(1)|A_i^θ(1),L'_i(1)), for subjects i such that T̃_i≥1. However, for all subjects such that T̃_i=0 and A_i^C(0)=0, Q̂_1 is set to Y_i(0). Note that the later prediction Q̂_1 is extrapolated to all subjects who were also right-censored at t=1. The first TMLE update modifies the initial estimate Q̂_1 with its targeted version Q̂_1^*. This update utilizes the estimates ĝ_A(j) of the joint exposure and censoring mechanism g_A(j) for time-points j=0,1. Furthermore, this update will be based on the least favorable univariate submodel (with respect to the target parameter ψ_0) {Q̂_1(ε^1):ε^1} through a current fit Q̂_1 at ε^1=0, were the estimate ε̂^1 of ε^1 is obtained with standard MLE. In practice, we define this parametric submodel through Q̂_1 as Q̂_1(ε^1)=Q̂_1+ε^1 and we use the following weighted loss function function for fitting ε^1:L(Q_1(ε^1))≡ H_1(ĝ)L_1(Q_1(ε^1)),where H_1(ĝ)=I(A̅(1)=A̅^θ(1))∏_j=0^1ĝ_A(j)(O)and the fit of ε^1 defined by ε̂^1=min_ε^1L(Q_1(ε^1)). Thus, the estimate ε̂^1 of ε^1 can be obtained by simply running the intercept-only weighted logistic regression using the sample of observations that were used for fitting Q̂_1, using the outcome Y(1), intercept ε^1, the offset Q̂_1 and the predicted weights H_1(ĝ). The fitted intercept is the maximum likelihood fit ε̂^1 for ε^1, yielding the model update which can be evaluated for any fixed (a,w), by first computing the initial model prediction Q̂_1(a,w) and then evaluating the model update Q̂_1(ε̂^1). This now constitutes the first TMLE step, which we define as Q̂_1^*=(Q̂_1+ε̂^1),for subjects i such that T̃_i≥1.The estimator Q̂_0 of Q_0 is then obtained by selecting subjects i∈{1,…,n} who were uncensored at t=0 (i.e., A_i^C(t)=0) and (optionally) had their observed exposure matching the values allocated by their dynamic treatment rule at t=0 (i.e., A_i^T(t)=A_i^θ,T(t)). The predicted outcomes Q̂_1^*(A_i^θ(1),L'_i(1)) are then regressed against (A_i(0),L'_i(0)) to obtain an estimate Q̂_0: (a(0),l'(0))→ E_n(Q̂_1^*|a(0),l'(0)). More generally, one can consider the following quasi-log-likelihood loss functions for Q_0: L_0,Q̂_1^*(Q_0)=-{Q̂_1^*log Q_0+(1-Q̂_1^*)log(1-Q_0)}and use data-adaptive techniques to obtain an estimate of Q_0. Finally, Q̂_0 is obtained from this regression function fit by evaluating E_n(Q̂_1^*|A^θ(0),L'(0)) for i=1,…,n, which yields n initial predictions Q̂_0 of Q_0. Similar to the update for Q̂_1, the updated estimate Q̂_0^* of Q̂_0 is obtained from the following least favorable univariate parametric submodel {Q̂_0(ε^0):ε^0} through the current fit Q̂_0:Q̂_0(ε^0)=Q̂_0+ε^0and using the following weighted loss function for Q̂_0(ε^0):L_Q̂_1^*(Q_0(ε^0))≡ H_0(ĝ)L_0,Q̂_1^*(Q_0(ε^0)),whereH_0(ĝ)=I(A(0)=A^θ(0))ĝ_A(0)(O). The corresponding TMLE Ψ̂^θ(t_0) of Ψ^θ(t_0) is given by the following substitution estimator Ψ̂^θ(t_0)=1/n∑_i=1^nQ̂_0^*(A_i^θ(0),L'_i(0)). The above loss functions and the corresponding least-favorable fluctuation submodels imply that the TMLE Ψ̂^θ(t_0) solves the the empirical score equation given by the efficient influence curve D^*(θ,t_0). That is, Ψ̂^θ(t_0) solves the estimating equation given by 1/n∑_i=1^nD^*(θ,t_0)(Q̂_1^*,Q̂_2^*,ĝ)=0, implying that Ψ̂^θ(t_0) also inherits the double robustness property of this efficient influence curve. Thus, the TMLE yields a substitution estimator that empirically solves the estimating equation corresponding to the efficient influence curve.§ WEB APPENDIX D. REGULARITY CONDITIONS AND INFERENCE§.§ Regularity conditions The following theorem states the regularity conditions for asymptotic normality of the TMLE Ψ̂^θ(t_0). In this discussion we limit ourselves to providing the regularity conditions in a more limited setting when both nuisance parameters, Q and g, are both assumed to converge to the truth “fast enough” rates, as clarified in conditions below. However, when some or all estimates in Q̂=(Q̂_1,…,Q̂_t_0) are incorrect, the asymptotic normality may still hold, e.g., when nuisance parameters in g converge to the truth at parametric rates. For a more general discussion of such cases and the corresponding technical conditions that guarantee the asymptotic normality of the TMLE we refer to <cit.> and <cit.>.For a real-valued function w↦ f(w), let the L^2(P)-norm of f(w) be denoted by f≡ E[f(𝐖)^2]^1/2. Define ℱ and 𝒢 as the classes of possible functions that can be used for estimating Q and g, respectively. The first assumption below states the empirical process conditions which ensure that the estimators of Q and g are well-behaved with probability approaching one <cit.>. For a K-dimensional vector of functions 𝐟=(f_1,…,f_K) we also define ||𝐟||≡max_k∈{1,…,K}{ ||f_k||}. Finally, we let Pf denote an expectation E_Pf(O) for any function f of O.If the below conditions hold then the TMLE Ψ̂^θ(t_0) is asymptotically linearly estimator of Ψ^θ(t_0) at true P (true distribution of observed data), with the influence curve D^*(θ,t_0)(Q,g) as defined in the previous section. Donsker class: Assume that {D^*(Q,g):Q,g} is P-Donsker class, for Q∈ℱ and g∈ G. Assume that g belongs to a fixed class 𝒢 with probability approaching one. Universal bound: Assume sup_f∈ F,O| f|(O)<∞, where the supremum of O is over a set that contains O with probability one. Note that this condition will be typically satisfied by the above Donsker class condition.Positivity: Assume 0<min_o∈𝒪{∏_j=0^tg_A(t)(o)}, for all t and 𝒪 representing the support of the random variable O.Consistent estimation of D^*: P(D^*(Q̂^*,ĝ)-D^*(Q,g))^2→0 in probability, as n→∞.Rate of the second order term:Define the following second-order term R_n(Q̂^*,ĝ,Q,g)≡ P{ D^*(Q̂^*,ĝ)-D^*(Q̂^*,g)} -P{ D^*(Q,ĝ)-D^*(Q,g)} .Assume that R_n(Q̂^*,ĝ,Q,g)=o_P(1/√(n)). Note that the above two conditions (rate and consistency) will be trivially satisfied if one assumes the following stronger condition: Consistency and rates for estimators of nuisance parameters: Assume that Q̂-Qĝ-g=o_P(n^-1/2), where the norm · for the k-dimensional function f is defined above. Note that these rates are achievable if these estimates Q̂ and ĝ are based on the corresponding correctly specified classes ℱ and 𝒢.§.§ Inference Estimation of the SEs for RDs can be based on the estimates of the efficient influence curve (EIC) D^*(P), where the EIC for our parameter of interest is defined in Web Supplement B. The inference for the parameter Ψ^θ(t_0)(P) can be based on the plug-in variance estimate D̂^*(θ,t_0)=D^*(θ,t_0)(Q̂^*,ĝ) of the EIC D^*(θ,t_0)(Q,g) defined in the Web Supplement B. That is, the asymptotic variance estimate of Ψ̂^θ(t_0) can be estimated as σ̂^2(θ,t_0)=1/n∑_i=1^n[D̂^*(θ,t_0)(O_i)]^2. Furthermore, from the delta method, we know that the EIC for the risk difference Ψ^θ_1(t_0)-Ψ^θ_2(t_0) and single observation O_i is given by D^RD(θ_1,θ_2,t_0)(O_i)=(D^*(θ_1,t_0)-D^*(θ_2,t_0))(O_i). Thus, the asymptotic variance of the TMLE RD Ψ_n^θ_1(t_0)-Ψ_n^θ_2(t_0) can be also estimated via the following plug-in estimator σ̂^2(θ_1,θ_2,t_0)=1/n∑_i=1^n[(D̂^*(θ_1,t_0)-D̂^*(θ_2,t_0))(O_i)]^2.Furthermore, Wald-type confidence intervals (CIs) can be now easily obtained from these variance estimates, as described in detail in <cit.>.§ WEB APPENDIX E. ADDITIONAL ANALYSES§.§ Summary of the IP-weights Note that all IP-weights are unstabilized.§.§.§ Summary of the IP-weights for the data-adaptive approach §.§.§ Summary of the IP-weights for the parametric approach biom | http://arxiv.org/abs/1705.09874v2 | {
"authors": [
"Oleg Sofrygin",
"Zheng Zhu",
"Julie A Schmittdiel",
"Alyce S. Adams",
"Richard W. Grant",
"Mark J. van der Laan",
"Romain Neugebauer"
],
"categories": [
"stat.AP",
"stat.CO",
"stat.ML"
],
"primary_category": "stat.AP",
"published": "20170527224308",
"title": "Targeted Learning with Daily EHR Data"
} |
Instituto de Astrofísica de Canarias, Vía Láctea, 38205 La Laguna, Tenerife, Spain Universidad de La Laguna, Departamento de Astrofísica, 38206 La Laguna, Tenerife, SpainConsejo Superior de Investigaciones Científicas, 28006 Madrid, Spain D.S. Aguado et al. EMP stars identified from SDSS and LAMOSTWe have identified several tens of extremely metal-poor star candidates from SDSS and LAMOST, which we follow up with the 4.2m William Herschel Telescope (WHT) telescope to confirm their metallicity. We followed a robust two-step methodology. We first analyzed the SDSS and LAMOST spectra.A first set of stellar parameters was derived from these spectra with the FERRE code, taking advantage of the continuum shape to determine the atmospheric parameters, in particular, the effective temperature. Second, we selected interesting targets for follow-up observations, some of them with very low-quality SDSS or LAMOST data. We then obtained and analyzed higher-quality medium-resolutionspectra obtained with the Intermediate dispersión Spectrograph and Imaging System (ISIS) on the WHT to arrive ata second more reliable set of atmospheric parameters. This allowed us to derive the metallicity with accuracy, and we confirm the extremely metal-poor nature in most cases. In this second step we also employed FERRE, but we took a running mean to normalize both the observed and the synthetic spectra, and therefore the final parameters do not rely on having an accurate flux calibration or continuum placement. We have analyzed with the same tools and following the same procedure six well-known metal-poor stars, five of them at [Fe/H]<-4 to verify our results. This showed that our methodology is able to derive accurate metallicity determinations down to [Fe/H]<-5.0. The results for these six reference stars give us confidence on the metallicity scale for the rest of the sample. In addition,we present 12 new extremely metal-poor candidates: 2 stars at [Fe/H] ≃ -4, 6 more in the range -4<[ Fe/H]<-3.5, and 4more at -3.5<[ Fe/H]<-3.0. We conclude that we can reliably determine metallicitiesfor extremely metal-poor stars with a precision of 0.2 dex frommedium-resolution spectroscopy with our improved methodology. This providesa highly effective way of verifying candidates from lower quality data.Our model spectra and the details of the fittingalgorithm are made public to facilitate the standardization of the analysis of spectra from the same or similar instruments.WHT follow-up observations of extremely metal-poor stars identified from SDSS and LAMOST D. S. Aguado1,2, J. I. González Hernández1,2, C. Allende Prieto1,2, R. Rebolo1,2,3 December 30, 2023 ========================================================================================§ INTRODUCTION The oldest stars in the Galactic halo contain information about the early Universe after the primordial nucleosynthesis. These stars are key for understanding how galaxies form, what the masses of the firstgeneration of stars were, and what the early chemical evolution of the Milky Way was like. The oldest stars are extremely poor in heavy elementsand havemost likely been preceded locally by only one massive Population III star <cit.>.Chemical abundances in extremely metal-poor (EMP) stars (with [ Fe/H]<-3), in particular dwarf EMP stars, are needed to study the difference between the primordial lithium abundance obtained from standard Big Bang nucleosynthesis, constrained by the baryonic density inferred from the Wilkinson Microwave Anisotropy Probe (WMAP) observations of the Cosmic microware background (CMB) <cit.>, andin situ atmospheric abundance measurements from old metal-poor turn-off stars <cit.>. Unfortunately,very metal-poor stars are extremely rare, with fewer than tenstars known in the [ Fe/H]<-4.5 regime. This seriously limits the number of lithium measurements and detections that are required to shed light on this problem.Detailed chemical abundance determinations of EMP stars require high-resolution spectroscopy <cit.>, but the identification of these stars is a difficult task. In the 1980's and 1990's several different techniques were designed and applied to search for metal-poor stars of the Galactic halo using various telescopes that were equipped with low- and medium-resolution spectrographs <cit.>.More recently, the stars with the lowest iron abundances have been discoveredfrom candidates identified in the Hamburg-ESO survey <cit.> or the SloanDigital Sky Survey <cit.>. In the past decade, additional stars have beenidentified by photometric surveys such as the one using the SkyMapper Telescope, see <cit.>, or the Pristine Survey <cit.>,and spectroscopic surveys such as the Radial Velocity Experiment (RAVE, ), the Sloan Extension for Galactic Understanding and Exploration (SEGUE,), or the large Sky AreaMulti-Object Fiber Spectroscopic Telescope (LAMOST, ).Extremely metal-poor stars tend to be located at large distances. For these stars, spectroscopic surveys usually do not provide spectra with sufficient quality, and it is therefore critical to examine and verify candidates with additional observations with higher quality.In this work, we follow a two-step methodology to identify newextremely metal-poor stars:We perform an improved analysis of SDSS and LAMOST datato select EMP candidates, and we carry outfollow-up observations of asubsample, which allows us to check the stellar parameters, metallicities, andcarbon abundances of these candidates. The paper is organized as follows. In Section <ref> we explain the candidate selection.Section <ref> is devoted to the follow-up observations and data reduction. Section <ref> describes in detail how theanalysis was carried out, including tests using well-known metal-poor stars(Section <ref>).We then discuss our carbon abundance determination from low- andmedium-resolution spectra (Section <ref>).Finally, Section <ref> summarizes our results and conclusions. § LOW SPECTRAL RESOLUTION ANALYSIS AND TARGET SELECTION We have analyzed the low-resolution spectra of more than 2.5 milliontargets from three different surveys:the Baryonic OscillationsSpectroscopic Survey (BOSS, ), SEGUE <cit.>,and LAMOST <cit.>.The analysis of the stars from these surveys was performed with[FERRE is availablefrom http://github.com/callendeprieto/ferre] <cit.>, which allowed usto derive the main stellar parameters, effective temperature T_ eff,and surface gravity log g, together with the metallicity [Fe/H][We use the bracket notationto report chemical abundances: [a/b]= log(N(a)/ N(b)) - log(N(a)/ N(b))_⊙, where N(x) represents number the density of nuclei of the element x.] and carbon abundance [C/Fe] <cit.>.As a result of the extremely low metal abundances of our targets, iron transitions are not detected, either individually or even as blended features, in medium-resolution spectra.We typically measured only the Caii resonance lines and used them asa proxy for the overall metal abundance of the stars, assuming the typicaliron-to-calcium ratio found in metal-poor stars, [Fe/H]=[Ca/H]-0.4. We analyzed about 1,700,000 (∼1,500,000 stars) spectrafrom LAMOST, 740,000 (∼660,000 stars) fromSEGUE and 340,000 (∼300,000 stars) from BOSS. A database with all candidates at [Fe/H]<-3.5 wasbuilt that contains ∼500 objects with a first set of stellar properties (T_ eff, log g, [Fe/H], and [C/Fe]).However, this first selection still contains many outliers and unreliable fits.We reevaluated the goodness of fit by measuring the χ^2 in a limited spectral region around the Balmer lines. This method prevents false positives even at low signal-to-noise ratios (S/N<20). Finally, a visual inspection of the spectra and their best fittingswas carried out to select, to skim the best several tens of candidates. The stars that passed this selection were then scheduled forobservations at medium-resolution and much higher S/N ratio. Table <ref> shows the derived parameters of the final sample that was analyzedin this work. Visual inspection helps us to detect promising candidates.The two LAMOST objects discussed below are good example for cases wherevisual inspection was critical. In addition to these stars, we chose some of the mostmetal-poor stars known ([Fe/H]<-4.0), with published determinationsfrom high-resolution spectroscopic data, in order to test our methodology. § OBSERVATIONS AND DATA REDUCTION The second step in our methodology makes use of follow-up medium-resolution spectroscopy obtained with the Intermediate dispersion Spectrograph and Imaging System (ISIS) <cit.> spectrographon the 4.2 m William Herschel Telescope (WHT) at the Observatorio delRoque de los Muchachos (La Palma, Spain). We used the R600B and R600R gratings, the GG495 filter in the red arm,and the default dichroic (5300 Å).The mean FWHM resolution with a one-arcsecond slit was R∼ 2400 inthe blue arm and R∼ 5200 in the red arm. More details are provided in <cit.>. The observations were carried out over the course of five observing runs:run I: Dec 31 - Jan 2, 2015 (three nights); run II: February 5-8, 2015 (four nights); run III:August 14-18,2015 (five nights); run IV: May 1-2, 2016 (two nights), and run V: July29 - 31, 2016 (three nights).A standard data reduction (bias substraction, flat-fielding and wavelengthcalibration, using CuNe + CuAr lamps) was performed with theonespec package in IRAF[IRAF is distributed by theNational Optical Astronomy Observatory, which is operated by theAssociation of Universities for Research in Astronomy(AURA) under cooperative agreement with the National ScienceFoundation] <cit.>.High-resolution spectra of well-known metal-poor stars wereextracted from the ESO archive[Based on data from theESO Science Archive Facility ] and obtained with the Ultraviolet andVisual Echelle Spectrograph (UVES, ) at the VLT.The spectral range of these data is 3300-4500 Å and the resolvingpower is about 47,000.The spectra were reduced using the automatic UVES pipeline <cit.>, offered to the community through the ESO ScienceArchive Facility.§ ANALYSIS AND DISCUSSION In order to derive the stellar parameters and chemical abundances agrid of syntheticspectra was computed with the ASSϵT code<cit.>, which uses the Barklem codes <cit.> to describe the broadening of the Balmer lines.This grid resembles the grid used by <cit.>, with somedifferences described below, and it is provided ready to be used with as an electronic table.The model atmospheres were computed with the same Kuruczcodes andstrategies as described by <cit.>.The abundance of α-elements was fixed to [α/ Fe]=+0.4, and the limits of the grid were -6≤[ Fe/H]≤-2; -1 ≤[ C/Fe]≤ 5,4750K≤ T_eff≤7000K and 1.0≤≤5.0, assuming a microturbulence of 2 km s^-1. We searched for the best fit using by simultaneously deriving the main three atmospheric parameters and the carbon abundance.Interstellar reddening and the instrument response distort the intrinsic shape of the stellar spectra. To minimize systematic differences between the synthetic spectra andthe observations,we worked with continuum-normalized fluxes. The observed and synthetic spectra were both normalized using a running-mean filter with a width of 30 pixels (about 10 Å). Although the precision in the derived parameters wasfound to be very sensitive to the normalization scheme, our testsshowed that when no reliable informationon the shape spectral energy distribution is available,the running-mean filter offers very good performance.With this modification we were able to retrieve T_ eff andinformation from ISIS spectra.In order to determine how robust theresults are, wecarried out a series of consistency tests by adding random noise witha normal distribution to an observed spectrum of the metal-poor starG64-12. The results are illustrated in Fig. <ref> and are described below. We find thatis able to recover the original set of parametersfrom high S/N values down to S/N =15 when the derived solutionbegins to drift away from the expected solution.This is consistent with the results obtained by <cit.>for SDSS spectroscopy.We used observations of well-known metal-poor stars to test our methodology.The comparison between the literature metallicitiesfrom high-resolution analyses and those we obtained using medium-resolution spectroscopy indicates that ourapproach works quite well. Nonetheless, our methodology does not allow us to separate theCaii K spectral lines from the star from the interstellar medium (ISM) contributions that arenot resolved in the ISIS spectra.Below, we describe tests in which we reanalyzde the originalUVES spectra (see Section <ref>) of several objects after smoothing to understand the effect of having a lower spectral resolution.is able to derive internal uncertainties in several ways. Afterextensive testing in our ISIS spectra, we find that the best optionis to inject noise using a normal distribution to the observed spectraseveral times, and search for the best-fitting parameters.Such Monte Carlo simulations were performed 50 times for eachobserved spectrum.The dispersion in the derived parameters was adopted as a measurement of the internal uncertainty on these parameters in our analysis. In addition to the final combined spectra for each target, we analyzed each individual exposure in the same way. Figure <ref> shows the relationship between our derived internalrandom uncertainties and the S/N.The uncertainties on the final combined spectra followthe same trend as was found in the analysis of individual exposures. The only giant star in the sample, J2045+1508, shows a higher uncertainty in its surface gravity determination, as expected (see Table <ref>). From these tests, we conclude that our random errors areconsistently and reliably determined, with the exception of the[C/Fe] abundance ratios, for which we tend to underestimate the random uncertainties, given that the dispersion from individual estimates of this parameter is larger than the random errorfrom individual exposures.This is particularly relevant at [C/Fe]<1.0 dex.In addition to random errors, other sources of systematic uncertaintycan affect our results. Stellar effective temperatures are not easy to determine accurately<cit.>.We adopted a 100 K uncertainty following the comparison between photometric and spectroscopic methods by <cit.>. In addition, the stellar isochrones depicted in Fig. <ref> suggest that systematic 0.3-0.5 dex error could be present in our spectroscopicgravities (see Sects. <ref> and <ref>).Based on these comparisons, we decided to adopt a systematic uncertaintyin surface gravity of 0.5 dex. The error bars shown for the data in Fig. <ref> are the quadraticcombination of our estimated random and systematic uncertainties. We adopted a 0.1 dex metallicity error, according to the comparisonsbetween our results and those in the high-resolution analyses inthe literature, but we caution that much larger systematic uncertainties could be present as a result of departures from local thermal equilibrium (LTE) andhydrostatic equilibrium (see <cit.> and references therein),which could add up to 0.8-0.9 dex. Based on other works (see Section <ref>) we estimate that ourderived carbon abundances could have a systematic uncertainty ofabout 0.3 dex. In Table <ref> we show the carbon abundances for well-knownmetal-poor stars whose uncertainties were computed by adding inquadrature the random error and the adopted systematic error. §.§ Well-known metal-poor stars §.§.§ G64-12<cit.> observed G64-12 with the High DispersionSpectrograph (HDS) at the Subaru telescope and derived itsmain atmospheric parameters asT_ eff=6390± 100 K, =4.4± 0.3, and [ Fe/H]=-3.2± 0.1. We obtained with ISIS at the WHT a high-quality spectrum (S/N∼300) ofthis object and derived a set of parameters that excellently agreewith those from <cit.>:T_ eff=6377± 104 K, =4.8± 0.7, and[ Fe/H]=-3.2± 0.2. §.§.§ SDSS J1313-0019 There is an open debate about the metallicity of this object, discovered by<cit.> using a low-resolution spectrum from the BOSS project<cit.>. <cit.> derived the following set of parameters:T_ eff=5300± 50 K, =3.0± 0.2,[ Fe/H]=-4.3± 0.1, and [ C/Fe]=2.5± 0.1.This temperature reproduces both the local continuum slope and the shape of theBalmer lines in the BOSS spectrum of the star.Shortly after the discovery, <cit.> obtained a high-resolution (R ∼ 35,000)spectrum with the MIKE spectrograph at the Magellan-Clay telescope. Their adopted effective temperature and surface gravity are T_ eff=5200± 150 K and =2.6± 0.7.They measured 37 FeI lines (FeII features were not detected)and estimated a metallicity of [ Fe/H]=-5.00± 0.28 and [ C/Fe]=2.96± 0.28. A high-S/N medium-resolution spectrum of J1313-0019 wasobtained as part of our WHT program. Figure <ref> shows the entire spectrum and the best-fitting spectrum from the analysis. We derive T_ eff=5525± 106 K, =3.6± 0.5,[ Fe/H]=-4.7± 0.2, and [ C/Fe]=2.8± 0.30.Our derived metallicity is in between those by <cit.>and <cit.>, but closer to the determination by <cit.>. When we use carbon-enhanced model atmospheres, consistent with the high[C/Fe] ratio adopted in the spectral synthesis, it does not introducesignificant changes in the derived values. The difference between our analysis and that in <cit.> ispartly explained by the fact that we assume [α/H]=0.4, whilethe authors derived [α/H]=0.2± 0.1.We checked that when we impose in theanalysisthe same effective temperature and surface gravity as adoptedby <cit.> (T_ eff=5200 and=2.6), we recovera metallicity and carbon abundance of [ Fe/H]=-4.9± 0.2and [ C/Fe]=2.7± 0.3, respectively.The adoption of an effective temperature 300 K cooler by <cit.>is responsible for an offset of about 0.2 dex in metallicity. Since our value of T_ eff (see Table <ref>) is mostly based on fitting all the Balmer lines available in the spectrumwe consider it reliable. §.§.§ HE 0233-0343 This star was studied by <cit.> using high-resolution spectra acquired with UVES on the VLT, and it is one of the rare stars at such a low metallicity where the lithium abundance wasmeasured, A(Li)=1.77. These objects are key for understanding the cosmological Liproblem <cit.>. Our effective temperature determination, T_ eff=6150± 103 K, is consistent with the temperature proposed by <cit.> of T_ eff=6100± 100 K. They assumed an age of 10 Gyr to infer a surface gravity of =3.4± 0.3 from isochrones, while we derive =4.9± 0.7 from the ISIS spectrum.Calcium ISM absorption is clearly visible in the UVES spectrumof this star, but it is not resolved at the ISIS resolution (see Fig. <ref>). An unresolved ISM contribution biases ourmetallicity determination from to [ Fe/H]=-4.0± 0.1,while <cit.> derived [ Fe/H]=-4.7± 0.2. To check the consistency of the high- and medium-resolution results, we smoothed the UVES spectrum to the ISIS resolution andreanalyzed it. We arrived at the same result as for the ISIS data: T_ eff=6207 K; =4.9, and [ Fe/H]=-4.0,as illustrated in Fig. <ref>.§.§.§ SDSS J1029+1729 The star SDSS J1029+1729 has no carbon (or nitrogen) detected, whichmakes it the most metal-poor star ever discovered <cit.>.This rare object challenges theoretical calculations that predict that nolow-mass stars can form at very low metallicities.Using UVES with a resolving power R∼38000, <cit.> derived the following set of parameters:T_ eff=5811± 150 K, =4.0± 0.5, and[ Fe/H]=-4.9± 0.2.Our analysis of the medium-resolution ISIS spectra arrives atT_ eff=5845± 105 K, =5.0± 0.8 and[ Fe/H]=-4.4± 0.2. Following a similar argumentas given in Section <ref>, the difference in metallicity is associated with the calcium ISM contribution (See Fig. <ref>).This is demonstrated by degrading the UVES spectrum to the resolutionof the ISIS data.This analysis gives a result that is fully consistent with the result fromthe ISIS spectrum:T_ eff=5867 K, =5.0, and [ Fe/H]=-4.5.§.§.§ HE 1327-2326 The study of this star is complicated because of the complex structure ofthe calcium ISM features and the extraordinary amount of carbon([C/Fe]=4.26, ) in the stellaratmosphere (see Fig. <ref>). The parameters from <cit.> from a high-resolution UVESspectrum are T_ eff=6180± 100 K, =3.7± 0.3, and[ Fe/H]=-5.6± 0.1 whilereturnsT_ eff=6150± 102 K, =4.3± 0.7, and [ Fe/H]=-4.8± 0.2 from the medium-resolution ISIS spectrum.We display inFig. <ref> the original UVES observation,the same data smoothed to the resolution of ISIS, and the ISIS spectrum. The UVES spectrum resolves multiple ISM features, while ISIS only uncoverssome of the blue components so that theanalysis of the ISIS spectrumreturns a higher metallicity.The smoothed spectrum solution is again consistent with ourown medium-resolution ISIS values: T_ eff=6119 K; =4.3, and [ Fe/H]=-4.9. Theuse of medium-resolution data inevitably leads to a biased metallicity because of theunresolved ISM calcium absorption. §.§.§ SDSS J1442-0015 The S/N ratio of this spectrum is the lowest in our sample since it is oneof the faintest metal-poor star in the [ Fe/H]<-4.0 regime.Even so, the original set parameters derived from UVES spectroscopy by <cit.>, T_ eff=5850± 150 K, =4.0± 0.5, and[ Fe/H]=-4.1± 0.2, are in fair agreement withour determinations from ISIS data: T_ eff=6036± 102 K, =4.9± 0.5, and[ Fe/H]=-4.4± 0.2.The spectrum of J1442-0015clearly shows an ISM contribution tothe observed calcium absorption, but this component is already resolvedin the ISIS data (see Fig. <ref>).The effective temperatureadopted by <cit.>, which is slightly lowerthan ourestimate, should lead to a difference of at least ∼0.4 dex between the two metallicity determinations but in the opposite direction to whatwe find. In addition, our analysis of the UVES spectrum smoothed to the resolution of ISIS provides a T_ eff=6167 K, =4.9 and [ Fe/H]=-4.2, consistent with the results from ISIS. §.§ New set of extremely metal-poor starsFollowing the same methodology as in Section <ref>, we analyzed spectra from the ISIS instrument for a sample of metal-poor star candidates identified from SDSS and LAMOST. The mean S/N of these spectra is around 75, so we expect theresultsto be reliable. Our effective temperatures are trustworthy, since we were able torecover the temperature for several well-known metal-poor stars whose metallicitiesare consistent with the literature values. In addition, the derived metallicitiesbyare also reliable and consistent with the results obtained from the analysis of SDSS spectra <cit.>. Only our surface gravities appear to be subject to a systematic error ofabout 0.5 dex.This is in contrast to the small random uncertainties we derive forour stellar parameters, and to the surface gravity in particular (Sect. <ref>):lower than 0.1 dex at S/N∼ 50.Figure <ref> shows DARMOUTH isochrones [The Dartmouth StellarEvolution Program (DSEP) is available from www.stellar.dartmouth.edu],HB,and AGB tracks compared to the stellar parameters derived using on the ISIS spectra and their derived error bars. Our mean uncertainty in the metallicity determination is 0.12 dex,whereas the mean uncertainty in T_ eff is 103 K. The mean metallicity difference between our first metallicity estimates from SDSS/LAMOST spectra and the second estimate we obtain from higher qualityISIS spectra is 0.31 dex with a standard deviation of 0.20 dex.In this computation we excluded the stars J134157+513534and J132917+542027, since the metallicity differences for them are much larger. The LAMOST spectra of these stars show several artifacts that donot allow us to perform a correct continuum normalization with ourrunning-mean algorithm, and consequently, we cannot derive reliable parameters from these data. However, our ISIS spectra have significantly higher quality,S/N∼ 82 and 76, and we finally derived areliable metallicityfor both stars, [Fe/H]=-2.7 and [Fe/H]=-3.5, respectively. In Fig. <ref> we compare the results that are summarized in Table <ref> with those from the low-resolution analysis (Table <ref>). In addition, we reanalyzed the stars studied in <cit.> with the improved methodology we presented here. We obtain a slightly higher dispersion for candidates with lower quality spectra. Our analysis uncovers two objects (J0304+3910 and J1055+2322)at [Fe/H]≃-4.0and six more at -3.5≥[Fe/H]>-4.0, inthe domain of the extremely metal-poor stars, and most of themappear to be dwarfs at ≥ 4.0. A deeper study of this sample using high-resolution spectroscopywould provide abundances for additional elements, which would help to constrainthe nature of the stars and the early chemical evolution of the Galaxy. §.§ Carbon abundances Carbon-enhanced metal-poor (CEMP) stars are defined by <cit.>as stars with [C/Fe]≥ +1.0. The fraction of CEMP/EMP starsincreases as metallicity decreases. Deriving reliable carbon abundancesin metal-poor stars using medium-resolution spectroscopy is not alwayspossible <cit.>. For the well-known EMP sample studied in Section <ref>, werecover carbon abundances that are compatible with the literature values forthree CEMP stars: SDSS J1313-0019, HE 1327-2326,and HE 0233-0343(see Table 3). Figure <ref> shows the carbon abundances derived from ISIS spectra for the targets studied in this work, together with those from <cit.>, with the improved methodology, and otherCEMP stars from the literature. The data suggest that the lower the metallicity, the stronger the carbon enhancement, as expected (see, e.g., <cit.>). Our ability to measure carbon abundances decreases as the effectivetemperature of the star increases because of CH dissociation.Empirically, to measure carbon abundances at the level of [C/Fe]≥ +0.5, the star should haveT_ eff∼5500/5700 K. Moreover, if the effective temperature lies at ∼5000/5100 K, weare able to measure carbon at[C/Fe]≥ +0.0. In addition, an S/N of at least 30 is required to derive reliable carbon abundances. Table <ref> summarizes our results and shows thatour sample includes at least three confirmedCEMP stars: SDSS J0151+1639, SDSS J1733+3329, andSDSS J2310+1210 with [Fe/H]=-3.8, -3.4, and -3.8, respectively,in perfect agreement with the statistics by <cit.>. In addition, five more stars appear to be very close toour detection limit of [C/Fe]∼+0.7-0.8.§ CONCLUSIONS We have demonstrated that our methodology for identifying metal-poorstars using low- and medium-resolution spectroscopy is effective.We were able to scrutinizemore than2.5 million spectra from SDSS and LAMOSTto identify extremely metal-poor star candidates, which wefollowed up with significantly higher S/N and slightlyhigher resolution ISIS observations. Our success in identifying metal-poor candidates is quite high,with only one star, J1329+5420, at [Fe/H]>-3. The use of thecode and a customgrid of synthetic spectra allowed us to simultaneously derivethe atmospheric parametersand the carbon abundance, which improved our values from SDSS or LAMOST. We showed thatcan be used efficientlyon extremely metal-poor stars. We make our tools publicly available to facilitate the cross-calibration of results from other teams.Our method offers an excellent way toidentify and analyze CEMP star candidates without the need forhigh-resolution spectroscopy. We presented a new EMP sample andreliable determinations of the metallicities and carbon abundances of these stars.These stars, especially the two at [Fe/H]<-4.0,are good candidates for follow-up high-resolution observations. This domain is sparsely populated, with fewer than 30 known stars, and it is of high interest for investigating the early chemical evolution of the Galaxy. The fact that we have used a grid of synthetic spectra including carbonas a free parameter not only in the synthesis but in the model atmosphereshelps us to detect promising extremely metal-poor stellarcandidates and derive their carbon abundances when[C/Fe]>0.7-1.0 and S/N>30/40, and even lower carbon enhancements and/or S/N values if the star is colder than 5500 K.We included in our work severalwell-known metal-poor stars that were previously analyzed in the literature.Astudy of these objects is useful to check for systematicdifferences that are due to multiple analysis methods. Our results shown that we are able to recover the effective temperaturewith an uncertainty of about 100 K and the metallicity within 0.2 dex.In addition, the comparison of our parameters with model isochrones suggests that our surface gravities have a systematic uncertainty of about 0.5 dex. Our study also confirms that we are able to recover thecarbon abundance for CEMP stars using medium-resolution spectra.Future work should include observations at higher resolution of the mostinteresting extremely metal-poor stars identified in this paper to study their chemical abundance patterns in detail. DA is thankful to the Spanish Ministry of Economy and Competitiveness(MINECO) for financial support received in the form of aSevero-Ochoa PhD fellowship, within the Severo-Ochoa International PhDProgram. DA, CAP, JIGH, and RR acknowledge the Spanish ministry projectMINECO AYA2014-56359-P.JIGH acknowledges financial support from the Spanish Ministry ofEconomy and Competitiveness (MINECO) under the 2013 Ramón y Cajalprogram MINECO RYC-2013-14875. This paper is based on observations made with the William Herschel Telescope, operated by the Isaac Newton Group at the Observatorio delRoque de los Muchachos, La Palma, Spain, of the Instituto deAstrofísica de Canarias. We thank the ING staff members for theirassistance and efficiency during the four observing runs in visitor mode.The author gratefully acknowledges the technical expertiseand assistance provided by the Spanish Supercomputing Network(Red Espanola de Supercomputacion), as well as the computer resources used: the LaPalma Supercomputer, located at the Instituto de Astrofisica de Canarias. | http://arxiv.org/abs/1705.09233v2 | {
"authors": [
"D. S. Aguado",
"J. I. González Hernández",
"C. Allende Prieto",
"R. Rebolo"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170525154159",
"title": "WHT follow-up observations of extremely metal-poor stars identified from SDSS and LAMOST"
} |
Núcleo de Investigación y Desarrollo Tecnológico Universidad Nacional de Asunción, Paraguay [email protected] A Block-Sensitivity Lower Bound for Quantum Testing Hamming Distance Marcos Villagra December 30, 2023 ==================================================================== The Gap-Hamming distance problem is the promise problem of deciding if the Hamming distance h between two strings of length n is greater than a or less than b, where the gap g=|a-b|≥ 1 and a and b could depend on n. In this short note, we give a lower bound of Ω( √(n/g)) on the quantum query complexity of computing the Gap-Hamming distance between two given strings of lenght n. The proof is a combinatorial argument based on block sensitivity and a reduction from a threshold function. § INTRODUCTION A generalized definition of the Hamming distance is the following: given two strings x and y, decide if the Hamming distance h(x,y) is greater than a or less than b, with the condition that b<a. Note that this definition gives a partial boolean function for the Hamming distance with a gap. There is a entire body of work on the computation of a particular case of this notion of Hamming distance in the decision tree and communication models known as the Gap-Hamming distance (GHD) problem, which asks to differentiate the cases h(x,y)≤ n/2-√(n) and h(x,y)≥ n/2+√(n) <cit.>. A lower bound on GHD implies a lower bound on the memory requirements of computing the number of distinct elements in a data stream <cit.>. Chakrabarti and Regev <cit.> give a tight lower bound of Ω(n); their proof was later improved by Vidick <cit.> and then by Sherstov <cit.>. For the Hamming distance with a gap of the form n/2± g for some given g, Chakrabarti and Regev also prove a tight lower bound of Ω(n^2/g^2). In the quantum setting, there is a communication protocol with cost 𝒪(√(n)log n) <cit.>.Suppose we are given oracle access to input strings x and y. In this note, we prove a lower bound on the number of queries to a quantum oracle to compute the Gap-Hamming distance with an arbitrary gap, that is, for any given g=a-b. Let x,y∈{0,1}^n and g=a-b with 0≤ b<a≤ n. Any quantum query algorithm for deciding if h(x,y)≥ a or h(x,y)≤ b with bounded-error, with the promise that one of the cases hold, makes at least Ω( √(n/g) ) quantum oracle queries. The proof is a combinatorial argument based on block sensitivity. The key ingredient is a reduction from a a threshold function. A previous result of Nayak and Wu <cit.> implies a tight lower bound of Ω(√(n/g)+√(h(n-h))/g); their proof, however, is based on the polynomial method of Beals et al. <cit.> and it is highly involved. The proof presented here, even though it is not tight, is simpler and requires no heavy machinery from the theory of polynomials. § PROOF OF THEOREM <REF> Let a,b be such that 0≤ b <a≤ n. Define the partial boolean function GapThr_a,b on {0,1}^n asGapThr_a,b(x)={ 1 if|x| ≥ a 0 if|x| ≤ b. . To compute GapThr_a,b for some input x, it suffices to compute the Hamming distance between x and the all 0 string. Thus, a lower bound for Gap-Hamming distancefollows from a lower bound for GapThr_a,b.Let f:{0,1}^n→{0,1} be a function, x∈{0,1}^n and B⊆{1,…,n} a set of indices called a block. Let x^B denote the string obtained from x by flipping the variables in B. We say that f is sensitive to B on x if f(x)≠ f(x^B). The block sensitivity bs_x(f) of f on x is the maximum number t for which there exist t disjoint sets of blocks B_1,…,B_t such that f is sensitive to each B_i on x. The block sensitivity bs(f) of f is the maximum of bs_x(f) over all x∈{0,1}^n.From Beals et al. <cit.> we know that the square root of block sensitivity is a lower bound on the bounded-error quantum query complexity. Thus, Theorem <ref> follows inmediately from the lemma below.bs(GapThr_a,b)=Θ(n/g). Let x∈{0,1}^n be such that GapThr_a,b(x)=0 and suppose that |x|=b. To obtain a 1-output from x we need to flip at least g=a-b bits of x. Hence, we divide the n-b least significant bits of x in non-intersecting blocks, where each block flips exactly g bits. The number of blocks is ⌊n-b/a-b⌋, which is at most bs_x(GapThr_a,b). To see that ⌊n-b/a-b⌋ is the maximum number of such non-intersecting blocks, consider what happens when the size of a block is different from g. If the size of a block is less that g, then we cannot obtain a 1-output from x; if the size of a block is greater than g, then the number of blocks decreases. Thus, we have that bs_x(GapThr_a,b)=⌊n-b/g⌋.For any x' with |x'|<b, we need to flip a-b bits plus b-|x'| bits. Using our argument of the previous paragraph, the size of each block is thus g+b-|x'|, and hence, bs_x'(GapThr_a,b)=⌊n-|x'|/g+b-|x'|⌋. Note that bs_x'(GapThr_a,b)≤ bs_x(GapThr_a,b).For the case when GapThr_a,b(x)=1 and |x|=a, to obtain a 0-output from x we need to flip at least g bits of x. Hence the same argument applies, and thus, bs_x(GapThr_a,b)=⌊n-a/g⌋.Taking the maximum between the cases when |x|=b and |x|=a, we have that bs(GapThr_a,b)=max{(n-b)/g,(n-a)/g}=Θ(n/g).splncs03 | http://arxiv.org/abs/1705.09710v1 | {
"authors": [
"Marcos Villagra"
],
"categories": [
"cs.CC",
"F.1.2; F.2.0"
],
"primary_category": "cs.CC",
"published": "20170526204140",
"title": "A Block-Sensitivity Lower Bound for Quantum Testing Hamming Distance"
} |
AnEfficient Keyless Fragmentation Algorithm for Data Protection Katarzyna Kapusta, Gerard Memmi, and Hassan NouraTélécom ParisTech, Université Paris-Saclay23, avenue d'Italie, 75013 Paris, France Email: {firstname.lastname}@telecom-paristech.frDecember 30, 2023 =================================================================================================================================================================================================The family of Information Dispersal Algorithms is applied to distributed systems for secure and reliable storage and transmission. In comparison with perfect secret sharing it achieves a significantly smaller memory overhead and better performance, but provides only incremental confidentiality.Therefore, even if it is not possible to explicitly reconstruct data from less than the required amount of fragments, it is still possible to deduce some information about the nature of data by looking at preserved data patterns inside a fragment.The idea behind this paper is to provide a lightweightdata fragmentation scheme, that would combine the space efficiency and simplicity that could be find in Information Dispersal Algorithms with a computational level of data confidentiality. § INTRODUCTION Fragmenting data and dispersing them over different physical locations through several transmission paths slows down an attacker from obtaining the totality of the original data. Data are usually transformed into fragments by the use of secret sharing <cit.>, information dispersal algorithms <cit.>, or data shredding <cit.>. The choice of the most appropriate fragmentation method depends on the particularity of a given use case. A user has to balance between memory use, performance and desired confidentiality level. Fragments obtained with perfect secret sharing are highly secure, but the technique is slow and very costly in memory. Information dispersal is resilient and relatively fast, but not secure. Combining symmetric encryption with fragmentation is secure and easily scalable, but in some circumstances may be less efficient than information dispersal <cit.>.A fragmentation technique for long-term archival storage of large data will usually differ from the one applied to disperse small data packets. In the last case, the choice would be to pick some lightweight fragmentation technique, like information dispersal or a fast computational secret sharing scheme, rather than to apply more complex mechanisms <cit.>.The idea of an Information Dispersal Algorithm(IDA) was first introduced by Rabin in the late 80s <cit.>. Broadly, the algorithm multiplies data chunks by a matrix in order to obtain fragments that are a linear combination of the chunks' data and matrix elements. The recovery is only possible when a certain amount of these fragments is being gathered. Rabin's IDA has several advantages: it adds resilience to data, produces almost no storage overhead and uses simple arithmetic operations. Although, the scheme guarantees only incremental confidentiality. An eavesdropper knowing the dispersal matrix can verify if a fragment has a predetermined value. Moreover, such attacker can guess the content of missing fragments when data have recognizable patterns. Despite of such problems, IDAs are still taken into consideration in the context of data protection, as the obstacle of not being able to explicitly reconstruct initial data from less than required amount of fragments may be sufficient in some scenarios <cit.>. Such scenarios include all use cases were memory or performance overhead caused by secret sharing or cryptographic operations may be a burden and data protection requirements are not too high. An example could be a fragmented database <cit.> where we would like to quickly retrieve records or a multipath transmission where we would like to defragment only the next destination address <cit.>.A new scheme situated between computational secret sharing and information dispersal algorithms was recently introduced in a three pages poster <cit.>. Its complexity and storage overhead are comparable to the one of the IDAs, but it provides higher level of data confidentiality. This paper significantly extends the previous proposal. It contains detailed descriptions, as well as presents more experimental security and performance analyses realized on industrial data.This paper is organized as follows. Section <ref> summarizes the notation used. Section <ref> contains a description of our contribution. Section <ref> presents related works. Section <ref> describes the scheme. Section <ref> presents its empirical security evaluation. Section <ref> contains a cryptanalysis discussion. Section <ref> shows performance results. An insight into future works ends the paper.§ NOTATION In order to unify descriptions, we are introducing the notation presented in Table <ref>. § OUR CONTRIBUTIONThe proposed algorithm fragments initial data d of size d_size into n fragments of a size close to d_size/k, any k of which are needed for data recovery. Initial data are processed by sets of k chunks. Data encoding is not based on a matrix multiplication, but on a modification of Shamir's secret sharing scheme <cit.> and depends on the encoding results of the previously processed k data chunks. A pseudo-random seed is used as the first set of data chunks and dispersed within the data. The main benefits of such scheme are:* No symmetric key encryption is applied, the processing makes use only of simple operations.* Data patterns are not being preserved inside fragments and the content verification is not straightforward(in contrast to IDAs).* Partial data defragmentation is possible(in contrast to schemes based on the all-or-nothing transform). § RELATED WORKSThis section presents most relevant works from the domains of secret sharing and information dispersal. §.§ Information Dispersal AlgorithmsAn Information Dispersal Algorithm <cit.> divides data d into n fragments of size d_size/k each, so that any k fragments suffice for reconstruction. More precisely, n data fragments are obtained by multiplying initial data by a k × n nonsingular generator matrix . Recovery consists in multiplying any k fragments by the inverse of a k × k matrix built from k rows of the generator matrix. Information dispersal adds redundancy to data and does not produce storage overhead. In <cit.>, Li analyzed the confidentiality of IDAs. For instance, Rabin'sIDA proposal was evaluated to have strong confidentiality, as the original data cannot be explicitly reconstructed from fewer than the k required fragments. However, even if it is not possible to directly recover the data, some information about the content of the initial data is leaked. Indeed, data patterns are preserved inside the fragments when the same matrix is reused to encode different data chunks. A similar problem occurs while using the Electronic Code Book block cipher mode for block cipher symmetric encryption <cit.>. §.§ Shamir's secret sharingShamir's perfect secret sharing scheme <cit.> takes as input data d and fragments them into n fragments f_1,...,f_n, of which at least k are needed for data recovery. The algorithm is based on the fact that given k unequal points x_1,...,x_k and arbitrary values f_1,...,f_k there is at most one polynomial y(x) of degree less or equal to k-1 such that y(x_i)=f_i, i=1,...,k. The algorithm provides with the highest level of confidentiality, but has quadratic complexity in function of k and produces fragments of size equal to the initial data. Therefore, it is usually applied for protection of smaller data like encryption keys. In such a use case, drawbacks of the scheme are acceptable, but for larger data they may be a major obstacle.§.§ Secret Sharing Made ShortKrawczyk's Secret Sharing Made Short (SSMS) <cit.> combines symmetric encryption with perfect secret sharing for protection of larger data. Data d are encrypted using a symmetric encryption algorithm, then fragmented using an Information Dispersal Algorithm. The encryption key is fragmented using a perfect secret sharing scheme and dispersed within data fragments. In consequence, the solution does not require explicit key management. The storage overhead does not depend on data size, but is equal to the size of the key per data fragment. The performance of the SSMS technique depends on the details of the chosen encryption and IDA techniques. §.§ AONT-RSThe AONT-RS technique <cit.> is similar to SSMS, as it combines symmetric encryption with data dispersal. It applies an all-or-nothing transform(AONT) <cit.> to create k fragments: encrypted data are divided into k-1 fragments and an additional fragment is generated by xor-ing hashes of these data fragments with the key used for encryption. Additional n-k fragments are produced using a systematic Reed-Solomon error correction code. Data integrity is ensured by the use of a canary that is dispersed within the fragments. §.§ Parakh's schemeA steganographic threshold scheme presented in <cit.> transforms k data chunks into n data fragments using a single polynomial of degree k-1. The size of produced fragments depends on the value of k: it decreases while the number of data chunks and the degree of the polynomial are growing. § FRAGMENTATION ALGORITHMThis section describes in details the processing core of the proposed fragmentation scheme. Algorithm <ref> presents the steps of the fragmentation procedure. Data defragmentation is not presented in the form of an algorithm, as it is basically a direct inverse of fragmentation.Data processing flow Initial data d are treated as a concatenation of l data chunks. These l data chunks are encoded one by one into l data shares. Further, n fragments are constructed from data shares in a way that k fragments are sufficient for the recovery of d. For a more convenient processing, data chunks are regrouped into Data Chunk Sets of k elements, where DCS_i(j) is the jth data chunk in the set i. Initial data d may be then presented as a concatenation of DCSs: DCS_1,...,DCS_m (m=l/k). Similarly, data shares are regrouped into Data Share Sets of k elements, so the result of encoding is a concatenation of DSSs: DSS_1,...,DSS_m. At the end, data shares are distributed to k final fragments and n-k redundant fragments are added.Encoding Data processing is done in a Shamir's like fashion: each data chunk is encoded as a constant term of a polynomial of degree k-1.More precisely, a data chunk is transformed into a data share inside the function Encode. Encode takes as parameters the value of the data chunk to be encoded, coefficients Coeffs of the polynomial, and a point x at which the polynomial will be evaluated. For each data chunk, x and Coeffs are calculated in function of DSS_i-1 and j. Therefore, to recover a single data chunk, a user has to possess the DSS_i containing the result of Encode for that data chunk, as well as the previous DSS_i-1. An example of that is shown in Figure <ref>. Reusing of data shares trades the perfect security of Shamir's scheme for a better performance and a smaller size of fragments: during processing the polynomial is evaluated only at one value. Varying not only the polynomial coefficients, but also the values of x, prevents the preservation of data patterns inside encoded fragments.Seed The Encode function transforming a data chunk into a data share takes as input k-1 values of previously encoded data shares. The first set of data chunks DCS_1 does not possess a predecessor. Thus, a seed composed of k pseudorandom values is introduced as DSS_0. Distributing to fragments After data encoding, data shares are distributed over k final fragments f_1,...,f_k inside the DistributeShares function, in a way that a data share DSS_i(j) goes to a fragment f_j.Adding redundancy In a final step n-k redundant fragments f_k+1,...,fn are added by the function AddRedundancy implementing a systematic version of a Reed-Solomon error correction code <cit.>. §.§ Main characteristics Complexity Data fragmentation into k fragments has linear complexity O(k), as encoding a single data chunk is equal to evaluating a value of a polynomial of degree k-1 at a single point.Processing redundant fragments depends on the implementation of the error correction code. Same for defragmentation. Parallelization Defragmentation can strongly benefit from parallelization, as each data chunk is recovered independently from others. Larger data are divided into blocks before applying Algorithm <ref> to partially parallelize also the fragmentation processing. Partial defragmentationAn interesting property of the scheme is its fine-grained granularity duringdefragmentation. Indeed, to defragment a single data chunk it is only required to know its position inside a fragment, as well as possess the previous data share set. Fragment size The size of produced fragments is close to the space efficient value of d_size/k, proposed by Rabin. The only data overhead comes from the seed, which is generated at the beginning of the fragmentation procedure and then attached to data fragments. Seed size depends on the chosen size of the data chunks, as seed values are of the same size than data chunks. Therefore, the fragmentation procedure produces a data overhead of size of one data chunk per fragment.§ EXPERIMENTAL SECURITY EVALUATIONAn experimental analysis of security characteristics of the scheme based on the methodology presented in <cit.> was performed. Tests were adapted to the fragmenting nature of the scheme. For instance, we measured the behavior of several parameters (entropy, correlation coefficient, probability density function) in function of number of fragments k.A secure fragmentation algorithm should ensure high level uniformity and independence of fragmented data. Section <ref> and <ref> analyze these two properties. Moreover, in Section <ref> we test the sensitivity of the scheme to changes inside the seed.All tests were performed using Matlab environment on textual data samples provided by LaPoste[http://www.laposte.fr/]. An example of one of such data sample is shown in Figure <ref>. Its corresponding fragment is presented in Figure <ref> and compared to the one obtained using an IDA (Figure <ref>). Matlab rand function was used for the generation of the pseudorandom seed. §.§ UniformityEncoded fragments should be characterized by high data uniformity, which is an essential property of a scheme resistant against frequency analysis.We measure fragments uniformity by visualizing their probability density functions, measuring their entropy, as well as applying the chi-squared test.§.§.§ Probability Density Function Frequency counts close to a uniform distribution testify data have a good level of mixing. This means that each byte value inside a fragment should have an occurrence probability close to 1/v=0.0039, where v is the number of possible values (256 for a byte). In Figure <ref>, the probability density function (PDF) of a data sample and one of its fragment (for k=2) are shown. Results for the fragment are spread over the space and have a distribution close to uniform. In Figure <ref>, the PDF function is also shown, but for different values of k (from 2 to 20). It demonstrates clearly that the occurrence probability of byte values is getting closer to 0.0039 as the value of k is increasing.§.§.§ Entropy Information entropy is a measure of unpredictability of information content <cit.>. In a good fragmentation scheme the entropy of the fragments should be as high as possible. Figure <ref> shows entropy variation for all data fragments compared to entropy variation of original data. The chosen data chunk size is equal to one byte, so the maximum entropy value is equal to 8. The average measured value for overall fragments (7.9926) is significantly higher than the one of original data (5.3498) and close to the maximum. This consequently demonstrates that our scheme ensures the uniformity property.Furthermore, Figure <ref> shows the entropy level in function of the number of fragments k. Each entropy value was obtained as an average of entropy of fragmentation results coming from 10 different data samples.The size of one data fragment was set to 1000 bytes. The resultshows that the entropy is growing with the number of fragments: it starts with a values close to 7.91 for a k=2 and achieves 7.99 for values of k close to 20. Moreover, the entropy level is in the same range of values for all k fragments coming from one fragmentation result. This demonstrates that the information is distributed evenly among the fragments. §.§.§ Chi-squared test Uniformity of data inside fragments was validated by applying a chi-squared test <cit.>. For a significance level of 0.05 the null hypothesis is not rejected and the distribution of the fragment data is uniform if χ^2_test≤χ^2_theory(255,0.05)≈293. The test was applied on fragmentation results of 15 different data samples for a k going up to 20 and fragment size of 1000 bytes. For all values of k, the tests was successful. There was no visible correlation between the number of fragments k and the results of chi-squared test. §.§ IndependenceFragmented data should be greatly different from its original form. To evaluate fragments' independence, we analyze recurrence plots, as well as correlation and bit difference between data and their fragments. §.§.§ Recurrence A recurrence plot serves to estimate correlation inside data <cit.>. Considering data vector x=x_1,x_2,...,x_m a vector with delay t ≥ 1 is constructed x(t)=x_1+t,x_2+t,...,x_m+t. A recurrence plot shows the variation between x and x(t). In Figure <ref>, such plots for a data sample and its fragments obtained by applying an IDA and the proposed scheme are shown. Using the proposed scheme, no clear pattern is obtained after data fragmentation. §.§.§ Correlation Correlation coefficient is used to evaluate the linear dependence between data <cit.>. A secure fragmentation algorithm should ensure as low correlation as possible between initial data and their fragments. Correlation coefficients were measured between 10 data samples and their corresponding fragments for different values of number of fragments k (from 2 to 20). The method used for the calculation was same as in <cit.>. Results are shown in Figure <ref>. Observed values of correlation coefficients are close to 0. This demonstrates that no detectable correlation exists between tested data samples and their fragments. Moreover, the correlation coefficients for higher values of k tend to have lower values. Then, the correlation between fragments coming from the same fragmentation results was measured. Correlation coefficients among fragments were also close to 0 (in a range of <-0.01,0.02>). It demonstrated that fragments are not correlated with each other and thus confirmed the independence property of the scheme. §.§.§ Difference Each fragment should be significantly different from the initial data and from other fragments of the same fragmentation result. Bit difference between a data sample and each of its fragments was measured and it was close to 50%. A similar result was obtained for the difference between fragments themselves.§.§ Sensitivity TestDifferential attacks study the relation between two ciphertexts resulting from a slight change, usually of one bit in the original plaintext or in the key. Inspired by this fact, a seed sensitivity test was realized in order to visualize the impact of a change in the seed on the fragmentation result. Indeed, two fragmentation results of same data should be different while obtained using two distinct seeds. For same data sample of size of 4000 bytes, two fragmentation results were obtained: one for a seed S_1 and one for a seed S_2 that differs S_1 by one random bit. The test has shown that such two fragmentation results are not correlated with each other (correlation coefficients close to 0) and significantly different (around 50% of bit difference). § CRYPTANALYSIS DISCUSSIONIn this Section, we discuss about the confidentiality level provided by the proposed algorithm ans its resistance against most known types of attack scenarios <cit.> (statistical, differential, chosen/known plain-text, and brute-force) in a situation when fewer than k fragments have been revealed. During such study, the steps of the algorithm are considered to be public in compliance with Kerckhoff's principle.§.§ General Security PropertiesThe threat model we use is the one where data fragments are physically dispersed over n different storage locations or transmission paths. Therefore, resulting data protection relies essentially on the difficulty to collect k of the fragments. Indeed, an attacker has to find the locations or transmission paths of dispersed fragments and then manage to access or eavesdrop them. In this paper, we do not deal with data dispersal questions, which includes ensuring the physical separation of fragments, as well as protecting the information describing the order in which fragments have to be assembled to reconstitute the original data. Nevertheless information about the defragmenation procedure should be stored in a secure location or dispersed within the fragments, as its importance could be in a sense compared to the one of the encryption key. The strength of our solution relies on the use of a seed in form of k pseudo-random values, as well as on unpredictability and high sensitivity of the fragments. Moreover, a user have the possibility to adapt the security level to their needs, not only by increasing the value of number of fragments k, but also by augmenting the data chunk size. The proposed scheme ensures forward and backward secrecy, but only in a situation when the seed values are not being repeated. §.§ Most known types of attacks In this Section we are analyzing the resistance of the scheme to the most known types of attacks.Statistical attacks The category of statistical attacks exploits the fact that encoded data may reveal some statistical properties. Immunity against such attacks requires that fragments achieve high level of randomness <cit.>. Therefore, in an ideal situation, the frequency analysis of data inside a fragment should be indistinguishable from the output of a pseudorandom generator. Results presented in Section <ref> have shown that our scheme achieve good uniformity and recurrence characteristics. Indeed, no useful information can be detected from the fragmented data. This demonstrates the high randomness of the scheme and its resistance against statistical attacks. Such property is not achieved by Information Dispersal Algorithms, where in the case of a known generator matrix and data with recognizable patterns it is possible to guess the content of missing fragments. Brute-force attack and content verification While possessing a p<k number of fragments, an attacker can attempt a brute-force attack by trying to guess the content of the missing (k-p) fragments. Intuitively, the difficulty level of such attack grows with the required number of fragments k and decreases with the number of possessed fragments p. The recovery of a set of k data shares of size w each implicates trying 2^w possibilities for k-p data shares of missing fragments. Therefore, an increasing of the size of a data chunk/share may harden the brute-force attack on a set of k data chunks. On the other hand, a way of facilitating the attack would be to make some assumptions about the content of the missing fragments that would limit the number of possibilities to verify.In a different scenario, an attacker with less than k fragments may like to verify whether the data inside fragments matches some predetermined value. In the case of an IDA, such verification is easy when the generator matrix is known. In the case of our scheme, it is harder, as the attacker will have to guess or the missing seed values or the missing data shares used to encode the part of the data they would like to verify.Known and chosen plain text attacks The knowledge of a part of the plaintext facilitates a brute-force attack. However, as each time a new seed is used, it does not help to recover fragments of other data.Linear and differential attacks A linear attack consists of exploiting the linear relations between the plaintext, the ciphertext and the key. The knowledge of the first data chunks from the plaintext and of the p fragments allows the recovery of the missing seed values. However, the recovery of the seed is not as critical, as the key leakage in symmetric encryption schemes. Because fragments are dependent of each others, to decode adata share it is necessary to possess its preceding data shares, also the one that are inside of the k-p missing fragments.In order to avoid differential attacks, seed values have to be change for each fragmentation procedure. A reuse of the seed could expose some relations between fragmentation results, for instance the fragmentation result of two identical plaintexts would be the same. As presented in Section <ref>, encoded data show high seed sensitivity, so even a single bit change is sufficient to obtain two different and not correlated fragmentation results of the same plain text.§ PERFORMANCE RESULTSProposed algorithm was implemented in JAVA using the following resources:JDK 1.8 on DELL Latitude E6540, X64-based PC running on Intel® CoreTM i7-4800MQ CPU @ 2.70 GHz with 8 GB RAM, under Windows 7. It was tested on data provided by La Poste, the French postal office. An implementation of java.security.SecureRandom is used for seed generation. The scheme can be implemented in any GF(2^Q) and is designed to use only logical operations. For integrity purposes, Q is selected according to word size of processorsand can be 8,16, 32 or 64-bit. As presented in Figure <ref>, the variation of average time for fragmentation is linear. Similar results were obtained for the defragmentation process. A multi-threaded version, optimized for 4 cores, sped up the performance by a factor of 3 that becomes close to 4 for more intensive computations.Another test was performed on the tera-memory platform TeraLab[https://www.teralab-datascience.fr] using following resources: 32GB RAM and 4 VCPUs. Figure <ref> shows the results for fragmentation of a data file of 100MB up to 100 fragments. This result exhibits the linear complexity of the fragmentation relatively to k the number of generated fragments. Figure <ref> presents performance measured in function of data size for several values of k. Similar results were obtained for the defragmentation process. Results demonstrate that the scheme achieves good scalability. § FUTURE WORKSIn future, we plan to benchmark our algorithm to compare its performance with most relevant works. A possibility of performance improvement is seen in parallelization of the fragmentation processing by partially limiting the sequential character of the encoding procedure. A more sophisticated way of distribution of encoded data chunks could be also envisioned in order to create fragments taking into account several levels of confidentiality. Such processing would require an additional step in form of the separation of initial data chunks regarding their level of confidentiality. It could be inspired by works done in <cit.>, that use wavelet transform to separate data into two parts, a private and a public one, without any user interaction.Another, more complex, research track would be to design a complete fragmentation architecture for data protection by means of fragmentation, encryption, and dispersion. This work would focus on secure management of information about the location and order of data fragments, as well as on secure distribution of fragments from the trusted site performing fragmentation procedure to their final storage destinations.§ CONCLUSIONA novel fragmentation algorithm for secure and resilient distributed data storage was described and analyzed. It combines the keyless property, computational simplicity and space efficient size of fragment of Information Dispersal Algorithms with an adjustable computational level of security. Security analysis shows that the produced fragments achieve good randomness and are not correlated neither with the initial data, nor among themselves. The use of a fresh seed for each fragmentation procedure ensures the backward and forward secrecy properties, as well as limits the possibility of differential attacks. Brute-force and chosen-plaintext attacks could be considered in some situations. Their attack resistance depends on the number of fragments in the possession of an attacker, as well as the quantity of knowledge about of the initial data.The scheme was implemented and tested on different samples of industrial data. Tests show good performance and scalability. The fragmentation procedure is linear in terms of number of fragments. We believe that the scheme could be applied to all application of data storage or transmission, where we would like to hide the nature of fragmented data without applying cryptographic mechanisms.§ ACKNOWLEDGMENTS The work is partially funded by the ITEA2 CAP project. abbrv | http://arxiv.org/abs/1705.09872v1 | {
"authors": [
"Katarzyna Kapusta",
"Gerard Memmi",
"Hassan Noura"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170527223524",
"title": "An Efficient Keyless Fragmentation Algorithm for Data Protection"
} |
http://arxiv.org/abs/1705.09631v3 | {
"authors": [
"Yubing Dong",
"Amand Faessler",
"Thomas Gutsche",
"Qifang Lü",
"Valery E. Lyubovitskij"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526160314",
"title": "Selected strong decays of $η(2225)$ and $φ(2170)$ as $Λ \\barΛ$ bound states"
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|
equationsection On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos Andreas Basse-O'Connor, Jan Pedersen, and Victor Rohde=========================================================================================Department of Mathematics, Aarhus University, E-mail adresses: [email protected] (A. Basse-O'Connor), [email protected] (J. Pedersen), [email protected] (V. Rohde)Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility is presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.Keywords: Sums of Gaussian squares; infinite divisibility; second Wiener chaosMSC 2010: 60E07;60G15; 62H05; 62H10§ INTRODUCTION Paul Lévy <cit.> raised the question of infinite divisibility of Gaussian squares, that is, for a centered Gaussian vector (X_1,…,X_n) when can (X_1^2,…,X_n^2) be written as a sum of m independent identical distributed random vectors for any m∈? Several authors have studied this problem. We refer to <cit.> and reference therein. These works include several novel approaches and gives a great understanding of when Gaussian squares are infinitely divisible. In this paper we will provide a characterization of infinite divisibility of sums of Gaussian squares which to the best of our knowledge has not been studied in the literature except in special cases. This problem is highly motivated by the fact that sums of Gaussian squares are the usual limits in many limit theorems in the presence of either long range dependence, see <cit.> or <cit.>, or degenerate U-statistics, see <cit.>. In the following we will go in more details.Let Y be random variable in the second (Gaussian) Wiener chaos, that is, the closed linear span in L^2 of {W(h)^2-1 : h ∈ H, ‖ h ‖ =1 } for a real separable Hilbert space H and an isonormal Gaussian process W. For convenience, we assume H is infinite-dimensional. Then there exists a sequence of independent standard Gaussian variables (ξ_i) and a sequence of real numbers (α_i) such that Y ∑_i=1^∞α_i (ξ_i^2 -1),where the sum converges in L^2 (see for example <cit.>). Since the ξ_i's are independent, (ξ_1^2,…,ξ^2_d) is infinitely divisible for any d ≥ 1 and therefore, Y is infinitely divisible. Such a sum of Gaussian squares appears as the limit of U-statistics in the degenerate case (see <cit.>). In this case the α_i are certain binomial coefficients times the eigenvalues of operators associated to the U-statistics. We note that the sequence (ξ_i) depends heavily on Y, so one can not deduce joint infinite divisibility of random vectors with components in the second Wiener chaos. In particular, for a vector with dimension greater than or equal to three and components in the second Wiener chaos it is well known (cf. Theorem <ref> below) that it need not be infinite divisibility. In between these two cases is the open question of infinite divisibility of a two-dimensional vector with components in the second Wiener chaos. Let (X_1,…, X_n_1+n_2) be a mean zero Gaussian vector for n_1,n_2 ∈. That any two-dimensional vector in the second Wiener chaos is infinitely divisible is equivalent to(d_1 X_1^2+… + d_n_1 X_n_1^2,d_n_1X_n_1 +1 ^2+… +d_n_1+n_2X_n_1+ n_2^2)being infinitely divisible for any d_1,…,d_n_1+n_2 = ± 1, any covariance structure of (X_1,…,X_n_1+n_2), and any n_1,n_2 ∈ (something that follows by the definition of the second Wiener chaos).The following theorem, which is due to Griffiths <cit.> and Bapat <cit.>, is an important first result related to infinite divisibility in the second Wiener chaos. We refer to Marcus and Rosen <cit.> for a proof. Let (X_1,…,X_n) be a mean zero Gaussian vector with positive definite covariance matrix Σ. Then (X_1^2,…,X_n^2) is infinitely divisible if and only if there exists an n× n matrix U on the form (± 1,…,± 1) such that U^t Σ^-1 U has non-positive off-diagonal elements.This theorem resolved the question of infinite divisibility of Gaussian squares.For n ≥ 3 there is an n × n positive definite matrix Σ where there does not exist an n × n matrix U on the form (± 1,… ,± 1) such that U^t Σ^-1 U has non-positive off-diagonal elements. Consequently, there are mean zero Gaussian vectors (X_1,…, X_n) such that (X_1^2,… ,X_n^2) is not infinite divisible whenever n ≥ 3. Eisenbaum <cit.> and Eisenbaum and Kapsi <cit.> found a connection between the condition of Griffiths and Bapat and the Green function of a Markov process. In particular, a Gaussian process has infinite divisible squares if and only if its covariance function (up to a constant function) can be associated with the Green function of a strongly symmetric transient Borel right Markov process.When discussing the infinite divisibility of the Wishart distribution Shanbhag <cit.> showed that for any covariance structure of a mean zero Gaussian vector (X_1,…,X_n), (X_1^2,X_2^2+ … + X_n^2)is infinitely divisible. Furthermore, it was found that infinite divisibility of any bivariate marginals of a centered Wishart distribution can be reduced to infinite divisibility of (X_1X_2,X_3X_4). By the polarization identity, (X_1X_2,X_3X_4) = 14 ( (X_1 +X_2)^2 - (X_1 - X_2)^2, (X_3 + X_4)^2 - (X_3 - X_4)^2). Consequently, infinite divisibility of any bivariate marginals of a centered Wishart distribution is again related to the question of infinite divisibility of a two-dimensional vector from the second Wiener chaos.We will be interested in the infinite divisibility of (X_1^2+… +X_n_1^2,X_n_1 +1^2 +… +X_n_1+n_2^2),i.e., the case d_1=…=d_n_1+n_2=1 in (<ref>). The general case, where d_i=-1 for at least one i, seems to require new ideas going beyond the present paper. We will have a special interest in the case n_1 = n_2 = 2. Despite the simplicity of the question, it has proven rather subtle, and a definite answer is not presented. Instead, we give easily verifiable conditions for infinite divisible in the case n_1 = n_2 =2 as well as more complicated necessary and sufficient conditions in the general case that may or may not always hold. We will, in addition, investigate the infinite divisibility of (X_1^2+X_2^2,X_3^2 +X_4^2) numerically which, together with Theorem <ref> (ii), leads us to conjecture that infinite divisibility of this vector always holds.The main results without proofs are presented in Section 2. Section 3 contains two examples and a small numerical discussion. We end with Section 4 where the proofs of the results stated in Section 2 are given.§ MAIN RESULTS We begin with a definition which is a natural extension to the present setup (see the proof of Corollary <ref>) of the terminology used by Bapat <cit.>.Let n_1,n_2 ∈. An (n_1+n_2) × (n_1+n_2) orthogonal matrix U is said to be an (n_1,n_2)-signature matrix if U = [ U_1 0; 0 U_2; ]where U_1 is an n_1× n_1 matrix and U_2 is an n_2× n_2 matrix, both orthogonal, and for 0's of suitable dimensions.Let n_1,n_2 ∈ and consider a mean zero Gaussian vector (X_1,…,X_n_1+n_2) with positive definite covariance matrix Σ. Now we present a necessary and sufficient condition for infinite divisibility of (X_1^2+… +X_n_1^2,X_n_1 +1^2 +… +X_n_1+n_2^2).For a > 0, let Q = I - (I+ a Σ)^-1 and write Q = [ Q_11 Q_12; Q_21 Q_22 ]where Q_11 is an n_1 × n_1 matrix, Q_22 is an n_2 × n_2 matrix, and Q_12= Q_21^t (where Q_21^t is the transpose of Q_21) is an n_1 × n_2 matrix. Note that if λ is an eigenvalue of Σ, a λ/1+aλ is an eigenvalue of Q. Since Q is symmetric and has positive eigenvalues, it is positive definite. The vector in (<ref>) is infinitely divisible if and only if for all k,m ∈_0 and for all a>0 sufficiently large,∑ Q_11^k_1 Q_12Q_22^m_1 Q_21 Q_11^k_2⋯ Q_11^k_dQ_12Q_22^m_d Q_21 Q_11^k_d+1+ ∑ Q_22^m_1 Q_21Q_11^k_1 Q_12 Q_22^m_2⋯ Q_22^m_d-1Q_21Q_11^k_d Q_12 Q_22^m_d+1≥ 0,where the first sum is over all k_1,…, k_d+1 and m_1, …, m_d such that k_1 + … + k_d+1 + d= k and m_1 + … + m_d + d = m,and the second sum is over all m_1,…, m_d+1 and k_1, …, k_d such that m_1 + … + m_d+1 + d = m andk_1 + … + k_d + d = k.By applying Theorem <ref> we can give a new and simple proof of Shanbhag's <cit.> result that (X_1^2,X_2^2+… + X_1+n_2^2) is infinite divisible. To see this, consider the case n_1=1 and n_2 ∈. Then Q_11 is a positive number and Q_12 Q_22^m Q_21 is a non-negative number for any m ∈. In particular, we have Q_11^k_1 Q_12Q_22^m_1 Q_21⋯ Q_12Q_22^m_d Q_21 Q_11^k_d+1= Q_11^k_1⋯ Q_11^k_d+1 Q_12Q_22^m_1 Q_21⋯ Q_12 Q_22^m_d Q_21≥ 0for any k_1,…, k_d+1, m_1, … , m_d ∈_0. Consequently, the first sum in (<ref>) is a sum of non-negative numbers. A similar argument gives that the other sum is non-negative too. We conclude that (X_1^2,X_2^2+… + X_1+n_2^2) is infinite divisible. In order to get a concise formulation of the following results we will need some terminology and conventions. To this end, consider a 2× 2 symmetric matrix A. Let v_1 and v_2 be the eigenvectors of A, and λ_1 and λ_2 be the corresponding eigenvalues. We say that v_i is associated with the largest eigenvalue if λ_i ≥λ_j for j=1,2. Furthermore, whenever A is a multiple of the identity matrix, we fix (1,0) to be the eigenvector associated with the largest eigenvalue.Now consider the special case n_1=n_2=2, i.e., the vector (X_1^2+X_2^2,X_3^2+X_4^2)where (X_1,X_2,X_3,X_4) is a mean zero Gaussian vector with a 4 × 4 positive definite covariance matrix Σ. We still let Q = I - (I+a Σ)^-1 and write Q = [ Q_11 Q_12; Q_21 Q_22 ]where Q_ij is a 2 × 2 matrix for i,j=1,2. Let W be a (2,2)-signature matrix such thatW^tQW = [ W_1^t Q_11 W_1W_1^t Q_12W_2; W_2^tQ_21W_1W_2^t Q_22W_2 ] =[ q_110 q_13 q_14;0 q_22 q_23 q_24; q_13 q_23 q_330; q_14 q_240 q_44 ],where q_11≥ q_22>0 and q_33≥ q_44>0 which exists by Lemma <ref>. Note that q_ij is not the (i,j)-th entry of Q but of W^t Q W. Let v_1 = (v_11,v_21) be the eigenvector of W_1^t Q_12Q_21 W_1 associated with the largest eigenvalue. If q_11=q_22 or q_33=q_44, any orthogonal W_1 or W_2 gives the desired form. In this case, we may always choose W_1 or W_2 such that v_11 q_13 ( v_11 q_13+ v_21 q_23) ≥ 0 (see the proof of Lemma <ref>, (ii) ⇒ (iii)), and it is such a choice we fix. The following theorem addresses the non-negativity of the sums in (<ref>) when n_1=n_2=2,.Let n_1=n_2=2. Then, in the notation above, we have the following. (i) For all d ∈_0 and k_1,…,k_d+1,m_1,…,m_d ∈_0,Q_11^k_1 Q_12Q_22^m_1 Q_21 Q_11^k_1⋯ Q_11^k_dQ_12Q_22^m_d Q_21 Q_11^k_d+1≥ 0 if and only if v_11 q_13 ( v_11 q_13+ v_21 q_23) ≥ 0. In particular, (<ref>) is infinitely divisible if the latter inequality is satisfied for all sufficiently large a. (ii) For any k,m ∈_0 such that at least one of the following inequalities is satisfied: (i) k ≤ 2, (ii) m≤ 2, or (iii) k+m ≤ 7, the sum in (<ref>) is non-negative.When v_11 q_13 ( v_11 q_13+ v_21 q_23) < 0, we know that there are k, m ∈_0 such that (<ref>) with n_1=n_2=2 contains negative terms cf. Theorem 2.4 (i). If k=0 or m=0 then Theorem <ref> (ii) gives that the sum in (<ref>) is non-negative. If k, m ≥ 1, the sum in (<ref>) always contains terms on the formQ_11^k_1 Q_12 Q_22^m_1 Q_21.Since Q_11 is positive definite and AB =BA for any matrices A and B such that both sides make sense,Q_11^k_1 Q_12 Q_22^m_1 Q_21 = Q_11^k_1/2 Q_12 Q_22^m_1 Q_21 Q_11^k_1/2.Using Q_12 =Q_21^t we conclude that (<ref>) is equal to the trace of a positive semi-definite matrix and therefore non-negative. Consequently, there are always non-negative terms in (<ref>). It is an open problem if there exists a positive definite matrix Q with eigenvalues less than 1 and k,m ∈_0 such that (<ref>) is negative, which would be an example of (<ref>) not being infinite divisible, or if the non-negative terms always compensate for possible negative terms, which is equivalent to (<ref>) always being infinitely divisible. Continue to consider the case n_1=n_2=2 and write Σ^-1 = [ Σ^11 Σ^12; Σ^21 Σ^22 ] where Σ^ij is a 2× 2 matrix for i,j=1,2.Let W be a (2,2)-signature matrix such that W^t Σ^-1 W = [W_1^tΣ^11 W_1 W_1^t Σ^12 W_2;W_2^t Σ^21W_1 W_2^t Σ^22 W_2 ] =[ σ_110 σ_13 σ_14;0 σ_22 σ_23 σ_24; σ_13 σ_23 σ_330; σ_14 σ_240 σ_44 ]where σ_11≥σ_22>0 and σ_33≥σ_44>0 which exists by Lemma <ref>. Note that σ_ij is not the (i,j)-th entry of Σ^-1 but of W^t Σ^-1 W. Let v_1 = (v_11,v_21) be the eigenvector of W_1^t Σ^12Σ^21 W_1 associated with the largest eigenvalue. If σ_11=σ_22 or σ_33=σ_44, any orthogonal W_1 or W_2 gives the desired form. In this case, we may chose W_1 or W_2 such that v_21σ_24 ( v_21σ_24+ v_11σ_14) ≥ 0, and it is such a choice we fix. Then we have the following theorem. The vector (X_1^2+X_2^2,X_3^2+X_4^2) is infinitely divisible if one of the following equivalent conditions is satisfied. * There exists a (2,2)-signature matrix U such that U^t Σ^-1 U has non-positive off-diagonal elements. * The inequality v_21σ_24 ( v_21σ_24+ v_11σ_14) ≥ 0 holds. Example <ref> builds intuition about condition (ii) above, in particular that the condition holds in cases where (X_1^2,X_2^2,X_3^2,X_4^2) is not infinitely divisible, but also that it is not always satisfied. Theorem 2.6 (i) holds for general n_1,n_2 ≥ 1 as the following result shows. We give the proof below since it is short and makes the need for signature matrices clear. The proof of the more applicable condition (ii) in Theorem <ref> is postponed to Section 4 since it relies on results that will be establish in that section.Let (X_1,…,X_n_1+n_2) be a mean zero Gaussian vector with positive definite covariance matrix Σ. Then (X_1^2+… +X_n_1^2,X_n_1 +1^2 +… +X_n_1+n_2^2)is infinitely divisible if there exists an (n_1,n_2)-signature matrix U such that U^t Σ^-1 U has non-positive off-diagonal elements. Write X= (X_1,…, X_n_1) and Y=(X_n_1 + 1,…, X_n_1+n_2), and note that (X_1^2+… +X_n_1^2,X_n_1+1^2+… +X_n_1+n_2^2) = (‖ X‖ ^2, ‖ Y ‖^2)= (‖ U_1X‖ ^2, ‖ U_2Y ‖^2) for any n_1× n_1 orthogonal matrix U_1 and n_2 × n_2 orthogonal matrix U_2. Consequently, any property of the distribution of (<ref>) is invariant under transformations of the form[ U_1^t 0; 0 U_2^t ]Σ[ U_1 0; 0 U_2 ]of the covariance matrix Σ. Therefore, when there exists an (n_1,n_2)-signature matrix U such that U^t Σ^-1 U has non-positive off-diagonal elements, Theorem <ref> ensures infinite divisibility of (<ref>).§ EXAMPLES AND NUMERICS We begin this section by presenting two examples treating the inequalities in Theorem <ref> (ii) and Theorem <ref> (ii) in special cases. Then we calculate the sums in Theorem <ref> numerically with n_1 =n_2 =2 for a specific value of Q for k and m less than 60.Fix a >0 and assume that Q is on the formQ = [ Q_11 Q_12; Q_21 Q_22 ] = [ q_1 0 ε ε; 0 q_2 ε-δ; ε ε q_3 0; ε - δ 0 q_4 ]where δ, ε > 0, q_1 > q_2>0, and q_3 > q_4>0. Let v_1= (v_11,v_21) be the eigenvector of Q_12 Q_21 = [2 ε^2 ε ( ε - δ); ε ( ε - δ)ε^2 + δ^2 ]associated with the largest eigenvalue λ_1. We will argue that the inequality in Theorem <ref> (i), which readsv_11(v_11 +v_21 ) ≥ 0in this case, holds if and only if δ≤ε. Then the same theorem will imply that Q_11^k_1 Q_12Q_22^m_1 Q_21 Q_11^k_1⋯ Q_11^k_dQ_12Q_22^m_d Q_21 Q_11^k_d+1≥ 0 for all d ∈_0 and k_1,…, k_d+1m_1,…,m_d ∈_0 if and only if δ≤ε, and therefore also that the sum in (<ref>) is non-negative whenever this is the case. Since -v_1 also is an eigenvector of Q_12 Q_21 associated with the largest eigenvalue, we assume v_11≥ 0 without loss of generality. Assume δ≤ε. If δ = ε, v_1 = (1,0) and the inequality in (<ref>) holds. Assume δ < ε. Since λ_1 is the largest eigenvalue, λ_1 = sup_| v | =1 v^tQ_12Q_21 v≥ 2 ε^2which implies that 2 ε^2 - λ_1≤ 0 ≤ε ( ε - δ).Since v_1 is an eigenvector, (Q - λ_1) v_1 = 0 and we therefore have that 0 = ( 2 ε^2 - λ_1) v_11 +ε (ε -δ ) v_21≤ε ( ε - δ) (v_11 + v_21 ).We conclude that (<ref>) holds. On the other hand, assume δ > ε and v_11≥ 0. Since λ_1 is the largest eigenvalue, λ_1≥δ^2 + ε^2 > δε + ε^2 and therefore,(λ_1 - 2 ε^2) > ε (δ -ε).Note that v_11 can not be zero since the off-diagonal element in Q_12Q_21 is non-zero. We conclude that 0 = ( λ_1 - 2 ε^2)v_11 +ε ( δ - ε)v_21>ε ( δ - ε)(v_11 + v_21).This implies that (<ref>) does not hold.Assume Σ^-1 is on the form Σ^-1 = [ Σ^11 Σ^12; Σ^21 Σ^22 ] =[ σ_1 0-δ ε; 0 σ_2 ε ε;-δ ε σ_3 0; ε ε 0 σ_4 ]where σ_1 >σ_2 >0, σ_3 > σ_4 >0, and δ,ε > 0. Let v_1 = (v_11,v_21) be the eigenvector of Σ^12Σ^21 associated with the largest eigenvalue. We will argue that the inequality in Theorem <ref> (ii) holds if and only if δ≤ε. Then the same theorem implies that (X_1^2+X_2^2,X_3^2+X_4^2) is infinitely divisible whenever δ≤ε. On the other hand, Theorem <ref> implies that (X_1^2,X_2^2,X_3^2,X_4^2) is never infinite divisible under (<ref>) since there does not exists a matrix D on the form (± 1,± 1, ± 1, ± 1) such that D Σ^-1D has non-positive off-diagonal elements. Indeed, for any two matrices D_1 and D_2 on the form (± 1, ± 1), D_1 Σ^12 D_2 has either three negative and one positive or one negative and three positive entrances. To see that v_21 (v_11 + v_21) ≥ 0 if and only if δ≤ε, let P =[ 0 1; 1 0 ]and Q_12 be given as in Example <ref>. Then P Σ^12 P = Q_12, implying that (v_21,v_11) is the eigenvector associated with the largest eigenvalue of Q_12 Q_21. We have argued in Example <ref> that v_21( v_11 + v_21) ≥ 0 holds if and only if δ≤ε which is the desired conclusion. Now we investigate infinite divisibility of (X_1^2+X_2^2,X_3^2+X_4^2) numerically. More specifically, we consider the sums in (<ref>) with n_1=n_2=2 for a specific choice of positive definite matrix and different values of k and m. We will scale Q to have its largest eigenvalue equal to one to avoid getting too close to zero. Due to Theorem <ref> the case where v_11 q_13 ( v_11 q_13+ v_21 q_23)<0 (in the notation from Theorem <ref>) is the only case where the infinite divisibility of (X_1^2+X_2^2,X_3^2+X_4^2) is open. Let Q = 1/λ[0.80 0.01 0.01;00.3 0.01 -0.2; 0.01 0.010.80; 0.01 -0.200.3 ]where λ>0 is chosen such that Q has its largest eigenvalue equal to 1. Note that by Example <ref>, v_11 q_13 ( v_11 q_13+ v_21 q_23)<0. In Figure <ref> the logarithm of the sums in (<ref>) for k and m between 0 and 60 is plotted. It is seen that the logarithm seems stable and therefore, that the sums in (<ref>) remain positive in this case. A similar analysis have been done for other positive definite matrices, and we have not encountered any k,m∈_0 such that (<ref>) is negative. This, together with Theorem <ref> (ii), leads us to conjecture that (X_1^2+X_2^2,X_3^2+X_4^2) is infinite divisible for any covariance structure of (X_1,X_2,X_3,X_4). § PROOFSWe start this section with two lemmas on linear algebra. Lemma <ref> will be very useful in the proofs that make up the rest of this section. Let A be a n × n positive definite matrix. Let n_1,n_2 ∈ be such that n_1+ n_2=n and write A = [ A_11 A_12; A_21 A_22 ]where A_11 is an n_1 × n_1 matrix, A_22 is an n_2 × n_2 matrix, and A_12 = A_21^t is an n_1 × n_2 matrix. Then there exists an (n_1,n_2)-signature matrix W such that W^t A W has the form[ Ã_11 Ã_12; Ã_21 Ã_22 ]where Ã_11= (a_1,…,a_n_1) and Ã_22= (a_n_1+1,…,a_n_1+n_2) with a_i > 0 for i=1,…, n_1+n_2, and where Ã_12= Ã_21^t. Furthermore, we may choose W such that a_1 ≥ a_2 ≥…≥ a_n_1 and a_n_1+1≥ a_n_1+2≥…≥ a_n_1+n_2. Since A is positive definite, A_11 and A_22 are positive definite. Consequently, by the spectral theorem (see for example <cit.>), there exists an n_1 × n_1 matrix W_1 and an n_2 × n_2 matrix W_2, both orthogonal, such that W_1^t A_11 W_1 and W_2^t A_22 W_2 are diagonal with positive diagonal entries. Since permutation matrices are orthogonal matrices, we may assume the diagonal is ordered by size in both W_1^t A_11 W_1 and W_2^t A_22 W_2. Consequently, lettingW =[ W_1 0; 0 W_2 ],implies that W^t A W has the right form.For a fixed eigenvector v_i we call the system A v_i = λ_i v_i, the system of eigenequations. The k'th equation in this system will be called the k'th eigenequation associated with v_i. Let A be a 4 × 4 positive definite matrix, and let W be a (2,2)-signature such that W^t A W= [ W_1^t A_11 W_1 W_1^t A_12 W_2; W_2^t A_21 W_1 W_2^t A_22 W_2 ] = [ a_110 a_13 a_14;0 a_22 a_23 a_24; a_13 a_23 a_330; a_14 a_240 a_44 ],where a_11≥ a_22 > 0 and a_33≥ a_44 >0 which exists by Lemma <ref>. Note that a_ij is not the (i,j)-th entry of A but of W^t A W. Let v_1= (v_11,v_21) be the eigenvector associated with the largest eigenvalue of W_1^t A_12A_21W_1. If a_11=a_22 or a_33=a_44, any orthogonal W_1 or W_2 give the desired form. In this case, we may chose W_1 or W_2 such that v_11 a_13 (v_11a_13+ v_21a_23 ) ≥ 0, and it is such a choice we fix. Then the lemma below will play a central role in the proofs of the previously stated results. In the notation above, the following are equivalent.* There exists a (2,2)-signature matrix U such that U^t A U has all entries non-negative.* For any d ∈ and k_1, …, k_d+1, m_1, … m_d ∈_0,A_11^k_1 A_12 A_22^m_1 A_21 A_11^k_2⋯A_11^k_d A_12 A_22^m_d A_21 A_11^k_d+1≥ 0.* The inequality v_11 a_13 (v_11a_13+ v_21a_23 ) ≥ 0 holds. (i) ⇒ (ii).Let U = [ U_1 0; 0 U_2 ]be such that B_ij = U_i^t A_ijU_j has non-negative entries for i,j =1,2. Then A_11^k_0 A_12A_22^m_1 A_21 A_11^k_1⋯ A_11^k_d-1A_12A_22^m_d A_21 A_11^k_d= B_11^k_0 B_12 B_22^m_1 B_21 B_11^k_1⋯ B_11^k_d-1 B_12 B_22^m_d B_21 B_11^k_d.This trace is non-negative since all matrices in the product only contain non-negative entries.(ii) ⇒ (iii). By the spectral theorem, we may write W_1^t A_12A_21 W_1= V Λ V^t where V is a 2 × 2 orthogonal matrix and Λ = (λ_1,λ_2) with λ_1 ≥λ_2 ≥ 0. Note that v_1, the eigenvector associated with largest eigenvalue of W_1^tA_12A_21W_1, is the first column of V. If λ_1 = λ_2, v_1 = (1,0) and the inequality holds. If a_11= a_22 or a_33=a_44, W_1^t A_11 W_1 = A_11 or W_2^t A_22W_2 = A_22, and choosing W_1 or W_2 such that a_23 = 0 then ensures the inequality in (iii) holds. Assume now that λ_1 > λ_2, a_11 > a_22, and a_33 > a_44. It follows by assumption that0≤1/a_11^k1/a_33^k1/λ_1^k A_11^k A_12 A_22^k A_21 (A_12A_21)^k = [ 1 0; 0 (a_22/a_11)^k ]W_1^t A_12 W_2[ 1 0; 0 (a_44/a_33)^k ] W_2^t A_21 W_1V [ 1 0; 0 (λ_1/λ_2)^k ] V^t →[ 1 0; 0 0 ] W_1^t A_12 W_2 [ 1 0; 0 0 ]W_2^t A_21W_1 V [ 1 0; 0 0 ] V^t as k →∞. This gives the inequality in (iii) since[ 1 0; 0 0 ] W_1^t A_12 W_2 [ 1 0; 0 0 ]W_2^tA_21W_1 V [ 1 0; 0 0 ] V^t= v_11 a_13 (v_11a_13+ v_21 a_23 ). (iii) ⇒ (i). To ease the notation and without loss of generality assume that W = I. We are then pursuing two 2 × 2 orthogonal matrices U_1 and U_2 such that U_1^t A_11 U_1, U_1^t A_12 U_2, and U_2^t A_22 U_2 all have non-negative entrances. Initially consider D_1 and D_2 on the form (± 1, ± 1). Then clearly,D_1 A_11 D_1 = A_11 and D_2 A_22 D_2=A_22 since A_11 and A_22 are diagonal matrices. Next, note that either it is possible to find D_1 and D_2such that D_1 A_12 D_2 has all entrances non-negative or such that D_1 A_12 D_2 = [r] a_13a_14a_23-a_24where a_13,a_23,a_14,a_24 > 0. Consequently, we will assume A_12 is on the form in (<ref>) since otherwise choosing U_1 = D_1 and U_2 = D_2 would be sufficient. As one of two cases, assume a_13a_23 - a_14a_24≥ 0, and define U_2 = [ αa_14 a_24/ a_23 β a_23; α a_14-β a_24 ]where α, β > 0 are chosen such that each column in U_2 has norm one. Then U_2 is orthogonal, A_12 U_2 = [ α (a_14^2 + a_13a_14a_24/a_23)β (a_13a_23-a_14a_24);0β (a_23^2+a_24^2) ],and U_2^t A_22 U_2 = [ α^2 ( a_33( a_14a_24/a_23)^2 + a_44 a_14^2 )αβ a_14a_24 (a_33 - a_44); αβ a_14 a_24 (a_33 - a_44)β^2 a_23^2 + β^2 a_24^2 ]. Since a_33≥ a_44, all entries in A_12 U_2 and U_2^t A_22 U_2 are non-negative. Choosing U_1= I then gives a pair of orthogonal matrices with the desired property. Now assume a_13a_23 - a_14a_24 < 0. Note that A_12 on the form (<ref>) can not be singular and consequently, there exists λ_1 ≥λ_2 > 0 and an orthogonal matrix V such that A_12A_21 = V Λ V^t, where Λ = (λ_1,λ_2). Furthermore, since V contains the eigenvectors of A_12A_21 we may assume v_11 and v_12 have the same sign where v_ij is the (i,j)-th component of V. Define W = A_21 V ( Λ^1/2 )^-1,and note that this is an orthogonal matrix which, together with V, decomposes A_12 into its singular value decomposition, that is, V^t A_12 W = Λ^1/2. ThenV^t A_11 V = [ a_11 v_11^2 + a_22 v_21^2 v_11 v_12 (a_11 - a_22); v_11 v_12 (a_11 - a_22) a_11 v_12^2 + a_22 v_22^2 ].All entries in V^t A_11 V are non-negative since we chose v_11 and v_12 to have the same sign, and since a_11≥ a_22>0. To see that W^t A_22 W also have all entries non-negative, consider the first line in the eigenequations for A_12A_21 associated with the eigenvector (v_12,v_22), the eigenvector associated with the smallest eigenvalue λ_2,(a_13^2+a_14^2 - λ_2) v_12 + (a_13a_23 -a_14a_24)v_22 = 0.Since λ_2 is the smallest eigenvalue of A_12A_21, λ_2 = inf_| v | =1 v^tA_12A_21 v, and since the off-diagonal elements in A_12A_21 are non-zero, (1,0) and (0,1) cannot be eigenvectors. Consequently, λ_2 is strictly smaller than any diagonal element of A_12A_21, and in particular a_13^2+a_14^2 - λ_2 > 0. Since we also have a_13a_23 -a_14a_24 < 0, (<ref>) gives that v_12 and v_22 need to have the same sign for the sum to equal zero. Let w_ij be the (i,j)-th component of W and note that by (<ref>), w_11w_12 = v_11 a_13 + v_21a_23/λ_1^1/2v_12 a_13 + v_22a_23/λ_2^1/2. The assumption v_11 a_13 (v_11 a_13 + v_21a_23) ≥ 0 implies that v_11 a_13 + v_21a_23 and v_11 have the same sign. Since v_11 and v_12 were chosen to have the same sign, and v_12 and v_22 have the same sign, we conclude that (v_11 a_13 + v_21a_23)(v_12 a_13 + v_22a_23) is non-negative and therefore, w_11w_12 is non-negative too. Then writing W^t A_22 W =[ a_33 w_11^2 + a_44 w_21^2 w_11 w_12 (a_33 - a_44); w_11 w_12 (a_33 - a_44) a_33 w_12^2 + a_44 w_22^2 ]makes it clear that W^t A_22 W has non-negative elements. Thus, letting U_1= V and U_2= W completes the proof. Let A and v_1 be given as in Lemma <ref>. Then there exists a (2,2)-signature matrix U such that U^t A U has non-positive off-diagonal elements if and only if v_21 a_24 ( v_21 a_24 + v_11 a_14) ≥ 0. Let W be defined as in Lemma <ref>. Define P_1 = [ 0 1; 1 0 ]andP = [ P_1 0; 0 P_1 ].Then P_1 v_1 = (v_21,v_11) is the eigenvector of P_1 W_1^t A_12 A_21 W_1 P_1 associated with the largest eigenvalue. Letà = [ W_1^t A_11 W_1P_1 W_1^t A_12W_2 P_1; P_1 W_2^t A_21 W_1 P_1W_2^t A_22W_2 ] =[ a_110 a_24 a_23;0 a_22 a_14 a_13; a_24 a_14 a_330; a_23 a_130 a_44 ].By Lemma <ref>, there exists a (2,2)-signature matrix Ũ = [ Ũ_1 0; 0 Ũ_2 ] such that Ũ^t ÃŨ has non-negative entries if and only if v_21 a_24 ( v_21 a_24 + v_11 a_14) ≥ 0. Define now the (2,2)-signature matrix U as U = [ U_1 0; 0 U_2 ] =[ - W_1 P_1 Ũ_1 0; 0 W_2 P_1 Ũ_2 ].Let ũ_ij be the (i,j)-th component of Ũ_1. Since Ũ_1 is orthogonal, ũ_12ũ_22 = -ũ_11ũ_21 implying that U_1^t A_11 U_1 = [ ũ_11^2 a_22 + ũ_21^2a_11ũ_11ũ_12(a_22-a_11);ũ_11ũ_12(a_22-a_11) ũ_12^2 a_22 + ũ_22^2a_11 ]and Ũ^t_1 W_1^tA_11W_1 Ũ_1=[ ũ_11^2 a_11 + ũ_21^2a_22ũ_11ũ_12(a_11-a_22);ũ_11ũ_12(a_11-a_22) ũ_12^2 a_11 + ũ_22^2a_22 ].Consequently Ũ^t_1 W_1^tA_11W_1 Ũ_1 has non-negative elements if and only if U_1^t A_11 U_1 has non-positive off-diagonal elements. Similarly, Ũ^t_2 W_2^tA_22W_2 Ũ_2 has non-negative elements if and only if U_2^t A_22 U_2 has non-positive off-diagonal elements by a similar argument. Finally we note that U_1^t A_12 U_2 = - Ũ^t_1 P_1 W_1^t A_12 W_2 P_1 Ũ_2,and it follows that U^t A U has non-positive off-diagonal elements if and only if Ũ_1^t P_1 W_1^t A_12 W_2 P_1 Ũ_2, Ũ_1^t W_1^t A_11 W_1 Ũ_1 and,Ũ_2^t W_2 A_22 W_2 Ũ_2have all entries non-negative. We conclude that we can find a (2,2)-signature matrix U such that U^t A U has non-positive off-diagonal element if and only if (<ref>) holds.The following lemma will be useful in the proof of Theorem <ref>. A proof can be found in <cit.>. Let ψ : _+^n → (0,∞) be a continuous function. Suppose that, for all a>0 sufficiently large, logψ (a(1-s_1),…,a(1-s_n)) has a power series expansion for s=(s_1,…,s_n) ∈ [0,1]^n around s=0 with all its coefficients non-negative, except for the constant term. Then ψ is the Laplace transform of an infinitely divisible random variable in _+^n. We now give the proof of Theorem <ref>, where all the main steps follow similar as in <cit.>, but with several modifications to adjust to a different setting. E.g. there is a difference in the S matrix appearing in the proof.By <cit.>,P(s_1,s_2)= exp{ - 12 a( (1-s_1) (X_1^2+… + X_n_1^2) + (1-s_2) (X_n_1+1^2+⋯ + X_n_2^2))} = 1/| I + Σ a (I - S) |^1/2,where S is the (n_1+n_2)× (n_1 +n_2) diagonal matrix with s_1 on the first n_1 diagonal entries and s_2 on the remaining n_2 diagonal entries. Recall that Q= I -(I+ aΣ)^-1. Then P(s_1,s_2)^2= | I + a Σ -a Σ S | ^-1= | (I-Q)^-1 - ((I-Q)^-1 - I)S | ^-1= | I-Q || I- QS | ^-1,from which it follows that2 log P(s_1,s_2)= log| I -Q | - log| I - QS |= log| I - Q | + ∑_n=1^∞{ (QS)^n }/n,where the last equality follows from <cit.>. Now assume that the vector (X_1^2+⋯ + X_n_1^2,X_n_1+1^2+⋯ + X_n_1+n_2^2) is infinitely divisible, and write (X_1^2+⋯ + X_n_1^2,X_n_1+1^2+⋯ + X_n_1+n_2^2) d= Y_1^n + … + Y_n^nwhere Y_1^n,… Y_n^n are 2-dimensional independent identically distributed stochastic vectors. Let Y_ij^n be the j-th component of Y_i^n and note that Y_ij^n ≥ 0 a.s. for all i,j,n. ThenP(s_1,s_2)^1/n = exp{ - 12 a( (1-s_1) Y_11^n + (1-s_2) Y_12^n )}.That P^1/n(s_1,s_2) has a power series expansion with all coefficient non-negative follows from writing exp{ - 12a ( (1-s_j) Y_1j^n)} = exp{ - 12 aY_1j^n)} ) ∑_k=0^∞(s_j a Y_1j^n)^k /2^k k! .We have thatlog P(s_1,s_2) = lim_n→∞ (n (P^1/n(s_1,s_2) -1)).Note that (s_1,s_2) ↦ n (P^1/n(s_1,s_2) -1) and all its derivatives converge uniformly on [0,1)× [0,1) by a Weierstrass M-test (see for example <cit.>). Consequently, we may use <cit.> to conclude that ∂^α + β/∂ s_1^α∂ s_2^βlim_n →∞ (n (P^1/n(s_1,s_2) -1))= lim_n →∞∂^α + β/∂ s_1^α∂ s_2^β (n (P^1/n(s_1,s_2) -1))for any α,β∈_0. Thus, that all the terms in the power series expansion of P^1/n(s_1,s_2) are non-negative implies that all the terms in the power series representation of log P(s_1,s_2) except the constant term are non-negative by (<ref>). By (<ref>) we conclude that any coefficient in front of s_1^k s_2^m in { (QS)^k+m} has to be non-negative for all k,m ∈ and a>0. Expanding out the trace then gives that this is equivalent to non-negativity of the sum in (<ref>) for all k,m ∈_0.On the other hand, if the sum in (<ref>) is non-negative for all k,m ∈_0 and a>0 sufficiently large, (<ref>) and Lemma <ref> imply that (X_1^2+⋯ + X_n_1^2,X_n_1+1^2+⋯ + X_n_1+n_2^2)is infinitely divisible.Lemma <ref> implies the equivalence in (i). Now we set out to show (ii), i.e., to show that the sum in Theorem <ref> is non-negative for k,m ∈_0 such that k≤ 2, m≤ 2, or k+m≤ 7 in the special case n_1 = n_2 = 2. To this end, consider a 4 × 4 positive definite matrix Q and write Q = [ Q_11 Q_12; Q_21 Q_22 ]where Q_ij is a 2 × 2 matrix for i,j =1,2. Let W_1 and W_2 be two 2 × 2 orthogonal matrices and define P_ij= W_i Q_ij W_j. ThenQ_11^k_1 Q_12Q_22^m_1 Q_21⋯ Q_12Q_22^m_d Q_21 Q_11^k_d+1 = P_11^k_1 P_12P_22^m_1 P_21⋯ P_12P_22^m_d P_21 P_11^k_d+1.Consequently (see Lemma <ref>), we may assume, without loss of generality, that Q_11 and Q_22 are diagonal with the first diagonal element greater than or equal the other and all entries non-negative. Either there exists D_1 and D_2 on the form (± 1, ± 1) such that D_1 Q_12 D_2 has all entries non-negative or such that D_1 Q_12 D_2 = [q_13q_23;q_14 -q_24 ]where q_13,q_23,q_14,q_24 >0. If D_1 Q_12 D_2 has all entries non-negative, writing as in (<ref>) with W_i replaced by D_i implies non-negativity of each individual trace. We conclude that we may assumeQ = [ λ_1 0q_13q_14; 0 λ_2q_23 -q_24;q_13q_23 λ_3 0;q_14 -q_24 0 λ_4 ],whereλ_1 ≥λ_2 ≥ 0 and λ_3 ≥λ_4 ≥ 0 andq_13,q_23,q_14,q_24 >0, without loss of generality. We now write out the traces in (<ref>) for specific values of k and m and show non-negativity in each case.§.§.§ k=0 or m=0 Assume k=0 and fix some m ∈. Then the terms in the sum in Theorem <ref> reduce to Q_22^m. Since Q_22 is positive definite, Q_22^m is positive definite. Consequently, Q_22^m >0. Similarly, when m=0 and k ∈, the terms in the sum in Theorem <ref> reduce to Q_11^k, which again is positive since Q_11 is positive definite. §.§.§ k=1 or m=1 Assume k=1 and fix some m∈. Then (<ref>) reduces to Q_12 Q_22^m Q_21 + ∑_m_1 = 0^m-1 Q_22^m_1 Q_21Q_12 Q_22^m-1-m_1,which equals(m+1)Q_12 Q_22^m Q_21.Since Q_12 = Q_21^t and Q_22 is positive definite, Q_12Q_22^m Q_21 is positive semi-definite. We conclude that Q_12 Q_22^m Q_21≥ 0.Assume m=1 and fix some k ∈. Similar to above, (<ref>) reduces to Q_21 Q_11^k Q_12 + ∑_k_1 = 0^k-1 Q_11^k_1 Q_12Q_21 Q_11^k-1-k_1.That this trace is non-negative follows by arguments similar to those above.§.§.§ k=2 or m=2 Assume that k=2 and let m ∈. The case m=1 is discussed above. Assume m ≥ 2. Then (<ref>) reduces to Q_11 Q_12 Q_22^m-1 Q_21+∑_m_1 + m_2 +1 = m Q_22^m_1 Q_21 Q_11 Q_12 Q_22^m_2+∑_m_1 + m_2 +2 = m Q_12 Q_22^m_1 Q_21 Q_12 Q_22^m_2 Q_21+∑_m_1 + m_2 + m_3 +2 = m Q_22^m_1 Q_21 Q_12 Q_22^m_2 Q_21 Q_12 Q_22^m_3.All the traces above are non-negative. To see this, consider for example Q_22^m_1 Q_21 Q_12 Q_22^m_2 Q_21 Q_12 Q_22^m_3for some m_1,m_2,m_3 ∈_0. Since Q_22 is positive definite it has a unique positive definite square root Q_22^1/2. We conclude that Q_22^m_1 Q_21 Q_12 Q_22^m_2 Q_21 Q_12 Q_22^m_3 =Q_22^(m_1+m_3)/2 Q_21 Q_12 Q_22^m_2 Q_21 Q_12 Q_22^(m_1+m_3)/2.Note that Q_22^(m_1+m_3)/2 Q_21 Q_12= (Q_21 Q_12 Q_22^(m_1+m_3)/2)^t,which implies that (<ref>) is the trace of positive semi-definite matrix and therefore non-negative.Non-negativity of the traces when m=2 and k∈ follows by symmetry. §.§.§ k=3 and m=3 In the following we will need to expand traces, and we therefore note thatQ_11^k Q_12 Q_22^m Q_21 = λ_1^k λ_3^m q_13^2 + λ_1^k λ_4^m q_14^2 + λ_2^k λ_3^m q_23^2 + λ_2^k λ_4^m q_24^2for any k,m ∈, andQ_11^k_1 Q_12 Q_22^m_1 Q_21 Q_11^k_2 Q_12 Q_22^m_2 Q_21=λ_1^k_1+k_2λ_3^m_1+m_2 q_13^4+ λ_1^k_1+k_2λ_4^m_1+m_2 q_14^4 + λ_2^k_1+k_2λ_3^m_1+m_2 q_23^4+ λ_2^k_1+k_2λ_4^m_1+m_2 q_24^4 +λ_1^k_1+ k_2( λ_3^m_1λ_4^m_2+ λ_4^m_1λ_3^m_2) q_13^2q_14^2+λ_2^k_1+ k_2( λ_3^m_1λ_4^m_2 + λ_3^m_2λ_4^m_1) q_23^2q_24^2+λ_3^m_1 + m_2(λ_1^ k_1λ_2^k_2 +λ_1^ k_2λ_2^k_1 ) q_13^2q_23^2+ λ_4^m_1+m_2(λ_1^k_1λ_2^k_2+λ_1^k_2λ_2^k_1) q_14^2q_24^2- (λ_1^k_1λ_2^k_2 + λ_1^k_2λ_2^k_1)(λ_3^m_1λ_4^m_2 + λ_3^m_2λ_4^m_1) q_13q_23q_14q_24for any k_1,k_2,m_1,m_2 ∈.Assume now k=3 and m=3 and consider the sum in Theorem <ref>. The sum contains all terms on the form Q_11^k_1 Q_12 Q_22^2 Q_21 Q_11^k_2where k_1+k_2 =2 and Q_22^m_1 Q_21 Q_11^2 Q_12 Q_22^m_2where m_1 + m_2 =2. All these traces equalQ_11^2 Q_12 Q_22^2Q_21,and there are all together 6 of these terms. Next, the sum in Theorem <ref> also contains all terms on the formQ_11^k_1 Q_12 Q_22^m_1 Q_21 Q_11^k_2 Q_12 Q_22^m_2 Q_21 Q_11^k_3where k_1 + k_2 + k_3 =1 and m_1 + m_2 = 1, and Q_22^m_1 Q_21 Q_11^k_1 Q_12 Q_22^m_2 Q_21 Q_11^k_2 Q_12 Q_22^m_3where m_1 + m_2 + m_3 =1 and k_1 + k_2 = 1. Using both that AB =BA and A^t =A for any two square matrices A and B of the same dimensions we get that all these traces share the common traceQ_11 Q_12Q_21 Q_12Q_22Q_21.All together there are 12 of these terms. Finally, the sum in Theorem <ref> contains the two terms (Q_12 Q_21)^3 and (Q_21 Q_12)^3, which share a common trace. We conclude that the sum in Theorem <ref> reads{6 Q_11^2 Q_12Q_22^2 Q_21 +12 Q_11 Q_12 Q_21 Q_12 Q_22 Q_21 + 2 (Q_12 Q_21)^3 }.Since Q_12 = Q_21^t, Q_12Q_21 is positive semi-definite and consequently, (Q_12 Q_21)^3≥ 0. Furthermore, we have Q_11^2 Q_12Q_22^2 Q_21 = Q_11 Q_12Q_22^2 Q_21 Q_11≥ 0.Contrarily, there exists a positive definite matrix Q such that Q_11 Q_12 Q_21 Q_12 Q_22 Q_21 < 0.(To see this, consider Q on the form in Example <ref> with ε small and δ large relative to ε.) We will now argue that despite this,(<ref>) remains non-negative. Initially we note that Q_11^k_i Q_12 Q_22^m_i Q_21 = [λ_1^k_i (λ_3^m_i q_13^2 + λ_4^m_i q_14^2) λ_1^k_i (λ_3^m_i q_13q_23 -λ_4^m_i q_14q_24);λ_2^k_i( λ_3^m_i q_13q_23 - λ_4^m_i q_14q_24)λ_2^k_i (λ_3^m_i q_23^2 +λ_4^m_i q_24^2 ) ]and Q_22^m_i Q_21 Q_11^k_i Q_12 = [λ_3^m_i (λ_1^k_i q_13^2 + λ_2^k_i q_23^2) λ_3^m_i (λ_1^k_i q_13q_14 -λ_2^k_i q_23q_24);λ_4^m_i( λ_1^k_i q_13q_14 - λ_2^k_i q_23q_24)λ_4^m_i (λ_1^k_i q_14^2 +λ_2^k_i q_24^2 ) ].Since λ_1 ≥λ_2 and λ_3 ≥λ_4, we see that if q_13q_14≥ q_23q_24 or q_13 q_23≥ q_14q_24, then one of two matrices above have only non-negative entrances for any k_i, m_i ∈_0. Consequently,Q_11^k_1 Q_12Q_22^m_1 Q_21Q_11^k_2 Q_12 Q_22^m_2 Q_21= Q_22^m_1 Q_21Q_11^k_1 Q_12 Q_22^m_2 Q_21 Q_11^k_2 Q_12would be non-negative if this was the case. Especially, we would haveQ_11 Q_12 Q_21 Q_12 Q_22 Q_21≥ 0.Assume now that q_13 q_14≤ q_23 q_24 and q_13 q_23≤ q_14q_24. By (<ref>) and (<ref>), {12 Q_11^2Q_12 Q_22^2 Q_21 + Q_11 Q_12Q_22Q_21Q_12Q_21}= 12λ_1^2 λ_3^2 q_13^2 +12λ_1^2 λ_4^2 q_14^2 +12λ_2^2 λ_3^2 q_23^2 +12λ_2^2 λ_4^2 q_24^2 + λ_1λ_3 q_13^4+ λ_1λ_4 q_14^4 + λ_2λ_3 q_23^4+ λ_2λ_4 q_24^4+λ_1( λ_3 + λ_4) q_13^2q_14^2 +λ_2( λ_3 +λ_4) q_23^2q_24^2+λ_3(λ_1+ λ_2 ) q_13^2q_23^2 + λ_4 (λ_1+ λ_2) q_14^2q_24^2-(λ_1 + λ_2)(λ_3 +λ_4) q_13q_23q_14q_24.We are going to bound the term (λ_1 + λ_2)(λ_3 +λ_4) q_13q_23q_14q_24 by thepositive terms to show non-negative of this trace. We recall that λ_1 ≥λ_2>0 and λ_3 ≥λ_4>0. Initially, note that λ_2 λ _3 q_13q_23q_14q_24 ≤λ_2 λ_3 q_13^2q_23^2λ_2 λ _4 q_13q_23q_14q_24 ≤λ_1 λ_4 q_13^2q_14^2λ_1 λ _4 q_13q_23q_14q_24 ≤λ_1 λ_3 q_13^2q_14^2.This leaves only λ_1 λ _3 q_13q_23q_14q_24 to be bounded. If λ_1 λ _3 q_13q_23q_14q_24≤1/2λ_1^2λ_3^2 q_13^2, we have a bounding term in (<ref>). Therefore, assume 2q_23q_14q_24≥λ_1λ_3 q_13. Since Q was assumed positive definite, λ_2 λ_4 ≥ q_24^2. Consequently,λ_1 λ _3 q_13q_23q_14q_24 ≤ 2 q_23^2q_14^2q_24^2 ≤ 2 λ_2 λ_4 q_23^2 q_13^2 ≤λ_2 λ_4 ( q_23^4 + q_13^4) ≤λ_2 λ_3 q_23^4 + λ_1 λ_3 q_13^4.We conclude that (<ref>) and hence (<ref>) is non-negative.§.§.§ k+m=7 Now consider k,m ∈ such that k+m=7. Whenever k,m=1,2, we already know that the sum in Theorem <ref> is non-negative. Let k=3 and m=4. Then the sum in Theorem <ref> reads { 14Q_11 Q_12 Q_21 Q_12 Q_22^2 Q_21 +7 Q_11^2 Q_12Q_22^3 Q_21 7 Q_11 (Q_12 Q_22 Q_21)^2 +7 Q_12 Q_22 Q_21 (Q_12 Q_21)^2 }.Initially we note that Q_11 (Q_12 Q_22 Q_21)^2≥ 0 and Q_12 Q_22 Q_21 (Q_12 Q_21)^2 ≥ 0since they both can be written as the trace of positive semi-definite matrices (see above for more details). Next, by (<ref>) and (<ref>),{12 Q_11^2 Q_12 Q_22^3 Q_21 + Q_11 Q_12Q_22^2Q_21Q_12Q_21} = 12λ_1^2 λ_3^3 q_13^2 +12λ_1^2 λ_4^3 q_14^2 +12λ_2^2 λ_3^3 q_23^2 +12λ_2^2 λ_4^3 q_24^2+λ_1( λ_3^2 + λ_4^2) q_13^2q_14^2 +λ_2( λ_3^2 +λ_4^2) q_23^2q_24^2+λ_3^2(λ_1+ λ_2 ) q_13^2q_23^2 + λ_4^2 (λ_1+ λ_2) q_14^2q_24^2+λ_1λ_3^2 q_13^4+ λ_1λ_4^2 q_14^4 + λ_2λ_3^2 q_23^4+ λ_2λ_4^2 q_24^4-(λ_1 + λ_2)(λ_3^2 +λ_4^2) q_13q_23q_14q_24.Again we bound the negative term by positive terms. Recall thatλ_1 ≥λ_2 and λ_3 ≥λ_4, and that we may assume q_23q_24≥ q_13q_14 andq_14q_24≥ q_13q_23 without loss of generality. Consequently, λ_1 λ_4^2 q_13q_23q_14q_24 ≤λ_1 λ_4^2q_14^2 q_24^2 λ_2 λ_3^2 q_13q_23q_14q_24 ≤λ_2 λ_3^2q_23^2q_24^2λ_2 λ_4^2 q_13q_23q_14q_24 ≤λ_2 λ_4^2q_14^2q_24^2,leavingλ_1 λ_3^2 q_13q_23q_14q_24 to be bounded. First note that12λ_1^2 λ_3^3 q_13^2- λ_1 λ_3^2 q_13q_23q_14q_24 =λ_1 λ_3^2 q_13 ( 12λ_1 λ_3 q_13- q_23q_14q_24),so that non-negativity holds if 12λ_1 λ_3 q_13≥ q_23q_14q_24. Assume λ_1 λ_3 q_13≤ 2q_23q_14q_24 and recall that λ_2 λ_4 ≥ q_24^2 since Q is positive definite. Then λ_1 λ_3^2 q_13q_23q_14q_24 ≤2λ_3 q_23^2 q_14^2 q_24^2 ≤ 2 λ_2 λ_3 λ_4 q_23^2 q_14^2 ≤λ_2 λ_3^2 q_23^4 + λ_2 λ_4^2 q_14^4 ≤λ_2 λ_3^2 q_23^4 + λ_1 λ_4^2 q_14^4 so we have found bounding terms for the last expression. We conclude that (<ref>) is non-negative and therefore, (<ref>) is non-negative too. The case k=4 and m=3 follows by symmetry. It follows that the sum in Theorem <ref> is non-negative for k+m=7.Corollary <ref> gives that (i) and (ii) are equivalent and Corollary <ref> gives that (i) implies infinite divisibility.Acknowledgments. This research was supported by the Danish Council for Independent Research (Grant DFF - 4002-00003). abbrv | http://arxiv.org/abs/1705.09508v1 | {
"authors": [
"Andreas Basse-O'Connor",
"Jan Pedersen",
"Victor Rohde"
],
"categories": [
"math.PR",
"60E07, 60G15, 62H05, 62H10"
],
"primary_category": "math.PR",
"published": "20170526100342",
"title": "On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos"
} |
∂̅J̅∂ | http://arxiv.org/abs/1705.09661v2 | {
"authors": [
"Georgios Itsios",
"Yolanda Lozano",
"Jesus Montero",
"Carlos Nunez"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170526180001",
"title": "The $AdS_5$ non-Abelian T-dual of Klebanov-Witten as a $\\mathcal{N} = 1$ linear quiver from M5-branes"
} |
^(a)IRFU, CEA, Universit Paris-Saclay, F-91191 Gif-sur-Yvette, France^(b)AstroParticule et Cosmologie, Universit Paris Diderot, CNRS/IN2P3, CEA/DRF/IRFU, Observatoire de Paris, Sorbonne Paris Cit, 75205 Paris Cedex 13, France.The Daya Bay, Double Chooz and RENO experiments recently observed a significant distortion in their detected reactor antineutrino spectra, being at odds with the current predictions. Although such a result suggests to revisit the current reactor antineutrino spectra modeling, an alternative scenario, which could potentially explain this anomaly, is explored in this letter. Using an appropriate statistical method, a study of the Daya Bay experiment energy scale is performed. While still being in agreement with the γ calibration data and ^12 B measured spectrum, it is shown that a O(1%) deviation of the energy scale reproduces the distortion observed in the Daya Bay spectrum, remaining within the quoted calibration uncertainties. Potential origins of such a deviation, which challenge the energy calibration of these detectors, are finally discussed. Reactor antineutrino spectra energy nonlinearity statistical analysis. § INTRODUCTION Reactor antineutrino experiments have played a leading role in neutrino physics starting with the discovery of the electron antineutrino in 1956 <cit.>, through the first observed oscillation pattern in KamLAND <cit.>, up to recent high precision measurements on the θ_13 mixing angle <cit.>. Future projects JUNO <cit.> and RENO50 <cit.> even aim at reaching sub-percent accuracy on θ_12 on top of solving the neutrino mass hierarchy puzzle. However, two anomalies in the measured antineutrino spectra are being observed. The first is an overall rate deficit around 6% known as “The reactor antineutrino anomaly” <cit.>. The second one is a shape distortion in the 4–6 MeV region, often quoted as a “bump” or “shoulder” in the spectra. It should be particularly stressed out that the relation between these two anomalies is not straightforward since shape distortion does not necessarily imply a change in the total rate.This letter focuses on the second anomaly. In Section <ref>, a quantitative comparison of four reactor antineutrino experiments (Bugey 3 <cit.>, Daya Bay <cit.>, Double Chooz <cit.> and RENO <cit.>) is performed to demonstrate their incompatibility, thus questioning nuclear effects as a common origin, as proposed in <cit.>. The next sections are dedicated to the study of an alternative scenario accounting for the observed distortion. Section 3 reviews the energy scale determination in such reactor antineutrino experiments. Section 4 introduces a combined analysis of the Daya Bay calibration and reactor antineutrino data. Results are presented in section 5 and show that a 1% unaccounted break at 4 MeV in the energy scale can reproduce the observed antineutrino spectrum and still comply with calibration data within uncertainties. Section 6 discusses possible origins of such an energy nonlinearity and especially questions calibration of such detectors.§ REACTOR SPECTRA COMPARISON§.§ On statistical compatibility of reactor spectra Among all existing reactor antineutrino experiments, four of them gives precise reactor spectra shape information. The Bugey 3 experiment (B3) <cit.> has until recently provided the finest reactor antineutrino spectrum. The B3 measurement was in very good agreement with previous predictions <cit.>. The comparison is here updated to the most recent predictions <cit.>. As indicated on Figure <ref>, the net effect is an additional 1%/MeV decrease through the full energy range. This update is still compatible with prediction within the 2% linear spectral uncertainty envelop quoted in <cit.>. New measurements have been provided by three experiments: Double Chooz (DC) <cit.>, Daya Bay (DB) <cit.> and RENO (RN) <cit.>. Their ratios to the state of the art prediction <cit.> are depicted on Figure <ref> and exhibit a significant deviation from unity around 5 MeV. At first glance they clearly show a common feature which is described as a bump in the 4 to 6 MeV region. Nevertheless, to our knowledge, no quantitative comparison is available in the literature.To gain quantitative insights on their compatibility, each spectrum having different bin centers and widths, a direct χ^2 comparison is not possible. A bespoke statistical test was constructed to assess if all the observed spectra could come from a common unique distribution. This shared distribution was estimated thanks to a χ^2 approach using a Gaussian mixture model <cit.>. Such an approach offers the advantages of being reactor modeling independent and flexible enough to accurately fit each single data set. To do so, a large number () of knots (x_1,…,x_K) were used, evenly distributed between 0 and 10 MeV. The spectral density, f(x)=1/h∑_k=1^K w_k ϕ(x-x_k/h), with weight parameters w_k, a bandwidth fixed to the inter-knot distance h=x_k+1-x_k, ϕ being the standard normal probability distribution function, was then integrated over each bin width to properly model their content. On top of statistical uncertainties, normalization and linear energy scale uncertainties were included in covariance matrices. For DC a 1% in normalization and scale was used, for DB a 2.1% and 1%, for RN a 1.5% and 1% and for B3 a 4% and 1.4% as published by the collaborations <cit.>. The χ^2 was expanded with a quadratic term penalizing higher local curvatures for smoothness: λ ∫ f^''(x)^2dx. The amplitude of the positive parameter λ controls the regularization strength or equivalently the smoothness of the fitting function. It was automatically determined from data through the generalized cross validation method <cit.>. This procedure is nearly equivalent to minimizing the model predictive error. The optimization of the global χ^2 was iteratively performed until reaching convergence, each step alternating between the χ^2 function minimization with respect to the w_k parameters and the generalized cross validation criterion minimization with respect to λ. The model was fitted to all possible subset of experiments. While our method was able to individually reproduce each spectrum accurately adjusting differently the w_k and λ coefficients, it prominently stressed out their inconsistency when combined. A parametric bootstrap procedure was used to estimate the χ_ min^2 distribution. For each experiment, the combined best fit model (gray curve on Figure <ref>) was used to draw 10^4 Monte Carlo simulations using their respective covariance matrix. Each of these 10^4 four experiment data set was fitted with the same global framework and the residual sum of squares was computed. Their distribution was found to follow with a good accuracy a Gamma probability density function with a shape parameter of 33 (1±9%) and a scale parameter of 2.8 (1±9%). The associated p-value was estimated using another 10^6 Monte Carlo simulations, because of the uncertainties on the Gamma PDF fitted parameters. The 99^ th percentile of the p-value distribution was found below 10^-5 while the median was located around 10^-8. As a prominent conclusion, the observed spectra cannot come from a unique distribution with a high statistical significance (more than 4.4 σ at 99% CL). This result is mostly driven by the mismatch between the DB, RN and B3 spectra since they have by far the highest statistics. Any combination of two of these three experiments is also not consistent. Note that DB and RN chose a different ad hoc normalization. A quick integration of their respective spectra seems to indicate a 2.9% offset in their respective normalization. Our aforementioned procedure outputs respectively a median 3.4 σ (4 σ) incompatibility with (without) this normalization adjustment. A normalization free fit (increasing normalization uncertainties to infinity in both DB and RN covariance matrices), yields a consistent result with a 3.4 σ incompatibility (p-value below 7 10^-4). Except for DC whose tests are compatible with all other experiments within 1.6 σ, the lowest rejection significance between pairs of experiments was found for DB and B3, with a 2.5 σ significance. A free norm fit still yielded a 2.3 σ significance rejection of compatibility hypothesis.§.§ On isotopic compatibility of reactor spectra On the antineutrino production side, all reactors are similar pressurized water reactors. Seeking for differences among experiments, the core isotopic composition is the main differing component which could play a major role in the antineutrino spectral shape. For a complete fuel burning (a 60-70 GWd/t burnup), the reactor antineutrino rates are known to roughly decrease by -10% and the spectra to tilt by approximately -4%/ MeV between the fresh beginning and the far end of nuclear fuel burning. This effect is reduced by a factor 3 to 4 taking into account reactor operations where fuel assemblies are refreshed by thirds or quarters of the whole core. As indicated in Table <ref>, the average isotopic compositions over the data taking periods are very close from each other <cit.>. On the one hand, DB and RN have isotopic compositions which differ upmost by 1.7% (mixture of 6 cores with comparable fuels), but disagree on the distortion amplitude by 60%. On the other hand DC has the most burnt makeup among the four experiments and displays a distortion amplitude comparable to DB, while B3 reactor composition is half-way between DC and DB but does not observe any distortion in the spectrum. There is no simple coherent pattern explaining these variations with isotopic compositions. In a more complex scenario if the antineutrino spectra were extremely sensitive to fission fractions from each reactor, the near/far relative measurement strategy for the θ_13 quest would have failed. With many cores and the disparity of solid angle exposures among each detector, the near/far ratios of DB and alike experiments would also be distorted from one another to a large extent. As a conclusion, the differences in the published DC, DB, RN and B3 data-to-prediction ratios cannot be explained by core isotopic compositions and therefore, must have another origin. Considering these distortions are proportional to reactor powers <cit.>, a remaining possibility is to investigate at potential detector effects. While not being a definitive argument, it seems more likely to distort an energy scale and produce warped observed spectra to explain DC, DB and RN <cit.> than to wipe out a suspected spectral distortion with the exact required compensation from the energy scale to explain the flatness of spectra ratio in B3 <cit.>. In the next sections unassessed residual nonlinearities in energy scales are studied as a potential scenario to explain not only the difference between experiments but also the mismatches in the data-to-prediction ratios. § ENERGY DETERMINATION The four aforementioned experiments detect antineutrinos in liquid scintillators through the inverse β decay reaction: ν̅_e+p→ e^++n. In this process, most of the antineutrino energy is directly conveyed to the positron with nuclear recoil effects totalizing an uppermost 0.1%/MeV correction on positron/antineutrino energy. Antineutrino spectra are therefore well acquired through counting and energy determination of the detected positron. Such positrons deposit their energy in the scintillator, with subsequent light emission through energy transfers between solvent, primary and secondary fluors. The light yield scales nonlinearly with the deposited energy (with a characteristic O(10%) energy distortion below and above 1 MeV <cit.>). Dedicated and careful laboratory measurements are required for every liquid scintillator to achieve an energy determination accuracy at O(1%) <cit.>. The faint scintillation light is converted using photomultipliers tube (PMTs) to measurable pC charge signals to estimate the energy of the incident particle. Because of the data acquisition systems, digitization process, detection threshold effects, width of time window for charge acquisition, the typical nonlinearity in charge collection amounts to O(10%) <cit.>. These effects are peculiar to each acquisition system <cit.> and special LED and γ calibration runs are used to characterize the full electronic acquisition chain.To study the robustness and flexibility of the energy scale calibration, the most stringent Daya Bay results along with their recent reactor antineutrino spectrum deconvolution <cit.>, corrected from θ_13 oscillation effect, have been considered. The same studies could however be directly applied to the other reactor experiments. The Daya Bay experiment used dedicated γ calibration sources as well as ^12 B β spectrum to estimate and constrain the energy scale nonlinearities with an empirical model <cit.>. This model is a twofold factor, with a first term accounting for scintillation physics, and a second one accounting for electronic nonlinearities. Such an arbitrary function was designed to fit well the available data points, as illustrated on Figures <ref> (a) (γ sources) and (b) (^12 B β spectrum). The γ data points correspond to the ratios between reconstructed energies and those estimated from the best fitted empirical model <cit.>. A gray shaded area indicates the uncertainty envelope estimated from the comparison of 5 empirical models describing the energy scale nonlinearities <cit.>. However, a fit to the relative energy resolution was also provided: σ_E/<E_ rec>=√(a^2+b^2/E+c^2/E^2) , with a=0.015, b=0.087 and c=0.027 <cit.> as the fitted parameters. While b is governed by photostatistics, a and c are related to systematic effects. Parameter a is driven by spatial and temporal variations throughout the detector while c arises from intrinsic PMT and electronic noise. They both assess that the energy scale uncertainty is rather above 1.5% than below 1% <cit.>. Double Chooz obtained a comparable larger value a=0.018 <cit.>. Taking into account these two systematic effect evaluations from a and c a more conservative systematic uncertainty corridor was also displayed on Figure <ref> (a).Because the scintillation quenching depends on the particle type, γ from radioactive sources and neutron captures, e^- from ^12 B β spectrum, e^+ from antineutrino inverse β decay reaction do not produce the same amount of light for a given deposited energy. Tackling further associated nonlinearities would require to have a complete Monte Carlo simulation with scintillator content and Cherenkov radiation modeling, which is not within the scope of this article. As opposed to scintillation light emission, electronic charge collection nonlinearity is identical whatever the particle type is, and will be our focus in what follows.§ RESIDUAL NONLINEARITY MATCHING The aim of the present study is to assess if a small residual nonlinearity (RNL) in the energy scale can both explain the observed reactor antineutrino spectra and calibration data. For this purpose, an RNL function φ, which adds more flexibility to the empirical model used by DB, was introduced. As such, the probability density functions (PDF) of the DB nominal reconstructed energies, f_E, and of the transformed energies, f_φ(E), are related through: f_E(x)dx=f_φ(E)(φ(x))dφ(x). A more straightforward expression of the transformed energy PDF is:f_φ(E)(x)= dF_E(φ^-1(x))/ dx ,where F_E is the cumulative distribution function (CDF) of the original energy variable. The RNL function may also be expressed through the relative energy scale distortion δ:φ(x)=x (1+δ(x)) .The transformation relationship between the PDFs allows to simultaneously determine φ (or equivalently δ) on γ calibration, ^12 B and ν̅_e data. The function φ was estimated using a global χ^2 fitting framework on the 3 independent data sets and an additional regularization term as in section <ref>:χ^2=χ_γ^2+χ_ν^2+χ_ B^2+χ_ R^2 .The χ^2 associated to γ calibration data was defined asχ_γ^2=∑_i=1^n^(γ)(y_i^(γ)-δ(x_i^(γ))/σ_i^(γ))^2 ,where x_i^(γ),y_i^(γ) correspond to the n^(γ)=12 data points with associated uncertainties σ_i^(γ), as illustrated on Figure <ref> (a). The χ^2 associated to ν̅_e data was included as χ_ν^2=∑_i=1^n^(ν)(y_i^(ν)-(1+α_ν) N_i^(ν)/σ_i^(ν))^2+ (α_ν/σ_ν)^2 ,where y_i^(ν) are the n^(ν)=24 ratios of the observed to predicted antineutrino spectra (θ_13 effect already removed) displayed on Figure <ref> (c), σ_i^(ν) the associated uncertainties, σ_ν an additional detector normalization uncertainty of 2.1% <cit.> and α_ν the corresponding nuisance parameter. Eventually, the χ^2 associated to ^12 B spectrum was constructed as follows:χ_B^2=∑_i=1^n^(B)(y_i^(B)-α_BN_i^(B)-α_NN_i^(N)/σ_i^(B))^2 ,where y_i^(B) are the n^(B)=52 experimental ^12 B spectrum bin values with uncertainties σ_i^(B). The data were freely fitted with both ^12 B and ^12 N components as in <cit.>, as no further prior rate information was available. Using Equation (<ref>), the ν̅_e (N^(ν)), ^12 B (N^(B)) and ^12 N (N^(N)) count rates are given by:N_i^(·)= N^(·) ∫_x_i^(·)-b_i^(·)/2^x_i^(·)+b_i^(·)/2f_φ(E)^(·)(x) dx = N^(·) F_E^(·)(φ^-1(x_i^(·)+b_i^(·)/2))- N^(·) F_E^(·)(φ^-1(x_i^(·)-b_i^(·)/2)) ,where x_i^(·) and b_i^(·) are the i^ th bin center and width respectively, F_E^(·) is the associated CDF in the original energy variable, E. N^(·) is a nominal normalization factor given in <cit.>. F_E^(·) was determined from interpolation of Monte Carlo spectra from <cit.>. The target data set is indicated in the superscript parentheses ^(·) as in Equations (<ref>) and (<ref>). The ν̅_e data-to-prediction ratio was then estimated using the quotient between the above expression and the nominal DB prediction <cit.>. For small RNL (δ(x)≪1), φ^-1 can be expressed as φ^-1(x)≃ x (1-δ(x)). A Gaussian mixture model <cit.> was chosen for modeling δ with K evenly spaced knots, x_k, between 0 and 16 MeV, a bandwidth, h, equal to the inter-knot distance: δ(x)=1/h∑_k=1^K w_k ϕ(x-x_k/h), where, ϕ is the standard normal probability distribution function. The relative distortion just defined is rather flexible. To avoid data overfitting, the last term in our global χ^2 Definition (<ref>) is a quadratic regularization term penalizing higher local curvatures in δ as described in section <ref>: χ_R^2=λ∫δ^''(x)^2dx. The regularization level, λ>0, was self-determined from data through the generalized cross validation method <cit.>. The χ^2 optimization was found to be still slightly sensitive to the number and the position of the knots. To further prevent this, an extra regularization procedure was used to complement the global χ_R^2 penalization term. While the bandwidth was kept fixed to the inter-knot distance, the number of knots, K, was chosen with respect to the quality of the standardized fit output residuals. It had to be large enough to correctly model the distortions (more than 10 knots) but small enough to avoid overfitting (less than 30 knots). An optimum of 18 knots was selected, corresponding to a standardized residual distribution the closest to a standard normal distribution among our investigations. § RESULTS Figure <ref> presents the output of the global χ^2 function (<ref>) optimization. The RNL best fit appears in orange thick solid line with a shaded orange area indicating the 1 σ uncertainty. The best fit result gives χ_min^2/ ndof=89.7/86 (p-value of 0.39) with a standardized residual distribution following the standard normal distribution. The fitted distortion correctly reproduces the γ calibrations, the ^12 B constraint and the observed antineutrino spectrum warping. The best fit RNL agrees within 1 σ with the calibration data points. The ^208 Tl and ^60 Co around 2.5 MeV as well as the ^16 O^⋆ at 6 MeV are slightly off by less than 2.4 σ from the best fit. It is worth noting that the obtained RNL is still compatible with the uncertainties originally quoted by the DB collaboration (gray shaded area on Figure <ref>(a)). Furthermore, the frequently mentioned constraint of the ^12 B spectrum is validated, with a fit comparable to the nominal DB one <cit.>. The ratio of observed to predicted antineutrino spectra is well reproduced with such a fitted RNL. As shown by Figure <ref>(c), all data points fall within the gray shaded area representing the reactor spectrum prediction uncertainty <cit.> from DB <cit.>. This study demonstrates that a 1% energy scale distortion could result in a 10% ν̅_e spectrum deformation.This result can be explained by a closer look at the PDF ratio of the ν̅_e reconstructed energies using the best fit RNL and the DB nominal one. To a good accuracy, such a ratio can be computed from a Taylor expansion of Equation (<ref>): f_φ(E)(x)/f_E(x)=1-x δ(x) f_E^'(x)/f_E(x)-δ(x)-x δ^'(x) .The first term corresponds to the case of no residual distortion, where φ(x)=x and δ(x)=0. The next term, -x δ(x) f_E^'(x)/f_E(x), expresses the relative change of shape in the PDF due to δ. As illustrated by the dashed green curve on Figure <ref>(c), it gives a maximum positive deviation around 4.5 MeV where δ(x)<0, along with a major negative contribution after 6 MeV where δ(x)>0. The red dotted curve on Figure <ref>(c) shows the small contribution of the third term, -δ(x), to the ratio (<ref>). Although the maximal 1.2% distortion from δ(x) occurs around 4 MeV, the fourth term, -xδ^'(x), introduces a 6% positive deviation which is de facto shifted to the 5 MeV region. It dominates by far the PDF ratio as defined by Equation (<ref>) in the shoulder region. It is also worth noting that this term explains as well a large fraction of the deficit in the 2–3 MeV energy region (already corrected from θ_13 effect <cit.>). The notable feature of this overall RNL is a +1% amplitude kink around 4 MeV. No calibration point constrains the nonlinearity in this region. The nearest one, from ^12 C, shows a slight excess around 5 MeV, which is perfectly reproduced by our best fit nonlinearity. As shown by Figure <ref>(b), the Boron 12 spectrum is mostly unaffected by this additional RNL.§ DISCUSSION The fit from previous section is the required RNL to best match the observed antineutrino spectrum from the predicted one. It should be understood as a residual artifact, or bias, after all calibration works have already been performed. Our study shows that a small 1% mismatch in energy scale model in a localized energy range around 4 MeV, where no calibration cross-checks are available, can be responsible of the large 10% distortion in the observed to predicted antineutrino spectra. This effect is thus a small one, within calibration uncertainties, which might have been overlooked, and the consequences in reactor spectra prediction are sizable. Bringing definitive statements about absolute reactor spectra measurements requires extremely careful work on energy scale to ensure its robustness. This indicates that shape systematics in observed antineutrino spectra might have been underestimated.Such a large distortion amplification process is rooted in two phenomena. First, the energy scale uncertainties are relative. The higher the energy, the higher the absolute uncertainty. A fixed relative energy scale error hence induces a bigger impact at higher energies. A distortion around 4 MeV in energy scale might thus be shifted above 5 MeV in spectra ratio. Second, histograms are representative of a probability density function. Under a change/distortion of the histogrammed variable, they behave as a density function. Consequently an extra “infinitesimal volume conservation factor” arises with the change of shape of the density function. From these two characteristic effects, a localized change of slope in energy scale around 4 MeV of less than 1%/MeV alters significantly the antineutrino counting distribution around 5 MeV, as extensively demonstrated in previous section.The origin of such an artifact in energy calibration models should be investigated and tested. It could be of statistical nature, such as a bias coming from an average of nonlinearly biased charges. All the current energy reconstruction strategies in DC, DB, RN use the total charge, received by all PMTs, as a proxy of the true deposited energy. This work hypothesis, while fully legitimate when energy and charge are proportional, starts to break down in scenarii accumulating nonlinearities. Especially, if raw charge nonlinearities are not corrected on each channel (a correction known as “flat-fielding” in digital imaging), a complex position/energy nonlinear mapping arises and simple calibration scheme, decoupling energy and position variables, might be ineffective to simultaneously correct these dependences. As soon as the raw charge distortion ψ(Q) is not a linear mapping, the average of the distorted charges is different from the distorted average charge, whatever the distribution of Q (⟨ψ(Q)⟩≠ψ(⟨ Q⟩), where ⟨.⟩ stands for the average over charge distribution). In DC and DB, the electronics nonlinearity correction function is a convex function of the reconstructed charge/energy. Therefore for a given distortion ψ, the calibrations are overestimating the true distortions since they use distorted charge averages to estimate this distortion in E scale: ψ(⟨ Q⟩)<⟨ψ(Q)⟩. The fitted distortion from calibration data is thus overestimating, by construction, the true raw charge distortion. The amount of overestimated distortion depends on the peculiar charge distribution involved in the average estimate, therefore on the energy and position inside the detector. A part of this effect is canceled through the current calibration procedures, especially below 3 MeV and by the fact that in essence reconstructed energies are also to some extent average of charges. However, because of the lack of calibration points around 4 MeV, a residual nonlinearity might still be present as we investigated in this work.This residual nonlinearity can be experimentally tested with dedicated calibrations around 4 MeV and further works on raw charge corrections. With the high statistics of the DB experiment it is also possible to build up local antineutrino spectra inside the detectors. If charge reconstructions are biased, the comparison of these local spectra should exhibit enlarged variances, especially in the 4-5 MeV range. The DB and DC fits to the energy resolution already support such a point, with an energy scale systematics evaluated in DB around 1.5% and 1.8% in DC, nearly two times bigger than the quoted precision of the energy scale models. A systematic of such an amplitude is by far sufficient to make observed and predicted antineutrino spectra consistent with each other.As a conclusion, current observed antineutrino spectra of DC, DB, RN and B3 are not compatible with each other. This study suggests that reactor ν̅_e spectral distortions might have their origin in detector calibration artifacts. Taking into account this systematic effect, reactor spectra predictions become fully compatible with all the observations. Our work is, to our knowledge, the single one able to reconcile current generation of reactor antineutrino experiments with older ones such as Bugey 3, to recover consistency with BILL experiments and their converted β spectra. Because an energy scale nonlinearity induces a migration of events across bins, it induces mostly no impact on the reactor rate anomaly. Thus, if this interpretation is correct, the current observed distortions in reactor antineutrino spectra might have no relation with the 6% reactor rate anomaly. Regarding currently observed antineutrino spectra, as opposed to <cit.>, spectra prediction uncertainties would remain unchanged, but observed reactor spectra uncertainties of Double Chooz <cit.>, Daya Bay <cit.> and RENO <cit.> should be enlarged. As demonstrated in this article, the 10% spectral distortion around 5 MeV is well within the 1 σ calibration uncertainty of the energy scale. The uncertainties of Bugey 3 <cit.> being already consistent with our fit (the slope on spectra ratio might be induced by an energy scale bias within the quoted uncertainties), no extra calibration error is justified for this experiment. Decisive tests have to be planned. Until then a diligent work on studying the energy response of detectors in the 3–5 MeV region is clearly indicated. 10 RC53F. Reines and C. L. Cowan, Jr. Phys. Rev. 92:830 (1953)RC59 F. Reines and C. L. Cowan, Jr. 1959, Phys. Rev. 113:273 (1959)DCY. Abe et al., Double Chooz collaboration, JHEP 1410 (2014) 086.DBF. P. An et al., Daya Bay collaboration, Phys.Rev.Lett. 115 (2015) no.11, 111802RNJ. H. Choi et al., RENO collaboration, arXiv:1610.04326 [hep-ex], 2016.MUT. A. Mueller et al., Phys. Rev. C83, 054615 (2011).HUP. Huber, Phys. Rev. C84, 024617 (2011).RAAG. Mention et al., Phys. Rev. D83 (2011) 073006.B3 B. Achkar et al. 1996, Phys. Lett. B374:243 (1996).HAA. C. Hayes et al. Phys. Rev. D 92:033015 (2015)KLA. Gando et al., KamLAND Collaboration, Phys. Rev. D 83, 052002.SNOQ. R. Ahmad et al., (SNO Collaboration) Phys. Rev. Lett. 89, 011301.JNF. An et al., J. Phys. G 43 (2016) 030401.RN50S. B. Kim, Nucl. Part. Phys. Proc. 265–266 (2015) 93FEF. von Feilitzsch et al., Phys. Lett 118B:365 (1982)SCK. Schreckenach et al., Phys. Lett 160B:325 (1985)AHA. A. Hahn et al., Phys. Lett 218B:365 (1989)ABC. Aberle, PhD thesis, Ruperto-Carola University of Heidelberg (2011).WAS. Wagner, PhD. thesis, Ruprecht-Karls-Universitaet, Heidelberg,(2014).CLC. A. Lewis, PhD. thesis, University of Wisconsin–Madison (2014).GOG. Golub et al., Technometrics, Volume 21, Issue 2 (1979).HVT. Hastie et al., The Elements of Statistical Learning, 2nd Edition, 2009, Springer-Verlag. | http://arxiv.org/abs/1705.09434v1 | {
"authors": [
"G. Mention",
"M. Vivier",
"J. Gaffiot",
"T. Lasserre",
"A. Letourneau",
"T. Materna"
],
"categories": [
"hep-ex",
"physics.ins-det"
],
"primary_category": "hep-ex",
"published": "20170526053252",
"title": "Reactor antineutrino shoulder explained by energy scale nonlinearities?"
} |
Multiple Source Domain Adaptation with Adversarial Learning Han Zhao^†The first two authors contributed equally to this work. [email protected] Shanghang Zhang^* [email protected] Guanhang Wu^♮ [email protected] JoãoP. Costeira^♭ [email protected] José M. F.Moura^ [email protected] Geoffrey J. Gordon^† [email protected]^†Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, USA^Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA^♮Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, USA^♭Department of Electrical and Computer Engineering, Instituto Superior Técnico, Lisbon, Portugal Received; accepted =======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================While domain adaptation has been actively researched in recent years, most theoretical results and algorithms focus on the single-source-single-target adaptation setting. Naive application of such algorithms on multiple source domain adaptation problem may lead to suboptimal solutions. As a step toward bridging the gap, we propose a new generalization bound for domain adaptation when there are multiple source domains with labeled instances and one target domain with unlabeled instances. Compared with existing bounds, the new bound does not require expert knowledge about the target distribution, nor the optimal combination rule for multisource domains. Interestingly, our theory also leads to an efficient learning strategy using adversarial neural networks: we show how to interpret it as learning feature representations that are invariant to the multiple domain shifts while still being discriminative for the learning task. To this end, we propose two models, both of which we call multisource domain adversarial networks (MDANs): the first model optimizes directly our bound, while the second model is a smoothed approximation of the first one, leading to a more data-efficient and task-adaptive model. The optimization tasks of both models are minimax saddle point problems that can be optimized by adversarial training. To demonstrate the effectiveness of MDANs, we conduct extensive experiments showing superior adaptation performance on three real-world datasets: sentiment analysis, digit classification, and vehicle counting.§ INTRODUCTION The success of machine learning algorithms has been partially attributed to rich datasets with abundant annotations <cit.>. Unfortunately, collecting and annotating such large-scale training data is prohibitively expensive and time-consuming. To solve these limitations, different labeled datasets can be combined to build a larger one, or synthetic training data can be generated with explicit yet inexpensive annotations <cit.>. However, due to the possible shift between training and test samples, learning algorithms based on these cheaper datasets still suffer from high generalization error. Domain adaptation (DA) focuses on such problems by establishing knowledge transfer from a labeled source domain to an unlabeled target domain, and by exploring domain-invariant structures and representations to bridge the gap <cit.>. Both theoretical results <cit.> and algorithms <cit.> for DA have been proposed. Recently, DA algorithms based on deep neural networks produce breakthrough performance by learning more transferable features <cit.>. Most theoretical results and algorithms with respect to DA focus on the single-source-single-target adaptation setting <cit.>. However, in many application scenarios, the labeled data available may come from multiple domains with different distributions. As a result, naive application of the single-source-single-target DA algorithms may lead to suboptimal solutions. Such problem calls for an efficient technique for multiple source domain adaptation. In this paper, we theoretically analyze the multiple source domain adaptation problem and propose an adversarial learning strategy based on our theoretical results. Specifically, we prove a new generalization bound for domain adaptation when there are multiple source domains with labeled instances and one target domain with unlabeled instances. Our theoretical results build on the seminal theoretical model for domain adaptation introduced by <cit.>, where a divergence measure, known as the ℋ-divergence, was proposed to measure the distance between two distributions based on a given hypothesis space ℋ. Our new result generalizes the bound <cit.> to the case when there are multiple source domains. The new bound has an interesting interpretation and reduces to <cit.> when there is only one source domain. Technically, we derive our bound by first proposing a generalizedℋ-divergence measure between two sets of distributions from multi-domains. We then prove a PAC bound <cit.> for the target risk by bounding it from empirical source risks, using tools from concentration inequalities and the VC theory <cit.>. Compared with existing bounds, the new bound does not require expert knowledge about the target domain distribution <cit.>, nor the optimal combination rule for multiple source domains <cit.>. Our results also imply that it is not always beneficial to naively incorporate more source domains into training, which we verify to be true in our experiments.Interestingly, our bound also leads to an efficient implementation using adversarial neural networks. This implementation learns both domain invariant and task discriminative feature representations under multiple domains. Specifically, we propose two models (both named MDANs) by using neural networks as rich function approximators to instantiate the generalization bound we derive (Fig. <ref>). After proper transformations, both models can be viewed as computationally efficient approximations of our generalization bound, so that the goal is to optimize the parameters of the networks in order to minimize the bound. The first model optimizes directly our generalization bound, while the second is a smoothed approximation of the first, leading to a more data-efficient and task-adaptive model. The optimization problem for each model is a minimax saddle point problem, which can be interpreted as a zero-sum game with two participants competing against each other to learn invariant features. Both models combine feature extraction, domain classification, and task learning in one training process. MDANs is generalization of the popular domain adversarial neural network (DANN) <cit.> and reduce to it when there is only one source domain. We propose to use stochastic optimization with simultaneous updates to optimize the parameters in each iteration. To demonstrate the effectiveness of MDANs as well as the relevance of our theoretical results, we conduct extensive experiments on real-world datasets, including both natural language and vision tasks. We achieve superior adaptation performances on all the tasks, validating the effectiveness of our models. § PRELIMINARY We first introduce the notation used in this paper and review a theoretical model for domain adaptation when there is only one source and one target domain <cit.>. The key idea is the ℋ-divergence to measure the discrepancy between two distributions. Other theoretical models for DA exist <cit.>; we choose to work with the above model because this distance measure has a particularly natural interpretation and can be well approximated using samples from both domains. Notations We use domain to represent a distribution 𝒟 on input space 𝒳 and a labeling function f:𝒳→ [0, 1]. In the setting of one source one target domain adaptation, we use ⟨𝒟_S, f_S⟩ and ⟨𝒟_T, f_T⟩ to denote the source and target domain, respectively. A hypothesis is a binary classification function h:𝒳→{0, 1}. The error of a hypothesis h w.r.t. a labeling function f under distribution 𝒟_S is defined as: _S(h, f)𝔼_𝐱∼𝒟_S[|h(𝐱) - f(𝐱)|]. When f is also a hypothesis, then this definition reduces to the probability that h disagrees with h under 𝒟_S: 𝔼_𝐱∼𝒟_S[|h(𝐱) - f(𝐱)|] = 𝔼_𝐱∼𝒟_S[𝕀(f(𝐱)≠ h(𝐱))] = _𝐱∼𝒟_S(f(𝐱)≠ h(𝐱)).We define the risk of hypothesis h as the error of h w.r.t. a true labeling function under domain 𝒟_S, i.e., _S(h)_S(h, f_S). As common notation in computational learning theory, we use _S(h) to denote the empirical risk of h on the source domain. Similarly, we use _T(h) and _T(h) to mean the true risk and the empirical risk on the target domain. ℋ-divergence is defined as follows:Let ℋ be a hypothesis class for instance space 𝒳, and 𝒜_ℋ be the collection of subsets of 𝒳 that are the support of some hypothesis in ℋ, i.e., 𝒜_ℋ{h^-1({1})| h∈ℋ}. The distance between two distributions 𝒟 and 𝒟' based on ℋ is: d_ℋ(𝒟, 𝒟') 2sup_A∈𝒜_ℋ|_𝒟(A) - _𝒟'(A)| When the hypothesis class ℋ contains all the possible measurable functions over 𝒳, d_ℋ(𝒟, 𝒟') reduces to the familiar total variation. Given a hypothesis class ℋ, we define its symmetric difference w.r.t. itself as: ℋΔℋ = {h(𝐱)⊕ h'(𝐱)| h, h'∈ℋ}, where ⊕ is the xor operation. Let h^* be the optimal hypothesis that achieves the minimum combined risk on both the source and the target domains: h^*_h∈ℋ_S(h) + _T(h)and use λ to denote the combined risk of the optimal hypothesis h^*:λ_S(h^*) + _T(h^*)<cit.> and <cit.> proved the following generalization bound on the target risk in terms of the source risk and the discrepancy between the source domain and the target domain: Let ℋ be a hypothesis space of VC-dimension d and 𝒰_S, 𝒰_T be unlabeled samples of size m each, drawn from 𝒟_S and 𝒟_T, respectively. Let d_ℋΔℋ be the empirical distance on 𝒰_S and 𝒰_T; then with probability at least 1 - δ over the choice of samples, for each h∈ℋ, _T(h)≤_S(h) + 1/2d_ℋΔℋ(𝒰_S, 𝒰_T) + 4√(2dlog(2m) + log(4/δ)/m) + λ The generalization bound depends on λ, the optimal combined risk that can be achieved by hypothesis in ℋ. The intuition is that if λ is large, then we cannot hope for a successful domain adaptation. One notable feature of this bound is that the empirical discrepancy distance between two samples 𝒰_S and 𝒰_T can usually be approximated by a discriminator to distinguish instances from these two domains. § A NEW GENERALIZATION BOUND FOR MULTIPLE SOURCE DOMAIN ADAPTATION In this section we first generalize the definition of the discrepancy function d_ℋ(·, ·) that is only appropriate when we have two domains. We will then use the generalized discrepancy function to derive a generalization bound for multisource domain adaptation. We conclude this section with a discussion and comparison of our bound and existing generalization bounds for multisource domain adaptation <cit.>. We refer readers to appendix for proof details and we mainly focus on discussing the interpretations and implications of the theorems.Let {𝒟_S_i}_i=1^k and 𝒟_T be k source domains and the target domain, respectively. We define the discrepancy function d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k) induced by ℋ to measure the distance between 𝒟_T and a set of domains {𝒟_S_i}_i=1^k as follows: d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k) max_i∈[k]d_ℋ(𝒟_T; 𝒟_S_i) = 2max_i∈[k]sup_A∈𝒜_ℋ|_D_T(A) - _D_S_i(A)| Again, let h^* be the optimal hypothesis that achieves the minimum combined risk: h^* _h∈ℋ(_T(h) + max_i∈[k]_S_i(h))and define λ_T(h^*) + max_i∈[k]_S_i(h^*)i.e., the minimum risk that is achieved by h^*. The following lemma holds for ∀ h∈ℋ: theorempopulation_T(h) ≤max_i∈[k]_S_i(h) + λ + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k).Remark. Let us take a closer look at the generalization bound: to make it small, the discrepancy measure between the target domain and the multiple source domains need to be small. Otherwise we cannot hope for successful adaptation by only using labeled instances from the source domains. In this case there will be no hypothesis that performs well on both the source domains and the target domain. It is worth pointing out here that the second term and the third term together introduce a tradeoff (regularization) on the complexity of our hypothesis class ℋ. Namely, if ℋ is too restricted, then the second term λ can be large while the discrepancy term can be small. On the other hand, if ℋ is very rich, then we expect the optimal error, λ, to be small, while the discrepancy measure d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) to be large. The first term is a standard source risk term that usually appears in generalization bounds under the PAC-learning framework <cit.>. Later we shall upper bound this term by its corresponding empirical risk.The discrepancy distance d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) is usually unknown. However, we can bound d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) from its empirical estimation using i.i.d. samples from 𝒟_T and {𝒟_S_i}_i=1^k: theoremdiscrepancyLet 𝒟_T and {𝒟_S_i}_i=1^k be the target distribution and k source distributions over 𝒳. Let ℋ be a hypothesis class where VCdim(ℋ) = d. If 𝒟_T and {𝒟_S_i}_i=1^k are the empirical distributions of 𝒟_T and {𝒟_S_i}_i=1^k generated with m i.i.d. samples from each domain, then for ϵ > 0, we have:(| d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k) - d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k)| ≥ϵ)≤ 4k(em/d)^d exp(-mϵ^2 / 8) The main idea of the proof is to use VC theory <cit.> to reduce the infinite hypothesis space to a finite space when acting on finite samples. The theorem then follows from standard union bound and concentration inequalities. Equivalently, the following corollary holds: Let 𝒟_T and {𝒟_S_i}_i=1^k be the target distribution and k source distributions over 𝒳. Let ℋ be a hypothesis class where VCdim(ℋ) = d. If 𝒟_T and {𝒟_S_i}_i=1^k are the empirical distributions of 𝒟_T and {𝒟_S_i}_i=1^k generated with m i.i.d. samples from each domain, then, for 0 < δ < 1, with probability at least 1-δ (over the choice of samples), we have:| d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k) - d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k)| ≤ 2√(2/m(log4k/δ + dlogem/d)) Note that multiple source domains do not increase the sample complexity too drastically: it is only the square root of a log term in Corollary. <ref> where k appears. Similarly, we do not usually have access to the true error max_i∈[k]_S_i(h) on the source domains, but we can often have an estimate (max_i∈[k]_S_i(h)) from training samples. We now provide a probabilistic guarantee to bound the difference between max_i∈[k]_S_i(h) and max_i∈[k]_S_i(h) uniformly for all h∈ℋ: theoremerrorLet {𝒟_S_i}_i=1^k be k source distributions over 𝒳. Let ℋ be a hypothesis class where VCdim(ℋ) = d. If {𝒟_S_i}_i=1^k are the empirical distributions of {𝒟_S_i}_i=1^k generated with m i.i.d. samples from each domain, then, for ϵ > 0, we have:(sup_h∈ℋ| max_i∈[k]_S_i(h) - max_i∈[k]_S_i(h) | ≥ϵ) ≤ 2k(me/d)^d exp(-2mϵ^2) Again, Thm. <ref> can be proved by a combination of concentration inequalities and a reduction from infinite space to finite space, along with the subadditivity of the max function. Equivalently, we have the following corollary hold: Let {𝒟_S_i}_i=1^k be k source distributions over 𝒳. Let ℋ be a hypothesis class where VCdim(ℋ) = d. If {𝒟_S_i}_i=1^k are the empirical distributions of {𝒟_S_i}_i=1^k generated with m i.i.d. samples from each domain, then, for 0 < δ < 1, with probability at least 1-δ (over the choice of samples), we have:sup_h∈ℋ|max_i∈[k]_S_i(h) - max_i∈[k]_S_i(h)| ≤√(1/2m(log2k/δ + dlogme/d)) Combining Thm. <ref> and Corollaries. <ref>, <ref> and realizing that VCdim(ℋΔℋ)≤ 2VCdim(ℋ) <cit.>, we have the following theorem: Let 𝒟_T and {𝒟_S_i}_i=1^k be the target distribution and k source distributions over 𝒳. Let ℋ be a hypothesis class where VCdim(ℋ) = d. If 𝒟_T and {𝒟_S_i}_i=1^k are the empirical distributions of 𝒟_T and {𝒟_S_i}_i=1^k generated with m i.i.d. samples from each domain, then, for 0 < δ < 1, with probability at least 1-δ (over the choice of samples), we have:_T(h)≤max_i∈[k]_S_i(h) + √(1/2m(log4k/δ + dlogme/d))+ 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) + √(2/m(log8k/δ + 2dlogme/2d))+ λ = max_i∈[k]_S_i(h) + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) + λ + O(√(1/m(logk/δ + dlogme/d))) Remark. Thm. <ref> has a nice interpretation for each term: the first term measures the worst case accuracy of hypothesis h on the k source domains, and the second term measures the discrepancy between the target domain and the k source domains. For domain adaptation to succeed in the multiple sources setting, we have to expect these two terms to be small: we pick our hypothesis h based on its source training errors, and it will generalize only if the discrepancy between sources and target is small. The third term λ is the optimal error we can hope to achieve. Hence, if λ is large, one should not hope the generalization error to be small by training on the source domains. [Of course it is still possible that _T(h) is small while λ is large, but in domain adaptation we do not have access to labeled samples from 𝒟_T.] The last term bounds the additional error we may incur because of the possible bias from finite samples. It is also worth pointing out that these four terms appearing in the generalization bound also capture the tradeoff between using a rich hypothesis class ℋ and a limited one as we discussed above: when using a richer hypothesis class, the first and the third terms in the bound will decrease, while the value of the second term will increase; on the other hand, choosing a limited hypothesis class can decrease the value of the second term, but we may incur additional source training errors and a large λ due to the simplicity of ℋ. One interesting prediction implied by Thm. <ref> is that the performance on the target domain depends on the worst empirical error among multiple source domains, i.e., it is not always beneficial to naively incorporate more source domains into training. As we will see in the experiment, this is indeed the case in many real-world problems. Comparison with Existing Bounds First, it is easy to see that, upto a multiplicative constant, our bound in (<ref>) reduces to the one in Thm. <ref> when there is only one source domain (k = 1). Hence Thm. <ref> can be treated as a generalization of Thm. <ref>. <cit.> give a generalization bound for semi-supervised multisource domain adaptation where, besides labeled instances from multiple source domains, the algorithm also has access to a fraction of labeled instances from the target domain. Although in general our bound and the one in <cit.> are incomparable, it is instructive to see the connections and differences between them: on one hand, the multiplicative constants of the discrepancy measure and the optimal error in our bound are half of those in <cit.>'s bound, leading to a tighter bound; on the other hand, because of the access to labeled instances from the target domain, their bound is expressed relative to the optimal error rate on the target domain, while ours is in terms of the empirical error on the source domain. Finally, thanks to our generalized definition of d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k), we do not need to manually specify the optimal combination vector α in <cit.>, which is unknown in practice. <cit.> also give a generalization bound for multisource domain adaptation under the assumption that the target distribution is a mixture of the k sources and the target hypothesis can be represented as a convex combination of the source hypotheses. While the distance measure we use assumes 0-1 loss function, their generalized discrepancy measure can also be applied for other losses functions <cit.>. § MULTISOURCE DOMAIN ADAPTATION WITH ADVERSARIAL NEURAL NETWORKS In this section we shall describe a neural network based implementation to minimize the generalization bound we derive in Thm. <ref>. The key idea is to reformulate the generalization bound by a minimax saddle point problem and optimize it via adversarial training.Suppose we are given samples drawn from k source domains {𝒟_S_i}, each of which contains m instance-label pairs. Additionally, we also have access to unlabeled instances sampled from the target domain 𝒟_T. Once we fix our hypothesis class ℋ, the last two terms in the generalization bound (<ref>) will be fixed; hence we can only hope to minimize the bound by minimizing the first two terms, i.e., the maximum source training error and the discrepancy between source domains and target domain. The idea is to train a neural network to learn a representation with the following two properties: 1). indistinguishable between the k source domains and the target domain; 2). informative enough for our desired task to succeed. Note that both requirements are necessary: without the second property, a neural network can learn trivial random noise representations for all the domains, and such representations cannot be distinguished by any discriminator; without the first property, the learned representation does not necessarily generalize to the unseen target domain. Taking these two properties into consideration, we propose the following optimization problem: minimizemax_i∈[k](_S_i(h) + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k))One key observation that leads to a practical approximation of d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) from <cit.> is that computing the discrepancy measure is closely related to learning a classifier that is able to disintuish samples from different domains: d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) = max_i∈[k](1 - 2min_h∈ℋΔℋ(1/2m∑_𝐱∼𝒟_T𝕀(h(𝐱) = 1) + 1/2m∑_𝐱∼𝒟_S_i𝕀(h(𝐱 = 0))))Let _T, S_i(h) be the empirical risk of hypothesis h in the domain discriminating task. Ignoring the constant terms that do not affect the optimization formulation, moving the max operator out, we can reformulate (<ref>) as:minimizemax_i∈[k]( _S_i(h) - min_h'∈ℋΔℋ_T, S_i(h'))The two terms in (<ref>) exactly correspond to the two criteria we just proposed: the first term asks for an informative feature representation for our desired task to succeed, while the second term captures the notion of invariant feature representations between different domains. Inspired by <cit.>, we use the gradient reversal layer to effectively implement (<ref>) by backpropagation. The network architecture is shown in Figure. <ref>. The pseudo-code is listed in Alg. <ref> (the hard version). One notable drawback of the hard version in Alg. <ref> is that in each iteration the algorithm only updates its parameter based on the gradient from one of the k domains. This is data inefficient and can waste our computational resources in the forward process. To improve this, we approximate the max function in (<ref>) by the log-sum-exp function, which is a frequently used smooth approximation of the max function. Define _i(h) _S_i(h) - min_h'∈ℋΔℋ_T, S_i(h'): max_i∈[k]_i(h)≈1/γlog∑_i∈[k]exp(γ_i(h))where γ > 0 is a parameter that controls the accuracy of this approximation. As γ→∞, 1/γlog∑_i∈[k]exp(γ_i(h)) →max_i∈[k]_i(h). Correspondingly, we can formulate a smoothed version of (<ref>) as:minimize1/γlog∑_i∈[k]exp(γ( _S_i(h) - min_h'∈ℋΔℋ_T, S_i(h')))During the optimization, (<ref>) naturally provides an adaptive weighting scheme for the k source domains depending on their relative error. Use θ to denote all the model parameters, then:∂/∂θ1/γlog∑_i∈[k]exp(γ(_S_i(h) - min_h'∈ℋΔℋ_T, S_i(h'))) = ∑_i∈[k]expγ_i(h)/∑_i'∈[k]expγ_i'(h)∂_i(h)/∂θThe approximation trick not only smooths the objective, but also provides a principled and adaptive way to combine all the gradients from the k source domains. In words, (<ref>) says that the gradient of MDAN is a convex combination of the gradients from all the domains. The larger the error from one domain, the larger the combination weight in the ensemble. We summarize this algorithm in the smoothed version of Alg. <ref>. Note that both algorithms, including the hard version and the smoothed version, reduce to the DANN algorithm <cit.> when there is only one source domain. § EXPERIMENTS We evaluate both hard and soft MDANs and compare them with state-of-the-art methods on three real-world datasets: the Amazon benchmark dataset <cit.> for sentiment analysis, a digit classification task that includes 4 datasets: MNIST <cit.>, MNIST-M <cit.>, SVHN <cit.>, and SynthDigits <cit.>, and a public, large-scale image dataset on vehicle counting from city cameras <cit.>. Details about network architecture and training parameters of proposed and baseline methods, and detailed dataset description will be introduced in the appendix. §.§ Amazon ReviewsDomains within the dataset consist of reviews on a specific kind of product (Books, DVDs, Electronics, and Kitchen appliances). Reviews are encoded as 5000 dimensional feature vectors of unigrams and bigrams, with binary labels indicating sentiment. We conduct 4 experiments: for each of them, we pick one product as target domain and the rest as source domains. Each source domain has 2000 labeled examples, and the target test set has 3000 to 6000 examples. During training, we randomly sample the same number of unlabeled target examples as the source examples in each mini-batch. We implement the Hard-Max and Soft-Max methods according to Alg. <ref>, and compare them with three baselines: MLPNet, marginalized stacked denoising autoencoders (mSDA) <cit.>, and DANN <cit.>. DANN cannot be directly applied in multiple source domains setting. In order to make a comparison, we use two protocols. The first one is to combine all the source domains into a single one and train it using DANN, which we denote as (cDANN). The second protocol is to train multiple DANNs separately, where each one corresponds to a source-target pair. Among all the DANNs, we report the one achieving the best performance on the target domain. We denote this experiment as (sDANN). For fair comparison, all these models are built on the same basic network structure with one input layer (5000 units) and three hidden layers (1000, 500, 100 units).Results and Analysis We show the accuracy of different methods in Table <ref>. Clearly, Soft-Max significantly outperforms all other methods in most settings. When Kitchen is the target domain, cDANN performs slightly better than Soft-Max, and all the methods perform close to each other. Hard-Max is typically slightly worse than Soft-Max. This is mainly due to the low data-efficiency of the Hard-Max model (Section <ref>, Eq. <ref>, Eq. <ref>). We argue that with more training iterations, the performance of Hard-Max can be further improved. These results verify the effectiveness of MDANs for multisource domain adaptation. To validate the statistical significance of the results, we run a non-parametric Wilcoxon signed-ranked test for each task to compare Soft-Max with the other competitors, as shown in Table <ref>. Each cell corresponds to the p-value of a Wilcoxon test between Soft-Max and one of the other methods, under the null hypothesis that the two paired samples have the same mean. From these p-values, we see Soft-Max is convincingly better than other methods.§.§ Digits DatasetsFollowing the setting in <cit.>, we combine four popular digits datasets (MNIST, MNIST-M, SVHN, and SynthDigits) to build the multisource domain dataset. We take each of MNIST-M, SVHN, and MNIST as target domain in turn, and the rest as sources. Each source domain has 20,000 labeled images and the target test set has 9,000 examples. We compare Hard-Max and Soft-Max of MDANs with five baselines: i). best-Single-Source. A basic network trained on each source domain (20,000 images) without domain adaptation and tested on the target domain. Among the three models, we report the one achieves the best performance on the test set. ii). Combine-Source. A basic network trained on a combination of three source domains (20,000 images for each) without domain adaptation and tested on the target domain. iii). best-Single-DANN. We train DANNs <cit.> on each source-target domain pair (20,000 images) and test it on target. Again, we report the best score among the three. iv). Combine-DANN. We train a single DANN on a combination of three source domains (20,000 images for each). v). Target-only. It is the basic network trained and tested on the target data. It serves as an upper bound of DA algorithms. All the MDANs and baseline methods are built on the same basic network structure to put them on a equal footing. Results and Analysis The classification accuracy is shown in Table <ref>. The results show that a naive combination of different training datasets can sometimes even decrease the performance. Furthermore, we observe that adaptation to the SVHN dataset (the third experiment) is hard. In this case, increasing the number of source domains does not help. We conjecture this is due to the large dissimilarity between the SVHN data to the others. For the combined sources, MDANs always perform better than the source-only baseline (MDANs vs. Combine-Source). However, directly training DANN on a combination of multiple sources leads to worse performance when compared with our approach (Combine-DANN vs. MDANs). In fact, this strategy may even lead to worse results than the source-only baseline (Combine-DANN vs. Combine-Source). Surprisingly, using a single domain (best-Single DANN) can sometimes achieve the best result. This means that in domain adaptation the quality of data (how close to the target data) is much more important than the quantity (how many source domains). As a conclusion, this experiment further demonstrates the effectiveness of MDANs when there are multiple source domains available, where a naive combination of multiple sources using DANN may hurt generalization. §.§ WebCamT Vehicle Counting DatasetWebCamT is a public dataset for vehicle counting from large-scale city camera videos, which has low resolution (352×240), low frame rate (1 frame/second), and high occlusion. It has 60,000 frames annotated with vehicle bounding box and count, divided into training and testing sets, with 42,200 and 17,800 frames, respectively. Here we demonstrate the effectiveness of MDANs to count vehicles from an unlabeled target camera by adapting from multiple labeled source cameras: we select 8 cameras that each has more than 2,000 labeled images for our evaluations. As shown in Fig. <ref>, they are located in different intersections of the city with different scenes. Among these 8 cameras, we randomly pick two cameras and take each camera as the target camera, with the other 7 cameras as sources. We compute the proxy 𝒜-distance (PAD) <cit.> between each source camera and the target camera to approximate the divergence between them. We then rank the source cameras by the PAD from low to high and choose the first k cameras to form the k source domains. Thus the proposed methods and baselines can be evaluated on different numbers of sources (from 2 to 7). We implement the Hard-Max and Soft-Max MDANs according to Alg. <ref>, based on the basic vehicle counting network FCN <cit.>. We compare our method with two baselines: FCN <cit.>, a basic network without domain adaptation, and DANN <cit.>, implemented on top of the same basic network. We record mean absolute error (MAE) between true count and estimated count. Results and Analysis The counting error of different methods is compared in Table <ref>. The Hard-Max version achieves lower error than DANN and FCN in most settings for both target cameras. The Soft-Max approximation outperforms all the baselines and the Hard-Max in most settings, demonstrating the effectiveness of the smooth and adaptative approximation. The lowest MAE achieved by Soft-Max is 1.1942. Such MAE means that there is only around one vehicle miscount for each frame (the average number of vehicles in one frame is around 20). Fig. <ref> shows the counting results of Soft-Max for the two target cameras under the 5 source cameras setting. We can see that the proposed method accurately counts the vehicles of each target camera for long time sequences. Does adding more source cameras always help improve the performance on the target camera? To answer this question, we analyze the counting error when we vary the number of source cameras as shown in Fig. <ref>. From the curves, we see the counting error goes down with more source cameras at the beginning, while it goes up when more sources are added at the end. This phenomenon corresponds to the prediction implied by Thm. <ref> (the last remark in Section <ref>): the performance on the target domain depends on the worst empirical error among multiple source domains, i.e., it is not always beneficial to naively incorporate more source domains into training. To illustrate this prediction better, we show the PAD of the newly added camera (when the source number increases by one) in Fig. <ref>. By observing the PAD and the counting error, we see the performance on the target can degrade when the newly added source camera has large divergence from the target camera. § RELATED WORK A number of adaptation approaches have been studied in recent years. From the theoretical aspect, several theoretical results have been derived in the form of upper bounds on the generalization target error by learning from the source data. A keypoint of the theoretical frameworks is estimating the distribution shift between source and target. <cit.> proposed the ℋ-divergence to measure the similarity between two domains and derived a generalization bound on the target domain using empirical error on the source domain and the ℋ-divergence between the source and the target. This idea has later been extended to multisource domain adaptation <cit.> and the corresponding generalization bound has been developed as well. <cit.> provide a generalization bound for domain adaptation on the target risk which generalizes the standard bound on the source risk. This work formalizes a natural intuition of DA: reducing the two distributions while ensuring a low error on the source domain and justifies many DA algorithms. Based on this work, <cit.> introduce a new divergence measure: discrepancy distance, whose empirical estimate is based on the Rademacher complexity <cit.> (rather than the VC-dim). Other theoretical works have also been studied such as <cit.> that derives the generalization bounds on the target error by taking use of the robustness properties introduced in <cit.>. See <cit.> for more details.Following the theoretical developments, many DA algorithms have been proposed, such as instance-based methods <cit.>; feature-based methods <cit.>; and parameter-based methods <cit.>. The general approach for domain adaptation starts from algorithms that focus on linear hypothesis class <cit.>. The linear assumption can be relaxed and extended to the non-linear setting using the kernel trick, leading to a reweighting scheme that can be efficiently solved via quadratic programming <cit.>. Recently, due to the availability of rich data and powerful computational resources, non-linear representations and hypothesis classes have been increasingly explored <cit.>. This line of work focuses on building common and robust feature representations among multiple domains using either supervised neural networks <cit.>, or unsupervised pretraining using denoising auto-encoders <cit.>. Recent studies have shown that deep neural networks can learn more transferable features for DA <cit.>. <cit.> develop domain separation networks to extract image representations that are partitioned into two subspaces: domain private component and cross-domain shared component. The partitioned representation is utilized to reconstruct the images from both domains, improving the DA performance. Reference <cit.> enables classifier adaptation by learning the residual function with reference to the target classifier. The main-task of this work is limited to the classification problem. <cit.> propose a domain-adversarial neural network to learn the domain indiscriminate but main-task discriminative features. Although these works generally outperform non-deep learning based methods, they only focus on the single-source-single-target DA problem, and much work is rather empirical design without statistical guarantees. <cit.> present a domain transform mixture model for multisource DA, which is based on non-deep architectures and is difficult to scale up.Adversarial training techniques that aim to build feature representations that are indistinguishable between source and target domains have been proposed in the last few years <cit.>. Specifically, one of the central ideas is to use neural networks, which are powerful function approximators, to approximate a distance measure known as the ℋ-divergence between two domains <cit.>. The overall algorithm can be viewed as a zero-sum two-player game: one network tries to learn feature representations that can fool the other network, whose goal is to distinguish representations generated from the source domain between those generated from the target domain. The goal of the algorithm is to find a Nash-equilibrium of the game, or the stationary point of the min-max saddle point problem. Ideally, at such equilibrium state, feature representations from the source domain will share the same distributions as those from the target domain, and, as a result, better generalization on the target domain can be expected by training models using only labeled instances from the source domain. § CONCLUSIONWe derive a new generalization bound for DA under the setting of multiple source domains with labeled instances and one target domain with unlabeled instances. The new bound has interesting interpretation and reduces to an existing bound when there is only one source domain. Following our theoretical results, we propose MDANs to learn feature representations that are invariant under multiple domain shifts while at the same time being discriminative for the learning task. Both hard and soft versions of MDANs are generalizations of the popular DANN to the case when multiple source domains are available. Empirically, MDANs outperform the state-of-the-art DA methods on three real-world datasets, including a sentiment analysis task, a digit classification task, and a visual vehicle counting task, demonstrating its effectiveness for multisource domain adaptation.§ OUTLINEOrganization of the appendix: 1). For the convenience of exposition in showing our technical proofs, we first introduce the technical tools that will be used during our proofs in Sec. <ref>. 2). We provide detailed proofs for all the claims, lemmas and theorems presented in the main paper in Sec. <ref>. 3). We describe more experiment details in Sec. <ref>, including dataset description, network architecture and training parameters of the proposed and baseline methods, and more analysis of the experimental results. § TECHNICAL TOOLSThe growth function Π_ℋ:→ for a hypothesis class ℋ is defined by:∀ m∈,Π_ℋ(m) = max_X_m⊆𝒳| {( h(x_1), …, h(x_m))| h∈ℋ}|where X_m = {x_1, …, x_m} is a subset of 𝒳 with size m. Roughly, the growth function Π_ℋ(m) computes the maximum number of distinct ways in which m points can be classified using hypothesis in ℋ. A closely related concept is the Vapnik–Chervonenkis dimension (VC dimension) <cit.>:The VC-dimension of a hypothesis class ℋ is defined as:VCdim(ℋ) = max{m: Π_ℋ(m) = 2^m} A well-known result relating VCdim(ℋ) and the growth function Π_ℋ(m) is the Sauer's lemma:Let ℋ be a hypothesis class with VCdim(ℋ) = d. Then, for m≥ d, the following inequality holds:Π_ℋ(m)≤∑_i=0^dmi≤(em/d)^dThe following concentration inequality will be used:Let X_1, …, X_n be independent random variables where each X_i is bounded by the interval [a_i, b_i]. Define the empirical mean of these random variables by X̅1/n∑_i=1^n X_i, then ∀ > 0:(|X̅ - [X̅]|≥)≤ 2exp(-2n^2^2/∑_i=1^n(b_i - a_i)^2)The VC inequality allows us to give a uniform bound on the binary classification error of a hypothesis class ℋ using growth function:Let Π_ℋ be the growth function of hypothesis class ℋ. For h∈ℋ, let (h) be the true risk of h w.r.t. the generation distribution 𝒟 and the true labeling function h^*. Similarly, let _n(h) be the empirical risk on a random i.i.d. sample containing n instances from 𝒟, then, for ∀ > 0, the following inequality hold:(sup_h∈ℋ|(h) - _n(h)|≥)≤ 8Π_ℋ(n)exp(-n^2/32) Although the above theorem is stated for binary classification error, we can extend it to any bounded error. This will only change the multiplicative constant of the bound. § PROOFS For all the proofs presented here, the following lemma shown by <cit.> will be repeatedly used:∀ h, h'∈ℋ, |_S(h, h') - _T(h, h')|≤1/2d_ℋΔℋ(𝒟_S, 𝒟_T).§.§ Proof of Thm. <ref>One technical lemma we will frequently use to prove Thm. <ref> is the triangular inequality w.r.t. _𝒟(h), ∀ h∈ℋ:For any hypothesis class ℋ and any distribution 𝒟 on 𝒳, the following triangular inequality holds:∀ h,h',f∈ℋ,_𝒟(h, h')≤_𝒟(h, f) + _𝒟(f, h')_𝒟(h, h') = 𝔼_𝐱∼𝒟[|h(𝐱) - h'(𝐱)|] ≤𝔼_𝐱∼𝒟[|h(𝐱) - f(𝐱)| + |f(𝐱) - f(𝐱)|] = _𝒟(h, f) + _𝒟(f, h')Now we are ready to prove Thm. <ref>: *∀ h∈ℋ, define i_h_i∈[k]_S_i(h, h^*):_T(h)≤_T(h^*) + _T(h, h^*) = _T(h^*) + _T(h, h^*) - max_i∈[k]_S_i(h, h^*) + max_i∈[k]_S_i(h, h^*)≤_T(h^*) + |_T(h, h^*) - _S_i_h(h, h^*)| + _S_i_h(h, h^*)≤_T(h^*) + 1/2d_ℋΔℋ(𝒟_T, 𝒟_S_i_h) + _S_i_h(h, h^*)≤_T(h^*) + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) + _S_i_h(h, h^*)≤_T(h^*) + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) + _S_i_h(h) + _S_i_h(h^*)≤_T(h^*) + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k) + max_i∈[k]_S_i(h) + max_i∈[k]_S_i(h^*)= max_i∈[k]_S_i(h) + λ + 1/2d_ℋΔℋ(𝒟_T; {𝒟_S_i}_i=1^k)The first and the fifth inequalities are due to the triangle inequality, and the third inequality is based on Lemma <ref>. The second holds due to the property of |·| and the others follow by the definition of ℋ-divergence. §.§ Proof of Thm. <ref>* =(| d_ℋ(𝒟_T; {𝒟_S_i}_i=1^k) - d_ℋ(𝒟̂_T; {𝒟̂_S_i}_i=1^k)| ≥ϵ) = (|max_i∈[k]sup_A∈𝒜_ℋ|_𝒟_T(A) - _𝒟_S_i(A)| - max_i∈[k]sup_A∈𝒜_ℋ|_𝒟̂_T(A) - _𝒟̂_S_i(A)| |≥ϵ/2)≤(max_i∈[k]sup_A∈𝒜_ℋ||_𝒟_T(A) - _𝒟_S_i(A)| - |_𝒟̂_T(A) - _𝒟̂_S_i(A)| |≥ϵ/2)= (∃ i∈[k], ∃ A∈𝒜_ℋ:||_𝒟_T(A) - _𝒟_S_i(A)| - |_𝒟̂_T(A) - _𝒟̂_S_i(A)| |≥ϵ/2)≤∑_i=1^k (∃ A∈𝒜_ℋ:||_𝒟_T(A) - _𝒟_S_i(A)| - |_𝒟̂_T(A) - _𝒟̂_S_i(A)| |≥ϵ/2)≤∑_i=1^k (∃ A∈𝒜_ℋ: |_𝒟_T(A) - _𝒟̂_T(A) | + |_𝒟_S_i(A) - _𝒟̂_S_i(A)| ≥ϵ/2)≤ 2k (∃ A∈𝒜_ℋ: |_𝒟_T(A) - _𝒟̂_T(A) |≥ϵ/4)≤ 2k·Π_𝒜_ℋ(m)(|_𝒟_T(A) - _𝒟̂_T(A) |≥ϵ/4)≤ 2k·Π_𝒜_ℋ(m)· 2exp(-2mϵ^2 / 16)≤ 4k(em/d)^d exp(-mϵ^2 / 8)The first inequality holds due to the sub-additivity of the max function, and the second inequality is due to the union bound. The third inequality holds because of the triangle inequality, and we use the averaging argument to establish the fourth inequality. The fifth inequality is an application of the VC-inequality, and the sixth is by the Hoeffding's inequality. Finally, we use the Sauer's lemma to prove the last inequality.§.§ Proof of Thm. <ref>We now show the detailed proof of Thm. <ref>. (sup_h∈ℋ| max_i∈[k]_S_i(h) - max_i∈[k]_S_i(h) | ≥ϵ)≤(sup_h∈ℋmax_i∈[k]| _S_i(h) - _S_i(h) | ≥ϵ)= (max_i∈[k]sup_h∈ℋ| _S_i(h) - _S_i(h) | ≥ϵ)≤∑_i=1^k (sup_h∈ℋ| _S_i(h) - _S_i(h) | ≥ϵ)≤ k·Π_ℋ(m)(| _S_i(h) - _S_i(h) | ≥ϵ)≤ k·Π_ℋ(m)· 2exp(-2mϵ^2)≤ 2k(me/d)^dexp(-2mϵ^2)Again, the first inequality is due to the subadditivity of the max function, and the second inequality holds due to the union bound. We apply the VC-inequality to bound the third inequality, and Hoeffding's inequality to bound the fourth. Again, the last one is due to Sauer's lemma.§.§ Derivation of the Discrepancy Distance as Classification ErrorWe show that the ℋ-divergence is equivalent to a binary classification accuracy in discriminating instances from different domains. Suppose 𝒜_ℋ is symmetric, i.e., A∈𝒜_ℋ⇔𝒳\ A∈𝒜_ℋ, and we have samples {S_i}_i=1^k and T from {𝒟_S_i}_i=1^k and 𝒟_T respectively, each of which is of size m, then:d_ℋΔℋ(𝒟̂_T; {𝒟̂_S_i}_i=1^k)= max_i∈[k]sup_A∈𝒜_ℋΔℋ|_𝒟̂_T(A) - _𝒟̂_S_i(A)| = max_i∈[k]sup_h∈ℋΔℋ|_𝐱∼𝒟̂_T(h(𝐱) = 1) - _𝐱∼𝒟̂_S_i(h(𝐱 = 1))| = max_i∈[k]sup_h∈ℋΔℋ 1- (_𝐱∼𝒟̂_T(h(𝐱) = 1) + _𝐱∼𝒟̂_S_i(h(𝐱 = 0))) = max_i∈[k](1 - 2min_h∈ℋΔℋ(1/2m∑_𝐱∼𝒟̂_T𝕀(h(𝐱) = 1) + 1/2m∑_𝐱∼𝒟̂_S_i𝕀(h(𝐱 = 0)))) § DETAILS ABOUT EXPERIMENTSIn this section, we describe more details about the datasets and the experimental settings. We extensively evaluate the proposed methods on three datasets: 1). We first evaluate our methods on Amazon Reviews dataset <cit.> for sentiment analysis. 2). We evaluate the proposed methods on the digits classification datasets including MNIST <cit.>, MNIST-M <cit.>, SVHN <cit.>, and SynthDigits <cit.>. 3). We further evaluate the proposed methods on the public dataset WebCamT <cit.> for vehicle counting. It contains 60,000 labeled images from 12 city cameras with different distributions. Due to the substantial difference between these datasets and their corresponding learning tasks, we will introduce more detailed dataset description, network architecture, and training parameters for each dataset respectively in the following subsections.§.§ Details on Amazon Reviews evaluation Amazon reviews dataset includes four domains, each one composed of reviews on a specific kind of product (Books, DVDs, Electronics, and Kitchen appliances). Reviews are encoded as 5000 dimensional feature vectors of unigrams and bigrams. The labels are binary: 0 if the product is ranked up to 3 stars, and 1 if the product is ranked 4 or 5 stars.We take one product domain as target and the other three as source domains. Each source domain has 2000 labeled examples and the target test set has 3000 to 6000 examples. We implement the Hard-Max and Soft-Max methods according to Alg. <ref>, based on a basic network with one input layer (5000 units) and three hidden layers (1000, 500, 100 units). The network is trained for 50 epochs with dropout rate 0.7. We compare Hard-Max and Soft-Max with three baselines: Baseline 1: MLPNet. It is the basic network of our methods (one input layer and three hidden layers), trained for 50 epochs with dropout rate 0.01. Baseline 2: Marginalized Stacked Denoising Autoencoders (mSDA) <cit.>. It takes the unlabeled parts of both source and target samples to learn a feature map from input space to a new representation space. As a denoising autoencoder algorithm, it finds a feature representation from which one can (approximately) reconstruct the original features of an example from its noisy counterpart. Baseline 3: DANN. We implement DANN based on the algorithm described in <cit.> with the same basic network as our methods. Hyper parameters of the proposed and baseline methods are selected by cross validation. Table <ref> summarizes the network architecture and some hyper parameters. §.§ Details on Digit Datasets evaluation We evaluate the proposed methods on the digits classification problem. Following the experiments in <cit.>, we combine four popular digits datasets-MNIST, MNIST-M, SVHN, and SynthDigits to build the multi-source domain dataset. MNIST is a handwritten digits database with 60,000 training examples, and 10,000 testing examples. The digits have been size-normalized and centered in a 28×28 image. MNIST-M is generated by blending digits from the original MNIST set over patches randomly extracted from color photos from BSDS500 <cit.>. It has 59,001 training images and 9,001 testing images with 32×32 resolution. An output sample is produced by taking a patch from a photo and inverting its pixels at positions corresponding to the pixels of a digit. For DA problems, this domain is quite distinct from MNIST, for the background and the strokes are no longer constant. SVHN is a real-world house number dataset with 73,257 training images and 26,032 testing images. It can be seen as similar to MNIST, but comes from a significantly harder, unsolved, real world problem. SynthDigits consists of 500;000 digit images generated by <cit.> from WindowsTM fonts by varying the text, positioning, orientation, background and stroke colors, and the amount of blur. The degrees of variation were chosen to simulate SVHN, but the two datasets are still rather distinct, with the biggest difference being the structured clutter in the background of SVHN images.We take MNIST-M, SVHN, and MNIST as target domain in turn, and the remaining three as sources. We implement the Hard-Max and Soft-Max versions according to Alg. <ref> based on a basic network, as shown in Fig. <ref>. The baseline methods are also built on the same basic network structure to put them on a equal footing. The network structure and parameters of MDANs are illustrated in Fig. <ref>.The learning rate is initialized by 0.01 and adjusted by the first and second order momentum in the training process. The domain adaptation parameter of MDANs is selected by cross validation. In each mini-batch of MDANs training process, we randomly sample the same number of unlabeled target images as the number of the source images.§.§ Details on WebCamT Vehicle Counting WebCamT is a public dataset for large-scale city camera videos, which have low resolution (352×240), low frame rate (1 frame/second), and high occlusion. WebCamT has 60,000 frames annotated with rich information: bounding box, vehicle type, vehicle orientation, vehicle count, vehicle re-identification, and weather condition. The dataset is divided into training and testing sets, with 42,200 and 17,800 frames, respectively, covering multiple cameras and different weather conditions. WebCamT is an appropriate dataset to evaluate domain adaptation methods, for it covers multiple city cameras and each camera is located in different intersection of the city with different perspectives and scenes. Thus, each camera data has different distribution from others. The dataset is quite challenging and in high demand of domain adaptation solutions, as it has 6,000,000 unlabeled images from 200 cameras with only 60,000 labeled images from 12 cameras. The experiments on WebCamT provide an interesting application of our proposed MDANs: when dealing with spatially and temporally large-scale dataset with much variations, it is prohibitively expensive and time-consuming to label large amount of instances covering all the variations. As a result, only a limited portion of the dataset can be annotated, which can not cover all the data domains in the dataset.MDAN provide an effective solution for this kind of application by adapting the deep model from multiple source domains to the unlabeled target domain. We evaluate the proposed methods on different numbers of source cameras. Each source camera provides 2000 labeled images for training and the test set has 2000 images from the target camera. In each mini-batch, we randomly sample the same number of unlabeled target images as the source images. We implement the Hard-Max and Soft-Max version of MDANs according to Alg. <ref>, based on the basic vehicle counting network FCN described in <cit.>. Please refer to <cit.> for detailed network architecture and parameters. The learning rate is initialized by 0.01 and adjusted by the first and second order momentum in the training process. The domain adaptation parameter is selected by cross validation. We compare our method with two baselines: Baseline 1: FCN. It is our basic network without domain adaptation as introduced in work <cit.>. Baseline 2: DANN. We implement DANN on top of the same basic network following the algorithm introduced in work <cit.>. | http://arxiv.org/abs/1705.09684v2 | {
"authors": [
"Han Zhao",
"Shanghang Zhang",
"Guanhang Wu",
"João P. Costeira",
"José M. F. Moura",
"Geoffrey J. Gordon"
],
"categories": [
"cs.LG",
"cs.AI",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170526191056",
"title": "Multiple Source Domain Adaptation with Adversarial Training of Neural Networks"
} |
Sezione INFN di Roma “Tor Vergata”, Roma, Italy [email protected] High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, USA [email protected] Graduate School of Science, Osaka City University, Osaka, Japan [email protected] Ultra-High Energy Cosmic Rays (UHECRs) are charged particles of energies above 10^18 eV that originate outside of the Galaxy. Because the flux of the UHECRs at Earth is very small, the only practical way of observing UHECRs is by measuring the extensive air showers (EAS) produced by UHECRs in the atmosphere. This is done by using air fluorescence detectors and giant arrays of particle detectors on the ground.The Pierre Auger Observatory (Auger) and Telescope Array (TA) are two large cosmic ray experiments which use such techniques and cover 3000 km^2 and 700 km^2 areas on the ground, respectively.In this paper, we present the UHECR spectrum reported by the TA, using an exposure of 6300 km^2 sr yr accumulated over 7 years of data taking, and the corresponding result of Auger, using 10 years of data with a total exposure exceeding 50000 km^2 sr yr. We review the astrophysical interpretation of the two measurements, and discuss their systematic uncertainties.Measurement of Energy Spectrum of Ultra-High Energy Cosmic Rays Yoshiki Tsunesada December 30, 2023 =============================================================== § INTRODUCTIONThe origin of cosmic rays (CRs) is an important problem in modern astrophysics. Extraterrestrial particles of energies greater than 10^18 and even exceeding 10^20 eV are measured at Earth, while their arrival directions seem to be distributed almost randomly.Since the energies of these CRs extend well beyond the energy range of the solar particles and energies attainable by the artificial accelerators, the mechanisms of the CR production and acceleration must be related to unknown, very energetic phenomena in the universe.The energy spectrum plays a key role in understanding the CRs <cit.>.The spectrum falls off with energy approximately as a power-law function, with an average power index of γ∼ 3, and the CR flux becomes very small above 10^14 eV.Above this energy, it is only practical to detect the CRs indirectly, by measuring secondary particles, or the extensive air showers (EAS), which are produced by the primary CR particles in the atmosphere.The ultra-high energy CR spectrum has been measured by a number of cosmic rayexperiments <cit.>, and is known to have 5 features over 10 orders of magnitude in energy. There is a steepening at ∼ 3 × 10^15 eV, known as the knee in the spectrum, a flattening at ∼ 10^16 eV, and a steepening at ∼ 4 × 10^17 eV, the so-called the second knee.The flattening at ∼ 10^16 eV is called the low-energy ankle, a feature analogous to the widely recognized ankle at 5 × 10^18 eV.At the highest energies, ∼ 5 × 10^19 eV, there is an abrupt suppression of the cosmic ray flux to the level of ∼ 1 particle/km^2/century. For a historical overview of observational studies of CRs and air showers, see e.g. <cit.>.Although there are indirect evidences from the detection of very high energy gamma-rays from the supernova remnants that suggest the CRs of energies below the knee originate within the Galaxy, the interpretation of CR energy spectrum features remains uncertain.The knee and the second knee are believed to be caused by the maximum acceleration energy available at the Galactic sources, and by the maximum energies of the magnetic confinement of protons and high-Z nuclei in the Galaxy. The gyro-radii of CRs with energies beyond the second knee in the galactic magnetic field become larger than the size of the Galaxy, and therefore the magnetic confinement of CRs in the Galaxy is no longer effective. Consequently, the CRs of energies above 10^18 eV, the so-called Ultra-High Energy Cosmic Rays (UHECRs), must be of extra-galactic origin.The UHECRs from distant sources travel tens or hundreds of Mpc before reaching the Earth, and therefore, interactions and propagation effects, and the chemical composition must be taken into account in the interpretation of the CR energy spectrum. Shortly after the discovery of the cosmic microwave background (CMB) radiation,Greisen <cit.>, Zatsepin, and Kuz'min <cit.> in 1966 independently predicted a suppression of cosmic ray flux at the highest energies, as a consequence of photo-pion production from the interaction of the CRs with the low energy CMB photons.This feature is now called the GZK cutoff.In the case of a pure proton composition, the ankle can be explained by the electron-positron pair-production frominteractions of the CR protons with the CMB photons <cit.>.In the case of a mixed composition, on the other hand, propagation effects are complicated by the fact that the primary nuclei also suffer interactions that cause a progressive reduction of their mass numbers. Other alternative models assert that a cut-off on the acceleration mechanism at the sources may play some role in the explanation of the observed suppression of the cosmic ray flux <cit.>. Historically, observation of the cut-off in the energy spectrum was a technically challenging task. Because the rate of CRs of energies greater than 10^20 eV is as low as 1 event per square kilometer per century, experiments with very large effective areas, long observation periods, and good energy resolution were required to see the effect.AGASA (Akeno Giant Air Shower Array) <cit.> and Hi-Resolution Fly's Eye (HiRes) <cit.> were the first cosmic ray detectors large enough to measure the energy spectrum of UHECRs above 10^19.0 eV, as shown in Fig. <ref>. There were two major differences in their results. First, there was a difference in the overall energy scale, which came from the difference in the techniques employed by the two experiments. AGASA used an array of scintillation counters that were detecting EAS particles at the ground level, while HiRes employed fluorescence detectors that were sensitive to fluorescence light emitted due to the energy deposition of the EAS particles in the atmosphere.The systematic uncertainties in determining the CR primary energy were ∼ 20 % in both experiments.The second important difference was in the shape of the spectrum above the ankle. The HiRes spectrum showed a steepening in the spectrum at 6 × 10^19 eV <cit.> as predicted by the GZK theory <cit.>, whereas the AGASA spectrum extended well beyond the cut-off energy <cit.>.The tension between the two major experiments in the 1990's led to an idea of the hybrid detection of UHECRs, where both surface detectors and fluorescence detectors can be used within a single experiment.The Telescope Array (TA) <cit.> and Pierre Auger Observatory (Auger) <cit.> are modern hybrid cosmic ray experiments.This paper describes recent measurements of the UHECR spectrum by the Telescope Array (TA) experiment <cit.> and the Pierre Auger Observatory (Auger) <cit.>.The TA is a cosmic ray observatory that covers an area of about 700 km^2 in the northern hemisphere, and Auger has an effective area of 3000 km^2 in the southern hemisphere.Both experiments use two types of instruments, surface detectors (SDs) and fluorescence detectors (FDs).The hybrid detection technique, where the CR showers aresimultaneously observed with the FDs and SDs at the same site,allows a very precise determination of the CR energies and arrival directions.The FDs measure fluorescence light emitted by the atmospheric molecules excited by the charged particles in the EAS, and observe the longitudinal development of the EAS using mirror telescopes coupled with clusters of photo-multiplier tubes.Because the FDs are mostly sensitive to the calorimetric energy deposition in the atmosphere, the energy determination of the primary CRs is nearly independent of thedetails of the hadronic interactions within the EAS, where there are considerable uncertainties in different models. The FDs operate at a ∼ 10% duty cycle because the FD data can be collected only during nights with low moonlight background and with dry air and clear skies. The SDs, on the other hand, directly measure EAS particles at the ground level at a nearly 100% duty cycle, regardless of the weather conditions. This paper is organized as follows.The TA and Auger detectors are described in Sec. <ref>. The details of the energy scale in the two experiments are summarized and discussed in Sec. <ref>.The latest results of the energy spectrum measurements and their astrophysical interpretations are given in Sec. <ref> and <ref>. Sec. <ref> is devoted to the discussion on the current status and future prospects of the UHECR field, as well as the existing plans of extension and upgrade of the detectors foreseen by the TA and Auger collaborations.§ TELESCOPE ARRAY AND AUGER INSTRUMENTS §.§ TA DetectorsThe Telescope Array experiment <cit.> is located in Millard County, Utah (USA) at a 39.3^∘ N latitude and ∼ 1400 m altitude above sea level.The TA detectors have been in full operation since May 2008.A sketch of the TA site is shown in the left panel of Fig. <ref>.The TA SD consists of 507 particle counters arranged on a 1.2 km spaced square grid and covers an area of ∼700 km^2 on the ground.Each surface detector unit, shown on the left panel of Fig. <ref>, consists of two layers of 3 m^2, 1.2 cm thick plastic scintillators <cit.>.Scintillation light in each layer is collected and directed to the photo-multiplier tube (PMT) by the wavelength-shifting fibers. There is one PMT for each layer. Outputs of the PMTs of the upper and the lower layers are individually digitized by 12 bit flash analog-to-digital converters (FADCs) at a 50 MHz sampling rate. The TA includes three fluorescence detector stations thatoverlook the surface array. Middle Drum (MD) FD is located in the northern part of the TA, and Black Rock Mesa (BR) and Long Ridge (LR) FDs are in the southern part.The MD station utilizes 14 refurbished telescopes previously used in the High-Resolution Fly's Eye (HiRes) experiment <cit.>. Each telescope consists of a ∼ 5 m^2 spherical mirror and an imagingcamera of 256 PMTs that uses a sample-and-hold readout system. The telescopes are logically arranged in two layers, called rings, that observe two different elevations. Physically, the MD telescopes are arranged in pairs: ring 2 telescopes that observe higher elevations are placed next to the ring 1 telescopes. The station covers 110^∘ in azimuth and 3^∘ to 31^∘ in elevation. Black Rock Mesa and Long Ridge stations have each 12 fluorescence telescopes that are also arranged in two rings.The telescopes use a new design shown in the right panel of Fig. <ref>. Each mirror of the BR and LR telescopes is composed of 18 hexagonal segments. The radius of curvature of each segment is 6067 mm, and the total effective area of the mirror is 6.8 m^2. The imaging camera of a BR-LR telescope consists of 256 PMTs that are read out by a 40 MHz FADC system. Each station covers 108^∘ in azimuth and 3^∘ to 33^∘ in elevation <cit.>.The calibration of the TA FDs was carried out by measuring the absolute gains of dozens of “standard” PMTs that are installed in each camera using the CRAYS (Calibration using Rayleigh Scattering) system in the laboratory <cit.>.The rest of the PMTs in the cameras are relatively calibrated to the standard PMTs by using Xe lamps installed at the center of each mirror <cit.>.§.§ Auger Detectors The Pierre Auger Observatory <cit.> is located in a region called Pampa Amarilla, near the small town of Malargüe in the province of Mendoza (Argentina), at ∼35^∘ S latitude and an ataltitude of 1400 m above sea level.The observatory is in operation since 2004 and its construction was completedin 2008. A sketch of the site is shown in the right panel of Fig. <ref>.The Auger SD consists of 1660 water-Cherenkov detectors (WCDs) arranged on a hexagonal grid of 1.5 km spacing. The effective area of the array is ∼3000 km^2. Each WCD unit, shown in the left panel of Fig. <ref>, is a plastic tank of a cylindrical shape, 10 m^2 × 1.2 m, filled with purifiedwater.Cherenkov radiation, produced by the passage of charged particles through the water, is detected by the three PMTs, 9” in diameter each. The signal of the PMTs is digitized by an FADCs at a 40 MHz sampling rate.Because WCDs extend 1.2 m in the vertical direction, the Auger SD is sensitive to the cosmic ray showers that are developing at large zenith angles.The Auger FD consists of 24 telescopes placed in four buildings located along the perimeter of the site. A sketch of a telescope is shown in the right panel of Fig.<ref>.Each telescope has a 3.5 m × 3.5 m spherical mirror with a curvature radius of 3.4 m.The coma aberration is eliminated using a Schmidtoptics device, which consists of a circular diaphragm of radius 1.10 m and aseries of corrector elements mounted in the outer part of the aperture.An ultraviolet transmitting filter is placed at the telescope entrance in order to reduce the background light and to provide the protection from the outside dust.The focal surface is covered by 440 PMTs, 22 rows x 20 columns, and the overall field of view of the telescope is 30^∘ in elevation and 28.6^∘ in azimuth. The PMTs use photocathodes of an hexagonal shape and are surrounded by light concentrators in order to maximize the light collection and to guarantee a smooth transition between the adjacent pixels.The signal from each PMT is digitized by a 10 MHz FADC with a 12 bit resolution. The Auger FDs are calibrated using a portable cylindrical diffuser, called the drum <cit.>. During the calibration process, the drum is mounted to the aperture of each telescope, and provides a uniform illumination of the entire surface area that is covered by the PMTs of the telescope.The drum is absolutely calibrated using a NIST-calibrated photodiode, and provides an absolute end-to-end calibration of all pixels and optical elements of every Auger FD telescope.The long-term time variations in the calibration of the telescopes are monitored using LED light sources that are installed in each building. §.§ Auxiliary Facilities at the TA and AugerIn order to reconstruct the shower energy from the FD information accurately, it is necessary to know the attenuation of the light due to the molecular and aerosol scattering as the light propagates from the shower to the detector.The molecular scattering can be calculated from the knowledge of the air density as a function of height, and the aerosol content of the atmosphere is monitored each night during the FD data collection.The aerosols are measured in the TA and Auger using similar instruments.These include central laser facilities (CLFs) placed in the middle of the arrays, andstandard LIDAR (LIght Detection and Ranging) stations <cit.>. Auger CLF has recently been upgraded to include a backscatter Raman LIDAR receiver. Other instruments like infra-red cameras are also employed in both experiments to continuously monitor the cloud coverage.Both the TA and Auger collaborations have enhanced the baseline configurations of their detectors to lower the minimum detectable shower energies.The TA low energy extension (TALE) consists of 10 additional fluorescence telescopes viewing higher elevation angles <cit.>, from 32^∘ to 59^∘, installed at the MD site, and an infill array of the same scintillation detectors as those used by the main TA SD array. In a similar way, Auger has installed three additional High Elevation Auger Telescopes (HEAT), viewing from 30^∘ to 60^∘ in elevation, at the Coihueco site (see Fig. <ref>). HEAT overlooks a 27 km^2 region on the ground that is filled with additional WCDs using 750 m spacing <cit.>.An important calibration facility, called the Electron Light Source (ELS) <cit.>, has been implemented by the TA collaboration. The ELS is a linear accelerator installed in front of the TA Black Rock Mesa FD station at a distance of 100 m from the detector.The ELS provides a pulsed beam of 40 MeV electrons that are injected into the FD field of view.The pulse frequency is 1 Hz and each pulse has a duration of 1 μ s and an intensity of about 10^9 electrons.The ELS beam mimics the cosmic ray air showers and provides an effective test not only for the FD calibration but also for the other kinds of detectors, such as radio antennas. § TA AND AUGER ENERGY SCALEBoth TA and Auger experiments use FD measurements to set their energy scale. The FD measures fluorescence photons produced by de-excitation of the atmospheric molecules (nitrogen and oxygen) that have been excited by the charged particles in the EAS, and provides a nearly calorimetric estimate of the total EAS energy.The fluorescence photons are emitted isotropically in the wavelength range between 290 and 430 nm.The most significant line emission at 337 nm contributes ∼ 25% of the total emission intensity.The number of emitted photons is proportional to the energy deposited by the charged particles in the EAS.The proportionality factor, called the fluorescence yield (FY), is measured by several experiments using accelerator beam and radioactive sources (see <cit.> for a review on this topic).As the EAS develops in the atmosphere, fluorescence light emitted at different altitudes triggers the FD pixels (PMTs) at different times. Pointing directions of the triggered pixels and the pixel time information are used to reconstruct the full geometry of the cosmic ray shower event, which includes the event arrival direction and the impact parameter (distance between the FD station and the EAS axis). Additionally, information from the triggered surface detector stations on the ground can be added to constrain the FD timing fit and improve the resolution of the EAS geometry.Events that are reconstructed using FD and SD information simultaneously are called the hybrid events.Examples of reconstructed FD longitudinal profiles are shown in Fig. <ref>.The energy deposition (dE/dX) is determined as a function of the slant depth X along the shower axis using the intensity of the signal of the triggered pixels. The dE/dX reconstruction procedure requires the absolute calibration of the FD telescopes, knowledge of the attenuation of the light due to the scattering by the air molecules and the aerosols, and the absolute fluorescence yield.The integral of the dE/dX profile gives the calorimetric energy of the shower:E_ cal = ∫(dE/dX) dX. The total (and final) energy of the primary cosmic ray is obtained from E_ cal after the addition of the so called invisible (or equivalently missing) energy E_ inv, which is the energy that is carried away by the high-energy muons and neutrinos that do not deposit their energies in the atmosphere and thus cannot be seen by the FD.For a typical EAS, E_ inv is of the order of 10 to 15% of the total primary energy. Further details on the energy scale and invisible energy corrections in TA and Auger will be discussed in Sec. <ref> and <ref>.§.§ Surface Detector Energy ReconstructionThe SD energy scale in both TA and Auger is calibrated by the FDs using well reconstructed hybrid events.This is done by comparing the SD energy estimators with the energies obtained from the corresponding FD longitudinal profiles on an event by event basis.The SD energy estimators in TA and Auger are obtained using conceptually similar analyses <cit.>.The energy of the primary CR particle, arriving at a given fixed zenith angle θ, is assumed to be directly related to the intensity of the shower front at a certain distance from the shower core.This relation depends on θ because the effective amount of the air material that the EAS propagates through, before reaching the ground level, increases as 1/ cos(θ).The shower axis and the point of impact on the ground are determined from the timing and the intensity of the signal in the triggered SD stations.The best energy estimator is obtained by evaluating the intensity of the shower front at an optimum distance r_ opt from the shower core. This is done using analytic functions with parameters determined from the fit to the intensity of the signal as a function of the distance from the shower core (see Fig. <ref>).The optimal energy determination distance for the TA SD with 1200 m spacing is r_ opt = 800 m.For Auger, whose spacing is 1500 m, r_ opt is 1000 m.A similar reconstruction technique is used for the Auger events detected by the 750 marray placed in front of the HEAT telescopes (see Sec. <ref>). In thiscase r_ opt is 450 m. Next, the intensity of the signal at the optimal distance r_opt is corrected for the zenith angle attenuation. In TA, this correction is made using a detailed Monte Carlo simulation <cit.>, and an energy estimator E_ SDMC is obtained. E_ SDMC represents the reconstructed TA SD energy prior to the calibration of the energy scale by the FD. Standard TA SD reconstruction uses events with θ < 45^∘.In the case of Auger, the zenith angle attenuation is derived from the data using the “Constant Integral Intensity Cut” method <cit.>. The resulting Auger energy estimators are called S_38 for the 1500 m array and S_35 for the 750 m and array, and represent the signal values the EAS would have produced if it arrived at the zenith angles of 38^∘ and 35^∘, respectively.These numbers correspond to the median values of the event zenith angle distributions of the Auger 1500 m and 750 m arrays.In Auger, the reconstruction technique described above is applied to the showers of θ < 60^∘ for the 1500 m array and θ < 55^∘ for the 750 m array.A different reconstruction technique is used for the Auger 1500 m array in the case of inclined showers (θ > 60^∘) <cit.>. In these showers, the electromagnetic component is largely absorbed by the atmosphere and the signal in the WCDs is dominated by muons. The muon patterns (maps) are asymmetric because of the deflections of the muons in the magnetic field of the Earth. These maps are calculated for different zenith and azimuth angles using Monte Carlo simulations. The normalization of the maps, called N_19, is fitted to the data and provides an energy estimator for the inclined showers.Correlations between the SD energy estimators and the FD energies are shown in Fig. <ref>.In TA, the final event energy is determined by scaling the result from the Monte Carlo simulations E_ SDMC by a factor <E_ SDMC/E_ FD> = 1.27, to match the FD energy scale <cit.>.The same factor is used for all energies in TA. In Auger, the correlations between the SD energy estimators and the FD energies are well described by a power law function E_FD = A S^B, where S is S_38, S_35, or N_19, depending on the type of the Auger SD analysis. The parameters A and B are obtained from the fits to the data <cit.>.Quantities relevant for the TA and Auger SD energy calibration are summarized in Tables <ref> and <ref>. The TA and Auger SD energy resolution is summarized in the last rows of the tables.§.§ Systematic Uncertainties of the Energy ScaleSince both experiments calibrate their surface detectors to the FDs, the systematic uncertainties of their energy scales reduce to those of the FDs.Therefore, an effort has been made in both TA and Auger collaborations to understand the uncertainties that affect the reconstruction of fluorescence detector events <cit.>. Table <ref> shows a summary of the TA and Auger FD systematic uncertainties in terms of five major contributions: fluorescence yield, atmospheric modeling, FD calibration, determination of the longitudinal profile of the shower, and the invisible energy correction.The fluorescence yield model used by the TA collaboration is based on a combination of the measurements of the absolute yield by Kakimoto et al. <cit.>, in the 300 to 400 nm range, and the fluorescence spectrum measured by FLASH <cit.>.Temperature and pressure dependencies of the absolute FY in TA are taken into account by the Kakimoto et al. <cit.> model also.Auger uses all FY measurements performed by the Airfly experiment. Airfly results include a precise measurement of the absolute intensity at the 337 nm emission band <cit.>, with an uncertainty of 4%, the wavelength spectrum <cit.>, and the dependence on pressure <cit.>, temperature, and humidity <cit.> of the emission bands at different wavelengths.Contributions of the FY models to the systematic uncertainty on the energy scale are 11% for TA and 3.6% for Auger. In both cases, the FY model contributions are dominated by the systematic uncertainties on the absolute FY.In TA, the aerosol transmission is estimated using a median value of the aerosol optical depth profiles measured by the LIDAR <cit.>.The uncertainty on the shower energy determination, obtained by propagating the standard deviation of the LIDAR measurements, is under 10%. The Auger collaboration uses hourly estimates of the aerosol profile provided by the laser facilities placed in the middle of the SD array <cit.>. Uncertainties of these measurements contribute less than 6% to the reconstructed shower energy.A minor contribution to the systematic uncertainty arises from an imprecise knowledge of the atmospheric density profiles.Both TA and Auger use the Global Data Assimilation System (GDAS) that provides atmospheric data in 1^∘× 1^∘ grid points in longitude and latitude (∼ 110 km× 110 km) all over the world, with a time resolution of 3 h <cit.>. A detailed discussion on the implementation of the GDAS atmospheric profiles in the Auger FD event reconstruction can be found in <cit.>. The uncertainties due to the calibration of the FD telescopes contribute ∼10% for both TA and Auger. They are dominated by the uncertainties on the absolute calibration described in Sec. <ref>. The uncertainties from the relative calibration systems, which allow one to track the short and long term changes of the detector response, are taken into account in both experiments, and are small in comparison to those due to the absolute calibration.The uncertainties arising from the reconstruction of the longitudinal shower profiles are obtained by comparing different reconstruction techniques, as well as from studying the energy reconstruction biases with Monte Carlo simulations.Contributions due to the shower profile reconstruction are ∼9% for TA and ∼6% for Auger.The last important contribution is the uncertainty due to the determination of the invisible energy E_ inv. The TA collaboration mainly estimates E_ inv from Monte Carlo simulations of the proton air showers, using the QGSJetII-03 hadronic interaction model. For TA, the contribution to the systematic uncertainty on E due to the missing energy correction is estimated to be 5%.The Auger collaboration derives the invisible energy correction using data <cit.>. This is done by exploiting the WCDs sensitivity to the muons of the showers.The muons are mostly originating from the pion decays, with an associated muon neutrino (or muon antineutrino), and therefore, the signal in the WCDs is sensitive to the muon size of the shower, and it is well correlated with the E_ inv. This analysis allows to keep the uncertainty from the invisible energy estimate on the Auger energy scale well under 3%.Recently, the TA collaboration has also performed a check of the missing energy calculation by using inclined showers of the data, following this method <cit.>.The total uncertainty on the energy scale is obtained by adding in quadrature all individual contributions.It is found to be 21% for TA, and 14% for Auger.In addition, for Auger, the total uncertainty includes a further contribution of 5%, which has been evaluated by studying the stability of the energy scale in different time periods and/or under different conditions. §.§ Energy Scale Comparison between the TA and AugerUnderstanding all contributions to the difference in the energy scale between the two experiments is a difficult task, since many factors are related to the performance of the detectors and to the differences of the analyses techniques used by the two collaborations. On the other hand, two important contributions, the fluorescence yield and the invisible energy, can be considered as external parameters of the experiments, in the sense that they are related to the general properties of the atmospheric showers and thus they can be easily implemented in the CR event reconstruction chains in both collaborations. Therefore, the difference in the energy assignment can be addressed by studying the differences in the fluorescence yield (FY) model and in the invisible energy corrections.The impact of the FY on the reconstruction of the fluorescence events has been studied in detail since many years <cit.>. In the left panel of Figure <ref>, we report the results of the studies performed in <cit.>. Red points describe the effect (on Auger shower energies) of changing the fluorescence yield model from the FY model used by Auger to the FY model used by TA. The energy shift is ∼12% at 1 EeV and is slightly smaller at the highest energies.This energy shift is the result of the combined effect to change the absolute intensity of the fluorescence yield and all parameters describing the relative intensities of the spectral lines and their dependence on the atmospheric conditions. The effects of the single components can be disentangled by the following argument. The absolute FY from Kakimoto et al. <cit.>, when normalized to the intensity of the 337 nm line, where the Airfly experiment made a precise measurement of the absolute FY, differs from that of the Airfly measurement <cit.> by ∼20% (Airfly FY is higher) <cit.>. If the absolute FY from Kakimoto et al. <cit.> was used in the reconstruction of the Auger events, while retaining all other parameters of the Airfly model, one would expect the Auger energies to increase by ∼20%. From this, we conclude that the effects of the FY parameters, other than the absolute FY, are of the order of - 10%. About half of this effect is due to the removal of the temperature and humidity dependence of the quenching cross sections (see also <cit.>), effects that are properly accounted for in Auger experiment. We note that the 20% difference between the Kakimoto et al. and Airfly absolute FYs is outside of the range defined by the uncertainties stated by the two measurements, 10% <cit.> and 3.9% <cit.>, respectively.The right panel of the Figure <ref> describes the effect of changing the fluorescence yield model in the reconstruction of the fluorescence detector events seen by TA <cit.>. If TA were to use the FY model of Auger, the TA energy scale would be reduced by ∼ 14%. The inverse of this energy shift is directly comparable with the energy shift that is expected in the case of Auger using the TA FY, as shown in the left panel of the Figure <ref> using black points.It is not surprising that the Δ E/E results of the TA and Auger (black and red points in figure <ref> on the left) are different. For each experiment, the spectrum of the fluorescence photons detected by the FD is necessarily different from the one emitted at the axis of the cosmic ray shower: the fluorescence photon spectrum is folded with the FD spectral response, and the atmospheric transmission also dependents on the wavelength. Since the Auger and TA FD spectral responses and atmospheric transmission conditions are generally different, we expect larger differences for the higher energy showers that are occurring farther away from the telescopes. A better agreement between the energy shifts can be obtained by correcting the Auger energy shift for the effects due to the different spectral response. The results of this analysis are shown in the left panel of Figure <ref> <cit.> with blue dots, which are now in a better agreement with the TA energy shift (black points).Following the above studies we conclude that, despite the above mentioned inconsistency between the Airfly <cit.> and Kakimoto et al. <cit.> absolute FYs, the difference in the energy scales of TA and Auger due to the use of a different FY model are at the level of 10 -15% and are roughly consistent with the estimated uncertainties presented in Sec. <ref>. The validity of the estimations of the uncertainties on the FY has been also addressed by the ELS facility at the TA experiment. Preliminary results of several ELS runs, under different atmospheric conditions, have been presented in <cit.>.The ELS results are in a better agreement with the Airfly FY model. The invisible energy (E_inv) corrections implemented by the TA and Auger are shown in Figure <ref> <cit.>. They are presented in terms of the percent contribution to the total shower energy E.At 10^19 eV, the TA invisible energy correction is 7%, while that of the Auger is 13%.The difference between the two corrections is about 6% (slightly smaller at higher energies).The two corrections agree within the systematic uncertainties quoted by the two collaborations that are shown using dashed bands in Fig. <ref>.As already addressed in Sec. <ref> the invisible energy of TA has been estimated using Monte Carlo simulations of proton primaries with the QGSJetII-03 hadronic interaction model.For a heavier composition, the invisible energy correction would be larger.The assumption of the proton primaries is consistent with the light composition observed by TA through the measurement of the mean value of the maximum of the shower development (⟨ X_max⟩) <cit.>. It is worthwhile noting that the inference on mass composition strongly depends on the hadronic interaction models used to interpret ⟨ X_max⟩ <cit.>.The Auger measurements of ⟨ X_max⟩ <cit.> are consistent with those of the TA <cit.>, but they generally support a heavier mass composition.The Auger invisible energy correction has the advantage to be essentially insensitive to the hadronic interaction models since it is derived from the data.It has rather high values, even higher than the one predicted by the simulations for iron primaries.These higher values are due to the excess of muons measured by Auger in highly inclined events <cit.>. We can conclude this section by estimating the energy shifts of the Auger and TA energy scales by changing both the FY and the invisible energy. As a first approximation they can be obtained by combining the two energy shifts previously presented.The energies of the TA events would be decreasedby about 9% ((1-14%) × (1+6%) = -0.91) while the energies of Auger would be increased by about +5% ((1+12%) × (1-6%) = 1.05). § TA AND AUGER ENERGY SPECTRUMEnergy spectrum is obtained by dividing the energy distribution of cosmic rays by the accumulated exposure of the detector.The calculation of the exposure for the surface detector is generally robust, especially above the energy threshold where the array is fully efficient regardless of the event arrival direction.For the fluorescence detector, on the other hand, the calculation of the exposure should take into account the detector response as a function of energy and distance between the shower and the telescope, conditions of the data collection, and the state of the atmosphere. Large exposures accumulated by the surface detectors of Auger and TA experiments make it possible to study the UHECR flux at very high energies in different declination bands, and the measurements can be used to constrain the astrophysical models.§.§ Energy Spectrum Measurement§.§.§ TA Data The TA collaboration has measured four independent energy spectra <cit.>. The highest energies are covered by the SD, the intermediate energies are covered by the BR and LR telescopes <cit.>, and the lowest energies are measured by the TALE telescopes using Cherenkov light. The TALE events have been divided into two categories, one in which the fluorescence light is dominating the flux of photons detected by the telescopes (TALE Bridge) <cit.>, and another one where the Cherenkov light is the dominant component (TALE Cherenkov) <cit.>.The exposures for the four different reconstruction methods are shown in the left panel of Fig. <ref> and the energy spectra are shown in the left panel of Fig. <ref>. The spectrum obtained by combining the four measurements is presented in the right panel of Fig. <ref>. The TA spectrum, including the TA low energy extension, covers over 4.7 orders of magnitude in energy, starting at 4 × 10^15 eV, just above the knee. The analysis of the TALE data has allowed to observe the low energy ankle at ∼ 2 × 10^16 eV and the second knee at ∼ 2 × 10^17 eV. The ankle and the cut-off of the UHECR spectrum are confirmed with the improved statistics by the BR and LR FD and by the SD.At the very high energies the combined spectrum is dominated by the SD measurements. The TA SD is fully efficient above 8 × 10^18 eV and its energy scale is fixed by the FD as described in Sec. <ref>.The TA SD exposure accumulated over 7 yearsof data taking is ∼6300 km^2 sr yr. This is estimated using a detailed Monte Carlo simulation that takes into account the detector effects and includes the unfolding corrections that have to be applied to the observed event energy distribution to take into account the bin-to-bin migrations due to the finite resolution of the detector <cit.>.Due to the steepness of the spectrum, the effects of the resolution would otherwise be causing a positive bias in the observed flux, since the upward fluctuations of the energies are not fully compensated by downward fluctuations. It is customary to characterize the shape of the spectra using suitable functional forms.As seen in Fig. <ref>,the TA collaboration uses power laws with break points that correspond to the energies at which the spectral indexes change their values. Above ∼ 3 × 10^17 eV the function is expressed as J(E)∝E^-γ_1E < E_ ankleJ(E)∝E^-γ_2E_ ankle < E < E_ breakJ(E)∝E^-γ_3E > E_ breakand the values of the fitted parameters are shown in Table <ref>. §.§.§ Auger Data Auger collaboration has measured the energy spectrum using four different techniques. The first two measurements cover the highest energies.The measurements consist of two data sets of vertical and inclined events seen by Auger surface detector array of 1500 m spacing.Intermediate energies are covered by a set of hybrid events seen by the Auger FD and SD.Auger 750 m spacing array covers the low energies down to 3 × 10^17 eV. The energy calibration of these showers is done using the fluorescence detector, as explained in Sec. <ref>.The four energy spectra are shown in the left panel of Fig. <ref> and the corresponding exposures are shown in the right panel of Fig. <ref>. The energy spectrum obtained by combining all four measurements is presented in the right panel of Fig. <ref>.Large size of Auger water tanks as well as the overall surface area coverage are the key factors that enabled the Auger collaboration to perform a high precision measurement of the UHECR energy spectrum with relatively high statistics.All four Auger spectra overlap in the region of the ankle. The cut-off is precisely measured by the 1500 m array with an exposure of 42500 km^2 sr yr for the vertical and 10900 km^2 sr yr for the inclined showers. The data covers a period of about 10 years. The SD exposure is a purely geometrical quantity, which is based on the calculation of the number of active elemental hexagon cells of the array as a function of time, with an uncertainty of better than 3% <cit.>. As can be seen in Fig.<ref>, Auger collaboration characterizes the energy spectrum using a functional form that is different from that used by the TA.The function used by Auger consists of a power law below the ankle and a power law with a smooth suppression at the highest energiesJ(E) = J_0 (E/E_ ankle)^-γ_1 E < E_ ankleJ(E) = J_0 (E/E_ ankle)^-γ_2[ 1 +( E_ ankle/E_s)^Δγ] [ 1 + ( E/E_s)^Δγ]^-1E > E_ ankleHere γ_1 and γ_2 are the spectral indexes below and above E_ ankle, respectively, and therefore they have the same meaning as the corresponding TA parameters.E_s is, with a good approximation, the energy at which the spectrum drops to a half of what would be expected in the absence of the cutoff, and Δγ is the increment of the spectral index beyond the suppression region. J_0 is the overall normalization factor, that is conventionally chosen to be the value of the flux at E = E_ ankle.The values of the parameters are shown in Table <ref>.Auger SD spectrum is corrected for the effects of the detector resolution using a forward-folding approach.First, a Monte-Carlo simulation of the detector is used to calculate the resolution bin-to-bin migration matrix.Next, the measured Auger spectrum (before it has been corrected for the effects of the resolution) is fitted to the convolution of the functional form (described above) and the bin-to-bin migration matrix.Once the best fit parameters (Table <ref>) are obtained, the resolution correction factor is calculated by dividing the fitted spectrum function by the convolution of the fitted spectrum function and the bin-to-bin migration matrix.The final Auger spectrum result is obtained by applying this resolution correction factor to the initial measurement of the spectrum.§.§ Comparison of the TA and Auger ResultsIt is customary, in both TA and Auger, to present the cosmic ray spectrum as flux J(E) multiplied by the third power of energy (E^3) (see Fig. <ref> and <ref>).In this representation, the low energy ankle and the ankle are clearly seen as the local minima, while the second knee and the high energy suppression appear as the local maxima. Figure <ref> shows superimposed TA and Auger spectra simply as J(E) vs E.Stronger features, ankle and the suppression, are still seen in the two results, even without multiplying them by E^3. Combined energy spectra of TA and Auger above 3 × 10^17 eV are presented in Fig. <ref> (left panel).There is clearly an overall energy scale difference between the two measurements, which is emphasized by the multiplication of the two results by the third power of the energy.The offset appears to be constant below the cut-off energy, above which the TA flux becomes significantly higher than that of Auger.A more quantitative statement can be made by considering the ratio of the Auger and TA fluxes, shown in the right panel of Fig. <ref>. Below ∼2×10^19 eV, the Auger flux is ∼20% lower than the TA flux and the difference between the two measurements becomes large for E > 2× 10^19 eV.It should be noted that below 2× 10^19 eV, the two spectra agree within the systematic uncertainties of the two experiments: a shift in the energy scale of less than 20% (a negative energy shift for TA or a positive energy shift for Auger) would bring the two measurements to an agreement.This shift is well within the uncertainties described in Sec. <ref>, and it can be attributed to the different models of the fluorescence yield and/or the invisible energy correction used by the two collaborations (see Sec. <ref>).Another way to address the differences between the two measurements is to compare the fitting parameters of the functional forms that describe the shapes of the spectra (see Sec. <ref>).The energy of the ankle and the spectral indices below (γ_1) and above (γ_2) the ankle presented in Table <ref> and <ref> can be compared directly. As expected, they are in good agreement. In the region of the cut-off, on the other hand, the comparison is more difficult, since the parameters that define the two functional forms have different meanings.However, an unambiguous comparison can be made using the parameter suggested in <cit.> that defines the position of the observed cutoff. This is the energy E_1/2, at which the integral spectrum drops by a factor of two below that which would be expected in the absence of the cutoff. E_1/2 has been calculated by both collaborations.For TA, E_1/2 = 60 ± 7 EeV (statistical error only) <cit.> and for Auger, E_1/2 = 24.7 ± 0.1 ^+8.2_-3.4 EeV <cit.> (statistical and systematic error).The two values of E_1/2 are significantly different, even after taking into account the systematic uncertainties in the energy scales of the two experiments.The difference between the TA and Auger spectra in the region of the cut-off is very intriguing.Because the TA experiment is in the Northern hemisphere and Auger is in the Southern hemisphere and the two experiments look at different parts of the sky, this could be a signature of anisotropy of the arrival directions of the ultra-high energy cosmic rays. Moreover the highest energies are the most promising for the identification of the sources of cosmic rays since the deflections of the trajectories of the primaries in the galactic and extra-galactic magnetic fields are minimized. However the measurement of the spectrum at the cut-off is affected by large uncertainties.In addition to the poor statistics, the analysis is complicated by the steepness of the flux: large spectral index amplifies the uncertainty of the energy scale and it increases the unfolding corrections required to take into account the bin-to-bin migrations due to the finite energy resolution. A continuous and increasing effort is being made by the two collaborations at establishing a better control of these effects and evaluation of the systematic uncertainties. § DISCUSSION The TA and Auger collaborations have developed analyses to constrain the astrophysical models using measurements of the energy spectrum. Observed features in the UHECR spectrum can reveal astrophysical mechanisms of production and propagation of the UHECRs.Moreover, thanks to the unprecedented statistics accumulated by the two experiments, the collaborations have started studying the energy spectrum in different regions of the sky. This represents a big step forward in the cosmic ray field: combined analyses of anisotropies of the arrival directions of cosmic rays, using high statistics whole-sky data, and the features in the energy spectrum can significantly improve our understanding of the nature of the UHECRs. §.§ Fitting Energy Spectrum to Astrophysical Models The basic assumption of the models developed by the TA and Auger is that UHECRs are accelerated at the astrophysical sources (the bottom-up models).In fact, most of the so-called top-down models, in which the primaries are generated by the decay of the super heavy dark matter, or topological defects, or exotic particles have been excluded by strong upper limits on ultra-high energy photons and neutrino fluxes <cit.>.The basic approach developed by TA and Auger to interpret the UHECR spectrum consists of assuming a distribution of identical sources, a mass composition and an energy spectrum at the sources. Then the spectrum at Earth is simulated taking into account the interactions of the primaries with the cosmic radiation (CMB, infrared, optical and ultraviolet) and the magnetic fields encountered during their path. The models are characterized by the parameters whose values are determined from the fit to the experimental data. In the TA model <cit.> the sources are distributed either uniformly or according to the large-scale structure (LSS) described by the distribution of the galaxies from the Two Micron All Sky Survey <cit.>. Only proton primaries are simulated.This composition assumption is justified by measurements of the mean X_ max made with the TA FD <cit.>.The spectrum of cosmic rays at the sources is parametrized using α E^-p (1+z)^3+m, where z is the redshift and the parameter m describes the cosmological evolution of the source density (for m = 0 the source density is constant per comoving volume).The maximum energy at which the primaries are accelerated is fixed at 10^21 eV, well above the energy of the GZK effect.The results of the TA analysis are shown in Fig. <ref>. The model that fits the SD spectrum <cit.> well is shown using solid and dashed lines, for uniform and LSS density distribution of the sources, respectively. The confidence regions of the model parameters are shown in the right figure. The fitted parameters are p ≈ 2.2 and m ≈ 7 <cit.>. The latter indicates a very strong evolution of the sources. The conclusion of the analysis is that the TA spectrum is well described by the interaction of the protons with the CMB: the GZK cut-off <cit.>, due to the photo-pion production, and the ankle due to the electron-positron pair production <cit.>. In Auger model <cit.>, the UHECR mass composition is not fixed, but is fitted to the Auger X_ max data <cit.>, simultaneously with the fit to the UHECR energy spectrum <cit.>.The sources have an isotropic distribution in a comoving volume. The nuclei are accelerated with a rigidity-dependent mechanism up to the maximum energy E_ max = Z R_ cut (Z is the charge of the nuclei and R_ cut is a free parameter of the model). The spectrum of the sources is parametrized with α E^-γ.The results of the analysis are presented in Fig. <ref>. The model that fits the measured spectrum and the mean and the standard deviation of X_ max best is shown using solid lines in the left panel.The model describes the measurements at energies above the ankle.The deviance (equivalent to a χ^2 per degree of freedom) as a function of the fitted parameters isshown in the right figure. The absolute minimum corresponds to a very hard injection spectrum (γ≲ 1) and a low maximum acceleration energy, which is below the energy of the GZK cut-off. This suggests that the observed break of the spectrum is mainly due to a cut-off at the sources rather than to the effects of propagation. There is another less significant minimum at γ≈ 2. In this case, the value of R_ cut is larger and the propagation effects contribute to the break in the spectrum.The TA and Auger analyses lead to different conclusions. This is due to the difference in energies at which the cut-off is observed and to the different primary mass composition assumptions in the models. In TA, the primaries are protons, while in Auger, the composition is mixed and has a trend with energy toward heavier elements in the suppression region.It is worth mentioning that the Auger and TA measurements of X_ max agree within the systematic uncertainties <cit.> but the inferred mass composition results are different because different hadronic interaction models and Monte Carlo codes have been used to interpret the data in the two experiments. Moreover, the sensitivity of the experiments to the mass composition measurements in the suppression region is strongly limited by the reduced FD duty cycle, a limitation that the Auger Collaboration plans to overcome with an upgrade of the SD detector as described in Sec <ref>. §.§ Study of the Declination Dependence of the Energy Spectrum The TA and Auger collaborations have started studying the energy spectrum in different declination bands. The exposures of the two SD detectors versus declination, for one year of data taking, are shown in Fig. <ref> <cit.>. For TA, the exposure refers to the events detected by the SD with zenith angles θ below 45^∘. For Auger, the exposures are for the 750 m (θ <55^∘) and 1500 m (θ <60^∘ and >60^∘) arrays. The Auger exposure obtained by adding the three contributions is also shown. The study of the spectrum in different declination bands became possible due to the large statistics accumulated by the two experiments.The study is motivated by the recent indications of anisotropy of the arrival direction of cosmic rays. The TA collaboration has found an excess of events of E > 5.7 × 10^19 eV in the so called hot spot, an angular region of radius 20^∘ in the direction of (α = 148.4^∘, δ = 44.5^∘ - right ascension and declination), near the Ursa Major <cit.>.The Auger collaboration reported an indication of a dipole amplitude in right ascension for the events of energies above 8 × 10^18 eV, which corresponds to a reconstructed dipole with (α,δ) = (95^∘± 13^∘ , -39^∘± 13^∘).Also, Auger has found another, less significant dipole amplitude, at the lower energies <cit.>. The TA collaboration has measured the SD energy spectrum in two declination bands, δ > 26^∘ and δ < 26^∘.For this analysis, TA events with zenith angle (θ) up to 55^∘ have been selected. In comparison to the standard SD spectrum calculation, which is done using events with θ < 45^∘, this analysis allows to increase the statistics and to lower the minimum declination of the events from about -6^∘ to -16^∘. However it requires an higher energy threshold at 10^19eV, which is above the ankle. It has been shown that the two TA spectra calculations are fully consistent above 10^19 eV <cit.>.The declination dependence of the TA spectrum using 6 years of data <cit.> is shown in the left panel of Fig. <ref>.Solid lines represent the (fitted) power laws with one breaking point. The definition of the so-called second break point energy (E_2) is equivalent to E_ break of Table <ref>.The corresponding values are: E_ break = (69 ± 5) EeV, for δ > 26^∘, and E_ break = (42 ± 6) EeV, for δ < 26^∘.Even if the sensitivity of the analysis is low due to the limited statistics, it is interesting to note that the tension with Auger data (which observes the suppression at a significantly lower energy - see Sec. <ref>) persists in the band at larger values of declinations, δ > 26^∘, while at the lower declinations, δ < 26^∘, where the TA and Auger fields of view partially overlap, the experiments see very similar energies of the suppression.The Auger collaboration has measured the energy spectrum in four declination bands with an exposure of about 42500/4 km^2 sr yr each <cit.>. The results are presented in the right panel of Fig. <ref>.There is no significant declination dependence of the flux. It has been demonstrated that the small differences between the fluxes are consistent with the expectation from the dipole anisotropy <cit.>. The analysis is limited to the declinations up to +24.8^∘ since it uses only the events detected by the 1500 m array with zenith angles < 60^∘. A systematic study of the difference of the spectra measured by the two experiments in the same declination band is of crucial importance, since it will help to understand whether the differences between the spectra addressed in Sec. <ref> have been caused by the systematic uncertainties of the experiments or these differences are due to an anisotropy signal.It is worth noting that, even if the spectra are compared in a declination band accessible by the two experiments, such analysis would not allow to arrive to a definitive conclusion if the shapes of the directional exposures in the common declination band are not similar, because the spectra would be affected by a potential anisotropy signal in different ways. As shown in Fig. <ref>, this is the case of the comparison of the TA spectrum with the Auger one obtained with the vertical events (θ < 60^∘). In fact, the two directional exposures have an opposite trend, increasing function of the declination for TA and decreasing for Auger.At the time of writing this paper, the Auger collaboration has not presented the declination dependence of the energy spectrum obtained using the inclined events (θ > 60^∘). We remark the importance of this measurement, since in the common declination band, the directional exposure for the Auger inclined events is of a similar shape to that of TA. Moreover, the comparison could be extended to higher declinations, up to 44.8^∘, whereas the vertical event Auger analysis goes only up to 24.8^∘ degrees. At the 2016 Conference on Ultra-High Energy Cosmic Rays, Kyoto (Japan) the two collaborations have presented a new and promising analysis method <cit.>, proposed by the members of the working group, aimed at combining the results of the anisotropy searches within the TA and Auger <cit.>. It consists of comparing the results of an alternative flux estimate, obtained by counting the numbers of events in the energy bins and weighting them by the inverse of the directional exposure. The resulting flux does not depend on the shape of the directional exposure and therefore, it should be same for TA and Auger. If a difference is found, it is to be ascribed to the experimental effects, and it should be consistent with the systematic uncertainties <ref>.The analysis presented in <cit.> is still preliminary. However, it has marked the road that should be followed to understand the differences in the measurements of the energy spectrum at the highest energies.It is worth to note that the application of this method requires a very good control of the systematic uncertainties. This alternative flux estimate should be consistent with the standard flux calculation if the arrival directions of cosmic rays are distributed isotropically, and this is possible only if the systematic uncertainties on the event reconstruction and the exposure calculations are well understood. The TA collaboration has shown that the two flux estimation methods are consistent in the declination band accessible by Auger with vertical events (δ < 24.8^∘) and in the full declination band -16^∘ < δ < 90^∘ (which includes the hot spot).§ CONCLUSIONS AND OUTLOOKThe Telescope Array and the Pierre Auger Observatory are the two largest cosmic ray detectors built so far.Their large exposures have allowed an observation of the suppression of the flux of cosmic rays at the very high energy with unprecedented statistics and precision.Both experiments combine the measurements of a surface array with the fluorescence detector telescopes. The hybrid system allows to measure the cosmic rays with an almost calorimetric energy estimation, which is less sensitive to the large and unknown uncertainties due to limited knowledge in the hadronic models, that are extrapolated well beyond the energies attainable in laboratory experiments. Having a precise estimate of the energy scale is of crucial importance for the measurement of the energy spectrum. In fact, the uncertainty in the energy estimation (Δ E/E), when propagated to the energy spectrum (J), is amplified by the power index (γ) with which the flux falls off with energy (Δ J/J ≈γ Δ E/E). The TA and Auger measure the cosmic rays in the northern and southern hemispheres, respectively. At energies below the suppression, the fluxes are expected to be the same because of the high level of isotropy in the arrival directions of the cosmic rays <cit.>. A good control of the systematic uncertainties of the energy scale of the two experiments is demonstrated by a remarkable agreement attained in the determination of the ankle at about 5 × 10^18 eV. The energy of the ankle measured by TA is only +8% larger than the one measured by Auger (see Tables <ref> and <ref>), which is roughly in agreement with the 20% difference in the flux normalization below the cut-off shown in Fig. <ref>. The difference in the ankle positions is fully consistent with the uncertainties in the energy scales quoted by the two experiments (21% and 14% for TA and Auger, respectively) and, it is expected to be reduced if the two collaborations adopt the same model for the fluorescence yield and for the invisible energy correction (see Sec. <ref>). Despite the good agreement in the region of the ankle and even at the lower energies, the TA and Auger spectra differ significantly in the region of the suppression (see Fig. <ref>). As discussed in Sec. <ref>, this discrepancy can be also quantified comparing the values of the E_1/2 parameter <cit.> that describes the position of the cut-off. The values reported by the two collaborations differ by a factor 2.5, which is well beyond the systematic uncertainties of the energy determination. Understanding the difference between the two spectra in the region of the cut-off is one of the major issues in the study of the UHECRs. At these extreme energies the deflections of the trajectories of the primaries in the galactic and extra-galactic magnetic fields are minimized, allowing the source identification, and therefore the spectra at Earth detected in the two different hemispheres could be different. The two collaborations have started studying their spectra in different declination bands.For TA, these studies are very relevant because of the hot spot near the Ursa Major constellation <cit.>.As shown in Sec. <ref>, these studies have a great potential but are currently limited by the statistics. Another important finding of these studies is that the declination range of the exposures of the two experiments partially overlap.This offers the possibility of making a comparison of the spectra in the same region of the sky <cit.>. Any discrepancy found would be indicative of an experimental effect that's due to the systematic uncertainties.One should note that the spectrum steepness in the energy region of the suppression amplifies the uncertainties in the energy scale and the event bin-to-bin migration that is due to the finite energy resolution. These effects, in addition to the limited statistics, make the measurement of the flux at the energies of the suppression very challenging.The features of the energy spectrum at very high energies are sensitive to the production and the propagation of the UHECRs and have been used to constrain astrophysical models. As shown in Sec. <ref>, the TA spectrum is well fitted by a model in which the primaries are protons (hypothesis consistent with the TA FD measurement of the mean X_ max <cit.>) and therefore the ankle is explained by the proton interactions with the CMB via electron-positron pair production <cit.> and the cut-off is explained by the GZK effect <cit.>.The Auger interpretation of their energy spectrum is more complicated. The inclusion of the trend toward heavier nuclei at the highest energies inferred from the FD measurements <cit.> leads to a scenario in which the observed break of the spectrum is not due to the effects of propagation. In this model the nuclei are accelerated by a rigidity-dependent mechanism with a cut-off that is observed in the spectrum measured at Earth.The studies presented by the TA and Auger demonstrate that the knowledge of the chemical composition plays an important role in the interpretation of the features of the energy spectrum. The results on <X_ max> of the two experiments are consistent <cit.>, but the inferred mass composition answers are different because the two collaborations have assumed different hadronic interaction models and used different Monte Carlo procedures.Extrapolation of the hadronic models beyond the energies attainable by accelerator physics is one of the major issues in understanding the air showers produced by the UHECRs. The shower development is mainly influenced by the particle production in the forward region, where the accelerator data are available only for energies up to a few hundreds of GeV <cit.>. A big improvement in this field will be possible by building a fixed target experiment using the beam of the LHC collider.The two collaborations will be taking data in the next years and are working on improving their detectors. The TA collaboration will quadruple the area of the SD array to approximately the current size of Auger, which is 3000 km^2. This extension is called the TA×4 <cit.> and it will be realized by adding 500 surface detectors using 2.08 km spacing.The aim is to improve the measurement of the cosmic rays beyond the suppression energy, as well as the sensitivity to the hot spot and other astrophysical sources.Also, two additional FD stations will be constructed to overlook the new SD array and to improve the composition studies at the highest energies.The Auger collaboration will upgrade the SD array by mounting scintillator detectors on the top of each WCD station.The upgrade of the Pierre Auger Observatory is called AugerPrime <cit.>. The combined analysis of the signal of the two detectors will allow to extract the muonic shower component and to extend the composition sensitivity of the detector into the flux suppression region, where the FD measurements are limited by the duty cycle. This will allow to improve the understanding of the origin of the cut-off and to select light primaries for the anisotropy studies. Even if the primary scope of TA and Auger is to study cosmic rays at the highest energies, an effort has been made with the TALE and Infill detectors to lower the minimum detectable shower energy threshold. The TALE FD energy spectrum has made it possible to observe the low energy ankle and the second knee. A similar result could be obtained with the HEAT telescopes of Auger. Building surface arrays of closer spacing is feasible for large collaborations such as TA and Auger and it would allow to extend the measurements down to the energies of the knee.The next decade will offer many opportunities to understand the origin of the UHECRs. TA×4 and Auger will view the full sky with a total collection area of 6000 km^2. The two collaborations are working together on combining their measurements.The declination band accessible by the two experiments is instrumental in achieving a better understanding of the systematic uncertainties and the differences in the energy scales.This will allow us to measure the energy spectrum from the knee up to the suppression and beyond in the entire sky with an unprecedented statistics and precision, which in turn will allows us to measure the energy spectra of cosmic rays in different declination bands or sky patches. So far, anisotropy studies using small (a few degree) or intermediate angular scales were carried out independently from the energy spectrum studies, although there were several studies of the energy dependencies of anisotropies. We emphasize here that the energy spectrum, the number of cosmic ray particles per time in a unit area from a given direction in a given energy range is, by definition, a function of the direction. The measurement of the full-sky energy spectrum by the future Auger and TA will make a crucial contribution to identifying the sources of ultra- high energy cosmic rays. We conclude this review with a compilation of recent experimental data on the energy spectrum presented in Fig. <ref>.9 RNC G. Matthiae and V. Verzi, Riv. Nuovo Cimento 38 N.2 (2015) 73 and references therein. Verzi-icrc15V. Verzi, Rapporteur Report of 34th ICRC 2015, The Hague, The Netherlands, PoS (ICRC2015) 015.ReviewKH-Watson K-H. Kampert and A. Watson, Euro. Phys. J. 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Sagawa, for the Telescope Array Collaboration, in Proc. 34th ICRC 2015, The Hague, The Netherlands, PoS (ICRC2015) 657.AugerPrime R. Engel, for the Pierre Auger Collaboration, in Proc. 34th ICRC 2015, The Hague, The Netherlands, PoS (ICRC2015) 686. EnSp-IceTop K. Rawlins and T. Feusels, for the IceCube Collaboration, in Proc. 34th ICRC 2015, The Hague, The Netherlands, PoS (ICRC2015) 334. EnSp-Yakutsk I. Petrov et al., for the IceCube Collaboration, in Proc. 34th ICRC 2015, The Hague, The Netherlands, PoS (ICRC2015) 252. EnSp-KASCADE The KASCADE-Grande Collaboration, Astropart. Phys. 36 (2012) 183. EnSp-Hires2008 R.U. Abbasi et al., Phys. Rev. Lett., 100 (2012) 101101. | http://arxiv.org/abs/1705.09111v1 | {
"authors": [
"Valerio Verzi",
"Dmitri Ivanov",
"Yoshiki Tsunesada"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170525095245",
"title": "Measurement of Energy Spectrum of Ultra-High Energy Cosmic Rays"
} |
-14mm -4mm=17.3cm =24cm | http://arxiv.org/abs/1705.09309v1 | {
"authors": [
"V. I. Yukalov",
"E. P. Yukalova"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170525180438",
"title": "Bose-Einstein condensation temperature of weakly interacting atoms"
} |
Department of Physics, Harvard University, Cambridge, MA 02138, USADepartment of Physics, University of California, Santa Barbara, CA 93106, USADepartment of Physics, Harvard University, Cambridge, MA 02138, USAMassless 2+1D Dirac fermions arise in a variety of systems from graphene to the surfaces of topological insulators, where generating a mass is typically associated with breaking a symmetry. However, with strong interactions, a symmetric gapped phase can arise for multiples of eight Dirac fermions. A continuous quantum phase transition from the massless Dirac phase to this massive phase, which we term Symmetric Mass Generation (SMG), is necessarily beyond the Landau paradigm and is hard to describe even at the conceptual level. Nevertheless, such transition has been consistently observed in several numerical studies recently. Here, we propose a theory for the SMG transition which is reminiscent of deconfined criticality and involves emergent non-Abelian gauge fields coupled both to Dirac fermions and to critical Higgs bosons.We motivate the theory using an explicit parton construction and discuss predictions for numerics. Additionally, we show that the fermion Green's function is expected to undergo a zero to pole transition across the critical point.Symmetric Fermion Mass Generation as Deconfined Quantum Criticality Ashvin Vishwanath December 30, 2023 =================================================================== Introduction— Recently, much attention has been lavished on band structures with symmetry protected nodal points (Dirac and Weyl semimetals)<cit.> in both two<cit.> and three spatial dimensions.<cit.> The paradigmatic example is graphene, where the band touching points are protected by symmetry, and the low energy dispersion around these points is captured by the massless 2D Dirac equation.<cit.> Similarly, massless Dirac fermions also appear on the surface of free fermion topological phases<cit.>. A key question pertains to the stability of the Dirac nodes in the presence of interactions. This controls whether the materials remains a semimetal or develops a gap leading to a semiconductor. Typically, this has been discussed in terms of interaction induced symmetry lowering, where interactions lead to a spontaneous symmetry breaking. The resulting lowering of symmetry allows for an energy gap. The physics in these settings can be modeled by a mean field “mass” term that is spontaneously generated on lowering the symmetry, and gaps out the Dirac fermions. This is the standard mass generation in the Gross-Neveu<cit.> and the Yukawa-Higgs models. The main challenge then is identifying the appropriate channel of symmetry breaking, following which one can utilize the Landau paradigm of order parameters to describe the mass generation. In this work we will discuss an altogether different mechanism of mass generation for Dirac fermions, that breaks no symmetries and cannot be modeled by a single-particle mass term at the free fermion level. The possibility of such a scenario is informed by recent developments in the theory of interacting fermionic symmetry protected topological (SPT) phases,<cit.> relating to the stability of free fermion topological insulators/superconductors to interactions. The paradigmatic example given by Fidkowski and Kitaev<cit.> is the 1+1D Majorana chain with an appropriately defined time reversal that protects edge Majorana modes regardless of their multiplicity. However interactions lead to an energy gap to these modes when they are multiples of eight, leading to a reduction of the free fermion classification →_8. More relevant to our purposes is the interaction reduction of 2+1D surface states of 3+1D topological phases, which contain Majorana or Dirac fermions. Indeed, here with the standard time reversal for electrons (class DIII),<cit.> there is a interaction reduced classification →_16 of topological superconductors, implying that sixteen surface Majorana fermions or equivalently eight Dirac fermions are unstable towards a massive (gapped) phase in the presence of strong interactions without breaking any symmetry. These considerations prompt us to look for a model of an electronic semimetal with eight Dirac nodes in 2D. A single layer of graphene with its two fold valley and two fold spin degeneracy leads to four Dirac nodes, hence we need to consider two layers[To preserve the Dirac dispersion we avoid discussing Bernal stacked bilayer graphene. Instead we have in mind twisted sheets of graphene that reduce interlayer tunneling.] of graphene to obtain eight Dirac cones in all (by combining the valley, spin and layer degeneracies). There is a simple way to see that at half filling it is possible to realize a symmetric insulating phase if interactions are included. Let us consider an antiferromagnetic Heisenberg spin interaction H_ int = J∑_i S_i1·S_i2 between the vertically displaced sites across the two layers.<cit.> Since on average we have one electron per site and each electron carries spin-1/2, the two electrons across the layer will pair into singlets and acquire an energy gap as long as the interlayer Heisenberg interaction is strong enough. This leads to a fully-gapped and non-degenerated ground state, which can be described as a direct product state of interlayer singlets. The state neither breaks any symmetry nor does it develop topological order.Therefore it is a featureless gapped phase in 2+1D.<cit.> The strong-coupling interaction mass (the many-body gap) that the electrons acquire in this phase is called the symmetric mass,<cit.> and the continuous phase transition (if it exists) between the Dirac semimetal and the featureless insulator will be calledsymmetric mass generation (SMG).<cit.> Note, one can also discuss the transition for a system with fewer Dirac fermions. For the surface of a fermionic SPT phase (eg. in class DIII or AIII), the gapped phases necessarily involve topological order <cit.> and constitute a rather different problem. For the intrinsically 2D system of graphene with an even number of sites in the unit cell, it is believed there is no intrinsic obstruction to realizing a trivial gapped phase (e.g. as shown for spinful single layer graphene in Ref. Kim:2016mz). However writing Hamiltonian that realize these gapped phases is itself a nontrivial task. Therefore we focus on the case of 8 Dirac nodes in 2+1D systems where the gapped phases are readily accessible and the numerical evidence for a single continuous transition is encouraging. What are the possible scenarios for the transition from the Dirac semimetal to the featureless gapped phase? At least for small J, it is known that short-ranged interactions are perturbatively irrelevant for 2D Dirac fermions thus the transition can only occur at finite interaction strengths. Unlike the lower dimensional cases, where the instability of 1+1D gapless fermions is manifest perturbatively and/or can be studied with powerful tools such as bosonization, the situation for the 2+1D problem is more challenging. On general grounds, there could be several scenarios as we step out of the Dirac semimetal phase. First, there could simply be a direct first-order transition to the featureless gapped phase, where the symmetric mass gap opens up discontinuously. Next, an intervening symmetry breaking phase may occur, leading to an energy gap to the fermions. Subsequently the symmetry could be restored accomplishing the phase change in a two step process, as illustrated in fig: scenarios(a). A different two step evolution involves the existence of an exotic critical phase that can be stable over a range of parameters, dubbed as the Bose semimetal phase<cit.> in fig: scenarios(c). It is a gapless quantum liquid of bosons and can be described as a generalized Gutzwiller projected Dirac semimetal. The most interesting possibility is shown in fig: scenarios(b), where the SMG occurs as a single continuous transition without any intermediate phases. Remarkably, numerical simulations of the problem in different models with various microscopic symmetries using different numerical methods<cit.> seem to uniformly point towards a single continuous SMG transition. All these models share one key common property that the weakly interacting semimetal phase should have exactly eight massless Dirac fermions. Even at the conceptual level it is unclear how to write down a theory for this putative transition. This is the problem addressed in this work. Since the SMG transition lies outside the Landau symmetry breaking paradigm, it would necessarily be exotic and require new ideas. The strategy we will adopt is to consider a form of fractionalization, where the symmetry quantum number of the electron is peeled off from their Dirac dispersion and carried away by a set of bosonic partons, while the Dirac cone structure is still maintained by a set of symmetry neutral fermionic partons. The process of fractionalization leads to an emergent gauge interaction between the bosonic and fermionic partons. In this framework, the semimetal phase corresponds to the condensed (Higgs) phase of the bosons. The featureless gapped phase corresponds to the symmetric gapped phaseof the bosons, which triggers gauge confinement of the remaining degrees of freedoms.This theory of SMG therefore falls in the category of deconfined quantum critical points,<cit.> that contains a non-Abelian (Yang-Mills) gauge field coupled to both the massless scalar (Higgs) fields and eight flavors of massless Dirac fermions. In the following, we will first introduce a minimal model for the SMG in 2D with (4) symmetry.We will develop an intuitive picture of the gapped phase as a paired superconductor in which fluctuations restore symmetry but leave the gap intact. This will motivate our parton construction and lead to a field theory description for SMG. Finally we discussed the implication of our theory for the fermion Green's function which can be tested in numerics. Model— Consider the honeycomb lattice with four flavors of fermions at half filling on each site. This is a model of two layers of graphene (each with two component spinful fermions). Previously we discussed how an interlayer spin interaction could lead to singlets, but it will be useful to enhance the symmetry and consider the fermions to be fully symmetric under the (4) rotation of the four flavors. A minimal model that captures this is given by:H=H_0+H_I H_0=-t∑_⟨ ij ⟩∑_a=1^4 (c^†_iac_ja + h.c.)where c_ia is the fermion operator on site i and of flavor a=1,2,3,4. Now consider the interaction term which preserves (4) symmetry:H_I =-V∑_i ( c^†_i1c^†_i2c^†_i3c^†_i4 + h.c. ). Note however the V term does not preserve the charge conservation of the fermions. The charge (1) symmetry is explicitly broken at the Hamiltonian level and excluded from our symmetry consideration. This may be interpreted as a proximity induced charge-`4e' superconductivity.<cit.>. If the regular (charge-2e) superconductivity was brought in proximity to graphene, the fermions would immediately be gapped. In contrastthe presence of weak four fermion terms V≪ t do not destabilize the Dirac cone and a finite interaction strength is needed for a transition to occur. On the other hand, in the strong coupling limit V ≫ t, the ground state is a simple product state of on-site fluctuating charge-4e quartets:|Ψ_c⟩ = ∏_i(1+c^†_i1c^†_i2c^†_i3c^†_i4)|0_c⟩,where |0_c⟩ denotes the fermion vacuum state. Therefore atransition is expected between the gapless Dirac semimetal and the gapped charge-4e superconductor, as we tune the interaction strength. Numerical simulations of the (4) symmetric model<cit.> point to a single continuous transition, i.e. the SMG transition. In the following, we will build a theory for it. One can also think that the charge-4e interaction H_I is related to an interlayer spin-spin interaction as motivated in the introduction by suitable particle hole transformation of two of the four fermion components. The only modification is that we need to only consider the XY components of the interlayer spin interaction. There are several other choices of interactions<cit.> that also drive the SMG transition, but for this work, we will only focus on the charge-4e interaction H_I described in eq: Hint.Symmetries— Symmetries of the model include not only the (4) internal symmetry but also the lattice symmetry and the particle-hole symmetry which fixes half filling. The lattice symmetry G_latt includes translation, rotation and reflection symmetries of the honeycomb lattice. The particle-hole symmetry _2^𝒮 acts as 𝒮:c_i→(-)^ic_i^† followed by complex conjugation, such that 𝒮^2=+1, which is also known as the chiral symmetry or the CT symmetry.<cit.>The combined symmetry G_latt×(4)⋊_2^𝒮 protects the Dirac semimetal from all fermion bilinear masses and the chemical potential shift. This can be seen from the field theory description for the Dirac semimetal ℒ=∑_Q=K,K'c̅_Qγ^μ∂_μ c_Q, where c_Q is a (4) fundamental spinor at each valley (Q=K,K'). The (4) symmetric bilinear mass terms must take the form of c̅_QM_QQ'c_Q' with a Hermitian 2× 2 matrix M in the valley sector. The space of M is spanned by four Pauli matrices (including σ^0) basis, which correspond respectively to the Chern insulator gap, the charge density wave gap, and two Kekulé dimerization gaps. The first two break the particle-hole and the reflection symmetries and the last two break the translation symmetry, so none of them is allowed by the full symmetry. So the remaining option to generate fermion masses without breaking any symmetry is to invoke fermion interactions, such as the charge-4e interaction H_I. Featureless Gapped Phase— To understand the SMG transition, we need to first understand both sides of the transition. The Dirac semimetal phase is relatively simple. As the interaction is weak and irrelevant, the semimetal phase is well described by thefermion band theory. The featureless gapped phase (the charge-4e superconductor) is more exotic. As the gap is of the many-body nature, it can not be described by the simple band theory picture. Nevertheless, much understanding of the charge-4e superconductor was obtained by disordering the charge-2e superconductor in previous studies.<cit.> We will take the same approach here. Let us consider fermion mass generation in two steps: we first gap the fermion by introducing the charge-2e pairing at the price of breaking thesymmetry, and then we restore the symmetry by disordering the pairing field. The discussion will lead to a parton construction for the featureless gapped phase, based on which we can further explore the possibility to merge the two steps of the mass generation into one single transition without the intermediate symmetry breaking phase.Let us start from the semimetal side and consider the (4) sextet pairing on each site (which has six components labeled by m=1,⋯,6) <cit.> Δ_i^m=1/2∑_a,bc_ia β^m_abc_ib, where β^m are antisymmetric 4×4 matrices given by β=(σ^12,σ^20,σ^32,σ^21,σ^02,σ^23), where σ^μν=σ^μ⊗σ^ν denotes the direct product of Pauli matrices σ^μ and σ^ν. The paring operator Δ_i rotates like an Ø(6) vector under the (4)≅(6) symmetry, and it transforms under the chiral symmetry as 𝒮: Δ_i→-Δ_i^†. Introducing such a flavor sextet pairing to the Hamiltonian H_M=-∑_iM·(Δ_i+Δ_i^†) will break the (4) symmetry (down to its (2)≅(5) subgroup) as well as the chiral symmetry _2^𝒮, and at the same time gap out all the Dirac fermions. In the limit that the paring gap |M|→∞, the fermion correlation length shrinks to zero, and the ground state wave function (of H_M) reads |Ψ_c,M⟩=∏_i(1+M̂·Δ_i^†+c_i1^†c_i2^†c_i3^†c_i4^†)|0_c⟩, where M̂=M/|M| is the unit vector that points out the “direction” of the sextet pairing. Comparing |Ψ_c,M⟩ with the wave function |Ψ_c⟩ for the featureless gapped phase in eq: SC4e, we can see that the most essential difference lies in the additional fermion bilinear term M̂·Δ_i^† in |Ψ_c,M⟩, which breaks the (4) symmetry.To restore the (4) symmetry, we need to remove the fermion bilinear term from the wave function. This amounts to symmetrizing the wave function |Ψ_c,M⟩ over all directions of M, or in other words, projecting the wave function |Ψ_c,M⟩ to the (4) symmetric subspace. Loosely speaking, we propose the following projective construction |Ψ_c⟩∼∫_S^5M|Ψ_c,M⟩.This construction will be made precise using the parton formalism shortly.But the lesson we learnt is that the fermion bilinear mass M serves as a convenient scaffold to construct the featureless gapped state, which can be removed by the symmetrization in the end.Parton Construction— The idea of carrying out the symmetrization on every site invites us to think about “gauging” the (4) symmetry as follows. Consider decomposing the physical fermion c_ia into bosonic B_iab and fermionic f_ib partons (a,b=1,2,3,4)c_ia=∑_b=1^4B_iabf_ib, with the “orthogonal constraint”[To preserve the fermion anticommutation relation, we may expect the bosonic parton B_i to form an “unitary matrix”: ∑_cB_ica^† B_icb=∑_cB_iacB_ibc^†=δ_ab. However this constraint turns out to be too strong, which restrict the boson Hilbert space to one dimension and effectively quench all the bosonic degrees of freedom, i.e. there is only one state (B_i^†)|0_B⟩ that satisfies the constraint. So we choose to relax the constraint by allowing the boson number to fluctuate, meaning that the we still keep the orthogonality but abandon the normalization of the B_i matrix. With this, the physical fermion and the fermionic parton are still related by a unitary transform but up to a constant. The constant may be interpreted as the quasiparticle weight.] on the bosonic parton Hilbert space ∀ a≠ b:∑_cB_ica^† B_icb=∑_cB_iacB_ibc^†=0. An (4) gauge freedom emerges from the fractionalization.[Naively one may expect a larger (4) gauge structure just by looking at the fractionalization scheme, but as we will see soon, the interaction terms of the bosonic partons can break the gauge group down to (4). In a sense, we choose to fix the (1) gauge.] On each site, thegauge transformation U_i∈(4) is implemented as B_iab→∑_c=1^4B_iacU_ibc^*,f_ia→∑_b=1^4 U_iabf_ib. Both the bosonic and the fermionic partons carry the (4) gauge charge. Besides the gauge charge, the (4) symmetry charge is carried solely by the bosonic parton. The chiral symmetry _2^𝒮 acts projectively on the partons as 𝒮:B_i→ - B_i^†, f_i→ (-)^i f_i^†, where the factor ± should be understood as the gauge transform in the _4 center of the (4) gauge group. So we have 𝒮^2=-1 for both bosonic and fermionic partons, in contrast to 𝒮^2=+1 for the physical fermion. As we will show later, such a projective _2^𝒮 action is required by the non-trivial projective symmetry group (PSG) of the parton mean field theory.Wavefunction from Partons— Motivated by the previous projective construction, we put the fermionic parton in a (4) gauge sextet superconducting state[Note that the partonground state |Ψ_f⟩ is not gauge invariant as it depends on the gauge sextet M. This is legitimate because |Ψ_f⟩ is not a physical state and M is not a variational parameter to appear in the final construction of the physical ground state after the gauge projection.] in analogy to eq: Psi_cM,|Ψ_f⟩=∏_i(1+M̂·Δ_i^†[f] +f_i1^†f_i2^†f_i3^†f_i4^†)|0_f⟩, where |0_f⟩ denotes the fermionic parton vacuum state and Δ_i[f]= 12∑_a,b f_iaβ_abf_ib is the gauge sextet paring operator of the fermionic parton f_ia, which is similar to the flavor sextet pairing of the physical fermion in eq: Delta. With this gauge sextet pairing, the (4)≅(6) gauge group is broken down to its (2)≅(5) subgroup. However the (4) symmetry remains untouched, because the symmetry charge is now carried by the bosonic parton. More importantly, the parton state |Ψ_f⟩ is also symmetric under the chiral symmetry _2^𝒮 in the PSG sense,[Under projective action of _2^𝒮, the fermionic parton is particle-hole conjugated followed by the _4 gauge transformation f_i^†→ -(-)^if_i, and the parton vacuum state is sent to the fully occupied state |0_f⟩→∏_i f_i1^† f_i2^† f_i3^† f_i4^†|0_f⟩.] which is in contrast to the physical fermion state |Ψ_c,M⟩ in eq: Psi_cM where _2^𝒮 is broken. Using the previously proposed PSG transformation 𝒮:f_i→ (-)^i f_i^†, it can be shown that the parton pairing operator transforms as 𝒮:Δ_i[f]↔Δ_i^† [f]. Hence the gauge sextet pairing term M·(Δ_i[f]+ Δ_i^† [f]) is _2^𝒮 symmetric, so as the resulting mean-field state in eq: Psi_f. To construct a (4) symmetric state, we consider putting the bosonic parton in a short-range correlated (4) singlet state. In the extreme limit of zero correlation length, an (4) symmetric many-body state takes the following form: |Ψ_B⟩=∏_i(1 + 1/4! ϵ_abcdB_ia1^†B_ib2^†B_ic3^†B_id4^†)|0_B⟩, where |0_B⟩ denotes the bosonic parton vacuum state. ϵ_abcd is the antisymmetric (Levi-Civita) tensor of four indices, such that the flavor indices are antisymmetrized to form the (4) singlet.Now we take both the bosonic and the fermionic parton wave functions and project them to the physical fermion Hilbert space,|Ψ_c⟩=𝒫|Ψ_B Ψ_f⟩, where the projection operator maps each parton Fock state to the corresponding Fock state of physical fermions 𝒫=∏_i,a(|0_c⟩⟨0_f 0_B|+c_ia^†|0_c⟩⟨0_f 0_B|∑_b=1^4B_iabf_ib). The resulting state |Ψ_c⟩ in eq: Psi_c proj is precisely the featureless gapped state in eq: SC4e. This parton construction provides us one plausible picture of the featureless gapped phase: the fermionic parton is in a gauge sextet paired state, while the bosonic parton is in a (4)symmetric gapped state, and the remaining gauge degrees of freedom are confined. On the other hand, the Dirac semimetal phase also admits a simple picture in the parton formalism: if we put the fermionic parton in the same Dirac band structure as the physical fermion and condense the bosonic parton to the state ⟨ B_iab⟩=Zδ_ab (with Z acting like the quasi-particle weight), then the physical fermion will be identified to the fermionic partonc_ia=Z f_ia and retrieve the Dirac band structure. We will implement these insights in a field theory below.Field Theory— What we learned from the parton construction is that the Dirac semimetal and the symmetric massive phase correspond respectively to the Higgs and the confined phases of an (4) gauge theory. Thus if there is a direct continuous transition between them, it is conceivable that the transition should be a deconfined critical point,<cit.> i.e. the gauge theory will be deconfined at and only at the transition point. Therefore we propose the following field theory description for the symmetric mass generation, ℒ= ℒ_f+ℒ_B,ℒ_f= ∑_Q=K,K'f̅_Qγ^μ(∂_μ-a_μ^mτ^m)f_Q+ℒ_int,ℒ_B= -B(∂_μ-a_μ^mτ^m)^2B^†+rB B^† +u_1(B B^†)^2+u_2(B B^†)^2 +u_3(B+h.c.)+⋯, which contains the matter fields of bosonic partons B and fermionic partons f_Q as well as the (4) gauge field a_μ^mτ^m. The matrices τ^m(m=1,⋯,15) are (4) generators (as 4×4 Hermitian traceless matrices), and (γ^0,γ^1,γ^2)=(σ^2,σ^1,σ^3). ℒ_int contains short-range interactions of the fermionic parton which will be specified later in eq: L_int. This interaction term is treated perturbatively, but it will play an important role to deform the fermionic sector from a pure quantum chromodynamics (QCD) theory, giving rise to possible instabilities of spontaneous mass generation for the fermionic parton in the symmetric gapped phase (as to be analyzed soon).Furthermore the emergent (1) symmetry corresponding to rotating the overall phase of the fermionic parton (f_Q→ e^θf_Q) will also be broken by the interaction ℒ_int. To reformulate the fractionalization scheme eq: c=bf at the field theory level, we start with the low-energy physical fermions c_Q (Q=K,K') around K and K' points of the Brillouin zone. Both of them transform under the(4) symmetry as fundamental representations. We can fractionalize the physical fermion field to the parton fields as c_Q=B· f_Q, where f_Q is a four component fermion field (transforming as a (4) gauge fundamental) for each valley Q, and B is a 4×4 matrix field that transforms under both the (4) symmetry (from left) and the (4) gauge symmetry (from right). Based on the fractionalization scheme of eq: c=bf, we expect the matrix field B to be unitary (up to normalization constant Z) on the lattice scale. The constraint may be imposed by a Lagrangian multiplier λ (B B^† -Z^2)^2, which, under renormalization, leads to an effective potential for B in the field theory, whose leading terms (r and u_1,2 terms of ℒ_B) are given in eq: L. The u_3 term is another (4) symmetric four-boson interaction, which explicitly breaks the (1) symmetry of B and can be viewed as a descendant of the charge-4e superconducting interaction H_I in eq: Hint.Symmetric Gapped Phase: In the field theory eq: L, the SMG transition is driven by r. When r>0, the bosonic parton is gapped, leaving the fermionic parton coupled to the (4) gauge field below the scale of the bosonic parton gap Δ_B, described by the N_f=2 (4) QCD theory. We assume that this (4) QCD theory is confining.<cit.> The resulting confined phase will depend on additional details. For example, if we considered a pure SU(4) QCD, with no additional four fermion interactions, a (1) symmetry of f_Q→ e^θf_Q (the baryon number conservation) will be present, and will not be broken in the confined phase, by the Vafa-Witten theorem.<cit.> Instead, chiral symmetry breaking is likely to occur, breaking the (2) valley symmetry. However, for our purposes it will be crucial to include the following four fermion interaction term in the form of the pair-pair interaction of the gauge sextet pairing Δ[f]=f_K^⊺γ^0βf_K', ℒ_int=g/2(Δ[f]·Δ[f]+h.c.). This interaction can be written as f_1f_2f_3f_4 +h.c. equivalently (which is (4) gauge neutral the same as Δ·Δ), reminiscent of the charge-4e interaction V between electrons in eq: Hint that drives the transition.Now, the (1) baryon number is no longer a global symmetry of the theory and Vafa-Witten does not forbid mass generation in the (1) breaking (gauge sextet pairing) channel.If the (4) gauge fluctuation were absent, the short-range interaction ℒ_int would be perturbatively irrelevant, given the negative engineering dimension [g]=-1<0 of the coupling g. As the (4) gauge fluctuation is included in the QCD theory, the scaling dimension [g] can receive anomalous dimension corrections. By a controlled renormalization group (RG) analysis based on the 1/N_f expansion, detailed in Appendix <ref>, we compute the scaling dimension [g]=-1+80/(π^2N_f) to the 1/N_f order, implying that the interaction ℒ_int could become relevant (i.e. [g]>0) as we push N_f to N_f=2. As the interaction flows strong under RG, it will drive the condensation of the gauge sextet pairing and lead to a mass term M·(Δ[f]+h.c.) which will gap the fermionic partons, at the scale of Δ_f∼|M|, and break the gauge group down to (2). In the absence of matter field fluctuations below the energy scale Δ_f, the non-Abelian (2) gauge field will confine itself (at a confinementscale Δ_a). However the (4) symmetry remains unbroken, since the bosonic parton is gapped and disordered. The particle-hole symmetry 𝒮 also remains unbroken because it is realized on the fermionic partons projectively, 𝒮:f_Q→ f_Q^†, where the _4 gauge transformation is crucial to undo the sign change of M originally caused by the particle-hole transformation. Thus the system is in the featureless gapped phase that preserves all symmetries. Massless Dirac Phase: When r<0, the bosonic parton condenses ⟨ B⟩≠0. A positive u_2>0 term would favor the condensate configuration ⟨ B⟩ to be a unitary matrix (up to an overall factor Z). Thus we can alway choose ⟨ B_ab⟩=Zδ_ab by (4) gauge transformations. This will identify the (4) symmetry with the (4) gauge group and Higgs out all gauge fluctuations. The system is then in the Dirac semimetal phase, described by ℒ=∑_Q=K,K'c̅_Qγ^μ∂_μ c_Q, where c_Q=Z f_Q and Z may be interpreted as the quasiparticle weight. Any short-range interaction ℒ_int among the fermionic parton in eq: L will become perturbatively irrelevant once the (4) gauge fluctuation is Higgs out by the condensation of the bosonic parton, and the corresponding interaction-driven instability (such as the gauge sextet pairing instability) will cease to exist. In this way, the RG relevance of the fermionic parton interaction ℒ_int (and the fermionic parton mass generation) is controlled by the mass r of the bosonic parton, so the SMG does not need fine tuning (other than the only driving parameter r).Critical Point: At r=0, bothfermionic (f_Q) and bosonic (B) partons are gapless. Together, they screen the (4) gauge field more efficiently, hence reducing the tendency to confinement. This opens up the possibility a stable deconfined (4) QCD-Higgs theory, which could describe the SGM critical point. We note that the similar behavior, namely the gauge confinement being irrelevant<cit.> at the critical point due to the gapless bosons, was also discussed in the deconfined phase transition<cit.> between the Néel and the valence bond solid phases. On moving away from the critical point into the phase where the Higgs fields are gapped, confinement takes over. Based on the above understanding, the energy scales Δ_B, Δ_f and Δ_a should catch up one after another as we enter the featureless gapped phase, as illustrated in fig: scales(a). This implies a hierarchy of length scales ξ_B,f,a∼Δ_B,f,a^-1 near the SMG transition from the side of the featureless gapped phase as shown in fig: scales(b). For example, the (4) multiplet fluctuation will be gapped with the bosonic parton at the length scale of ξ_B. However the (4) singlet fluctuation can persist to a longer length scale ξ_a until the (2) gluon gets confined.The above field theory description is in parallel with the parton construction as discussed previously. But the field theory also provides other possible scenarios for the transition(s) between the Dirac semimetal and the symmetric gapped phase, when the gaps Δ_B, Δ_f, Δ_a fail to open up together as the interaction V/t increases. For example, by tuning the short-range interactions of the fermionic parton f_Q in eq: L, it is possible that the fermionic parton may develop the bilinear mass before the bosonic parton is gapped, an intermediate (4) symmetry breaking charge-2esuperconducting phase will set in, with the condensation of the (4) sextet Cooper pairs of the physical fermion, as shown in fig: scenarios(a). Such a charge-2e superconducting phase was also observed in numerical simulations if the lattice model in Eq. (<ref>,<ref>) is deformed by the attractive Hubbard interaction<cit.> or by doping the chemical potential away from the Dirac point<cit.>. A more exotic scenario occurs if an extended deconfined phase is present, leading to an intermediate gapless quantum liquid. That is, if the fermion parton mass generation and the gauge confinement happens after the gapping of the bosonic parton,as shown in fig: scenarios(c). In this phase, the physical fermions are gapped, and the low-energy bosonic fluctuations are described by a wave function obtained from the gauge projection of the fermionic parton semimetal state. Therefore we may called it a Bose semimetal(BSM) phase.<cit.> The gapless bosonic fluctuation should be (4) singlets and transform only under the lattice symmetry. A possible candidate is the valence bond solid order fluctuation. Dynamically, which of these scenarios are more favorable should depend on the details of parton interactions and gauge dynamics. Numerical evidence from the lattice model seems to support a direct continuous transition without either of the intermediate phases, as shown in fig: scenarios(b).Fermion Green's Function— One smoking gun “feature” of the featureless gapped phase is the existence of zeroes in the fermion Green's function at zero frequency.<cit.> To be precise, let us define the Green's function of the physical fermion to be G_ab(x)=-⟨ c_a(x)c̅_b(0)⟩ where x=(t,x) is the spacetime coordinate. Fourier transform to the momentum-frequency space, we have G(k)=∫^3x G(x)e^ k_μ x^μ with k=(ω,k). In the Dirac semimetal phase, poles of the Green's function appear along the band dispersion. In particular, at the K and K' points of the Brillouin zone where the fermion becomes gapless, the pole is pushed to zero frequency, and hence G(k)∼ω^-1. However in the symmetric gapped phase, as proven in Ref. You:2014br, the poles will be replaced by zeroes: G(k)∼ω as ω→0 at k=K,K'. In fact the Green's function zeros are symmetry protected in the featureless gapped phase, which was proved in Ref. You:2014br. Let us focus in the vicinity of the K and K' points and redefine k to be the momentum deviation from them. The SMG transition is also a zero-pole transition in the fermion Green's function at k→0. Let us see how this is reproduced by the parton theory. Using the parton construction outlined in eq: projective, we can calculate the fermion Green's function deep in the featureless gapped phase. The result is (see Appendix <ref> for details) G_ab(k)=γ^μk_μ/k^μk_μ+M^2δ_ab. where |M| corresponds to the sextet pairing gap of the fermionic parton.In the featureless gapped phase (where |M| is finite), G(k) approaches zero analytically at k→0 as expected, see fig: zero-pole(a). This lends confidence to our parton construction of the featureless gapped phase. The fact that the quasiparticle weight approaches to unity deep in the featureless gapped phase is also consistent with expectation. Because the charge-4e superconducting ground state is a fully gapped symmetric short-range entangled state (similar to a vacuum state), a physical fermion c doped into the system will just propagate as a quasiparticle above its spectral gap set by the mass scale |M| without any fractionalization (which is also consistent with the picture that the (4) gauge theory is confining in the featureless gapped phase).The Green's function G(k) in eq: G(k) also provides a plausible scenario for the zero-pole transition. As the gap |M| decreases, two branches of poles are brought down from high energy, as shown in fig: zero-pole(b). They approach the line of zero asymptotically and eventually annihilate with the zero at the SMG transition where |M|→0. Then only a line of pole is left in the Dirac semimetal phase in fig: zero-pole(c). A similar mechanism for the zero-pole transition was proposed in Ref. Gurarie:2011gl. However we also note that the Green's function G(k) in eq: G(k) can not describe the fermion correlation close to the SMG critical point, where the bosonic parton and the gauge fluctuation also become important, such that one needs to go beyond the variational approach to describe the vanishing quasiparticle weight and the continuum spectral function as a result of the fermion fractionalization.Conclusion and Discussion—The SMG is an exotic quantum phase transition between the Dirac semimetal phase and a symmetric gapped phase, which cannot be understood within conventional theories of Dirac mass generation such as the Gross-Neveau mechanism. However, it has been numerically observed in different models with different interaction and symmetries. We propose a theoretical framework for SMG broadly as a deconfined quantum critical point, where the physical fermion is fractionalized into bosonic and fermionic partons with emergent gauge interaction. The Dirac semimetal phase corresponds to the Higgs phase and the featureless gapped phase corresponds to the confined phase. The gauge group, the parton flavor number and the symmetry assignment are flexible components of the theoretical framework that can be adapt to the model details, including the interaction parameters and the model symmetries. More work is required to understand what determines these parameters. In this work, we propose thatSMG in 2+1D with (4) global symmetry can be described by an (4) QCD_3-Higgs theory. Analyzing the nonperturbative dynamics of such a strongly coupled critical point is currently beyond our analytical capabilities, but a few statements can be made based on the basic structure of the theory following the approach used in (1) deconfined quantum criticality<cit.>. At the critical point we expect an emergent SU(2) symmetry in the valley space, relating a pair of valence bond solid orders and staggered A/B sublattice order on the honeycomb lattice.Potentially, there is an additional charge U(1) symmetry that could emerge at the critical point if either the u_3(B) in the bosonic parton sector or ℒ_int of the fermionic partons is irrelevant at the transition. However this appears unlikely since the term that drives the transition, the four electron charge 4e superconductor term, itself breaks this symmetry. It will be interesting to test the enlarged symmetry of the SMG critical point in numerics.Another prediction is that the anomalous dimension of the electrons is large at criticality since it decays into a pair of partons.Similarly, the (4) symmetry order parameters which are bilinears in the B fieldshould also have large anomalous dimension. For example, the quantum Monte Carlo simulation in Ref. He:2016qy obtained the anomalous dimension η_SMG = 0.7 ± 0.1 for the Ø(6) order parameter, much larger than η_WF=0.035 at the Ø(6) Wilson-Fisher fixed point. The theoretical framework proposed in this work may be applied to the SMG with other symmetry groups and in other dimensions. For example, in a upcoming work,<cit.> we will study SMG in a model with lower symmetry,(2)×(2)×(2) which will be described by a (2) QCD_3-Higgs theory. The advantage of the lower symmetry model is that it will give us access to more phases, and we can check if our critical theory can be perturbed to obtain the larger phase diagram. The largest symmetry group for 2+1D SMG is (7), where we can still consider a honeycomb lattice model with eight Majorana fermion on each site, transforming like a (7) spinor. The (7) SMG can be driven by applying the(7) symmetric Fidkowski-Kitaevinteraction<cit.> to each site. All the lower symmetry SMGs in 2D can be considered asdescendants of the (7) SMG by partially breaking the (7) symmetry down to its subgroup. An interesting direction is to consider SMG in various dimensions. In 3+1D, once again it is readily shown that eight Dirac fermions (or 16 Weyl fermions) can be gapped to produce a featureless state <cit.>. Whether this can proceed through a single continuous transition remains to be seen, numerics on one microscopic model appear to give an intervening symmetry breaking phase <cit.>.One may also discuss SMGin1+1D, where we need four Dirac fermions. In fact this is closely related to the interaction reduction of topological phases in 1+1D described by Fidkowski-Kitaev <cit.>, where they show that edge states with eight Majorana modes are unstable despite the presence of time reversal symmetry that forbids a quadratic gapping term. Therefore, the transition between the trivial phase and a phase with eight Majorana (or four Dirac) edge zero modes can be circumvented by interactions, which is related to symmetric mass generation for four Dirac fermions in 1+1D. In Appendix. <ref>, we review and reinterpret the Fidkowski-Kitaev transition in 1+1D within an (7) SMG<cit.> in the parton language. There, the problem was solved using an alternate set of fermion variables informed by SO(8) triality, within which the transition is simply described. Could such a change of variable or duality transformation be constructed to describe SMG in higher dimensions? These are questions for future work. We would like to acknowledge the helpful discussions with Max Metlitski, N. Seiberg, T. Senthil, Chong Wang, Andreas Ludwig, John McGreevy, Tarun Grover, Xiao-Liang Qi, Yingfei Gu, Aneesh Manohar, Subir Sachdev and Leon Balents. AV and YZY was supported by a Simons Investigator grant. YCH is supported by a postdoctoral fellowship from the Gordon and Betty Moore Foundation, under the EPiQS initiative, GBMF4306, at Harvard University. CX is supported by the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208.§ RENORMALIZATION GROUP ANALYSISIn this appendix, we present the renormalization group (RG) analysis of the N_f=2 (4) QCD theory with short-range fermion interaction. The theory arises from the SMG field theory eq: L after gapping out the bosonic field B. To control the RG calculation, we can generalize the theory to the large-N_f limit. The Lagrangian in consideration readsℒ_f=∑_a=1^N_f∑_i,j∑_α,βf̅_aαiγ^μ_αβ(∂_μ-a_μ^mτ^m_ij)f_aβj+ℒ_int. The fermionic parton field f_aα i is label by the favor index a=1,⋯,N_f, the Dirac index α=1,2 and the color index i=1,2,3,4. The flavor indices are transformed under the flavor symmetry group (N_f/2) and the color indices are transformed under the gauge group (4). The case of N_f=2 is relevant to our discussion in the main text. a_μ^m is the (4) gauge field that couples to the fermion via the (4) generators τ^m acting in the color subspace. The (2) rotation in the Dirac subspace corresponds to the space-time rotation. The γ^μ matrices are chosen as (γ^0,γ^1,γ^2)=(σ^2,σ^1,σ^3) and f̅_aα i=f_aβ i^†γ^0_βα.ℒ_int denotes the short-range four-fermion interaction. The charge-4e superconducting interaction for the physical fermion will naturally induce a similar interaction for the fermionic parton with the same symmetry properties. It can be verified that the following interaction is the only four-fermion interaction that is invariant under the space-time rotation, the (N_f/2) symmetry and the (4) gauge transformations, but breaks the (1) symmetry in the same manner as the physical fermion interaction. ℒ_int =g(V^abcd_αβγδϵ_ijklf_aαif_bβjf_cγkf_dδl+h.c.),V^abcd_αβγδ =J_abJ_cdϵ_αβϵ_γδ+J_acJ_bdϵ_αγϵ_βδ = +J_adJ_bcϵ_αδϵ_βγ, where ϵ_ijkl is the totally antisymmetric tensor in the color subspace, ϵ_αβ is the antisymmetric matrix in the Dirac subspace, and J_ab is the symplectic form of the (N_f/2) group in the flavor subspace (such that the generator A of (N_f/2) preserves JA+A^⊺ J=0).fig: RG diagrams concludes the diagrams that contributes to the linear order (in g) of the RG equation at the 1/N_f order. Following Ref. Xu:2008rz,Xu:2008jc, the RG flow equation is given by g/ℓ=-(1-80/π^2N_f)g. So the interaction strength g has the scaling dimension [g]=-1+80/(π^2N_f)+𝒪(N_f^-2). At N_f=2, we have [g]≈ 3.05>0, implying that the short-range interaction g is relevant. To analyze theinstabilities due to this interaction, we recall the β=(σ^12,σ^20,σ^32,σ^21,σ^02,σ^23) matrices defined in the main text, then the antisymmetric tensor ϵ_ijkl can be decomposed as ϵ_ijkl=1/2∑_mβ_ij^mβ_kl^m. Thus at N_f=2, the interaction in eq: L_int large N becomes ℒ_int=g/2((f_K^⊺γ^0βf_K')^2+h.c.). So the strong charge-4e interaction could drive the spontaneous generation of the gauge sextet pairing Δ=f_K^⊺γ^0βf_K' regardless of the sign of g. The sign of g only determines whether the instability is in the Δ channel (g<0) or in the Δ channel (g>0). But either case will lead to the fermionic parton mass generation and the gauge confinement, so the sign of g is not important.§ DERIVATION OF THE GREEN'S FUNCTION As we have shown in eq: projective, in the extreme limit of zero correlation length, the featureless gapped state |Ψ_c⟩ in eq: SC4e can be obtained exactly by projecting the mean field state |Ψ_c,M⟩ to the (4) symmetric sector. In this appendix, we would like to generalize this construction to the case of finite correlation length. Although the projective construction will not be exact as we go away from the zero correlation length limit, yet it still provide us an useful variational wave function which has a controlled asymptotically exact limit, based on which we can evaluate the fermion Green's function.The idea to increase the fermion correlation in the wave function |Ψ_c⟩ is to allow the fermion to move around on the lattice. So we turn on the fermion hopping term in the mean field Hamiltonian, H_MF =H_0+H_M = -t∑_⟨ij ⟩ c^†_ic_j-∑_iM·Δ_i+h.c.. Let us still denote the mean field ground state as |Ψ_c,M⟩. Switch to the momentum-frequency space and use the Nambu spinor basis c_k=(c_K+k,c_K'+k^†)^⊺, the fermion correlation on the mean field state is given by G_M(k) =-⟨Ψ_c,M| c_k c̅_k|Ψ_c,M⟩ ≃γ^μk_μ-M·βM·β -γ^μk_μ^-1 =1/k^μk_μ+M^2γ^μk_μ-M·βM·β -γ^μk_μ at low energy. We propose an (4) symmetric wave function |Ψ_c⟩ by symmetrizing |Ψ_c,M⟩ following eq: projective, as |Ψ_c⟩=∫_S^5M|Ψ_c,M⟩ (assuming the measure is (6) symmetric and is properly normalized), where S^5 denotes a sphere of radius |M|. Then the Green's function on the symmetric state can be obtained by symmetrizing the mean field Green's function. To see this, we start with G(k) =-⟨Ψ_c|c_kc̅_k|Ψ_c⟩ =-∫MM'⟨Ψ_c,M|c_kc̅_k|Ψ_c,M'⟩. The overlap ⟨Ψ_c,M|c_kc̅_k|Ψ_c,M'⟩ vanishes if M≠M', due to the orthogonality catastrophe of fermion many-body states. Therefore we have G(k) =-∫M⟨Ψ_c,M|c_kc̅_k|Ψ_c,M⟩ =∫M G_M(k). The symmetrization will remove the M·β terms in the numerator but leave the M^2 term in denominator untouched. Switching back from the Nambu basis, we arrived at the Green's function in eq: G(k). § FIDKOWSKI-KITAEV SO(7) SYMMETRIC MASS GENERATION In this appendix, we will review the (1+1)D Fidkowski-Kitaev (7) symmetric mass generation<cit.> from the perspective of the parton construction. The model is defined on a 1D lattice. On each site, there are eight Majorana fermion modes χ_ia (a=1,⋯,8) forming the 8-dimensional real spinor representation of an (7) group. The Hamiltonian reads H=H_0+H_I withH_0 =∑_i∑_a=1^8 χ_i,aχ_i+1,a,H_I =-V/4!∑_i∑_m=1^7Δ_i^m Δ_i^m, where Δ_i^m=χ_iaΓ^m_abχ_ib are the seven components of the (7) vector. The Gamma matrices can be chosen as Γ=(σ^123,σ^203,σ^323,σ^211,σ^021,σ^231,σ^002), which form a set of purely imaginary, antisymmetric and anticommuting matrices. The Hamiltonian H is manifestly (7) invariant. Besides the internal (7) symmetry, the model also possess the translation symmetry T:χ_i→χ_i+1 and the chiral symmetry 𝒮:χ_i→(-)^iχ_i,→-. One can see 𝒮^2=+1 and T^-1𝒮T𝒮=-1 acting on the fermions.In the non-interacting limit (V→0), H_0 simply describes eight decoupled and gapless Majorana chains, whose field theory description is ℒ_0=1/2∑_a=1^8χ̅_aγ^μ∂_μχ_a, where χ_a=(χ_La,χ_Ra)^⊺ and χ̅_a=χ_a^⊺γ^0 contains both left- and right-moving Majorana modes with the gamma matrices (γ^0,γ^1)=(σ^2,σ^1). Since the Majorana coupling along the chain is purely imaginary, the Fermi points are located at momentum k=0,π, so χ_L,R fields are related to the real space fermion χ_i by χ_La=∑_iχ_ia,χ_Ra=∑_i(-)^iχ_ia. Both χ_L and χ_R transform as (7) spinors. The translation and the chiral symmetry acts as T:χ_Ra→-χ_Ra,𝒮:χ_La↔χ_Ra. All fermion bilinear mass terms (such as mχ̅χ) are forbidden by these symmetries. In fact, the translation symmetry is the most important protecting symmetry which is sufficient to rule out all bilinear masses (i.e. all back scattering terms between χ_L and χ_R). So the Majorana chain can not be symmetrically gapped on the free fermion level.However, it is possible to symmetrically gap out eightMajorana chains by fermion interactions. One possible interaction proposed by Fidkowski and Kitaev<cit.> is the (7) symmetric interaction V in eq: FK (abbreviated as the FK interaction hereinafter). In the strong interaction limit (V→+∞), the Hamiltonian is dominated by H_I, which decouples to each single site. Diagonalizing the on-site Hamiltonian, one finds a unique ground state separated from the excited states by a finitegap Δ=14V. If we pair up the on-site Majorana fermions into Dirac fermions as c_ia=χ_i,2a-1+χ_i,2a (a=1,⋯,4), the ground state wave function can be expressed as |Ψ_c⟩=∏_i(1+c_i1^†c_i2^†c_i3^†c_i4^†)|0_c⟩, which is the 1D version of the charge-4e superconducting state in eq: SC4e. It can be verified that |Ψ_c⟩ preserves the full (7), translation and chiral symmetries, and hence a featureless gapped state. The only difference with the (2+1)D case is that in (1+1)D the interaction V is marginal at the free fermion fixed point. Depending on the sign of V, it is marginally relevant if V>0 and marginally irrelevant if V<0. So as long as we turn on an infinitesimal but positive V, the system undergoes a 1D version of the symmetric mass generation (SMG) to the featureless gapped phase.A better understanding of this 1D SMG physics would greatly help us to understand the SMG transitions in all higher dimensions. In the following we will present a parton construction following Ref. Fidkowski:2010bf using the (8) triality. The (8) triality is a property that the (8) vector 8, left-spinor 8_+ and right-spinor 8_- representations can fuse to the trivial representation under the trilinear map t:𝒱_8×𝒱_8_+×𝒱_8_-→. The trality map t can also be written as a three-leg tensor t^m_ab, where the tensor indices m,a,b labels the basis of 8, 8_+ and 8_- respectively. Without interactions, H_0 has the full (8) symmetry that rotates the eight Majorana flavors. The interaction H_I breaks the (8) symmetry down to its (7) subgroup and at the same time fixes χ_a to be one of the spinor representations 8_±, say 8_+. Using the triality tensor t^m_ab, we can construct the physical fermion χ_a (as 8_+ spinor) by fusing the bosonic parton ϕ_b (as 8_- spinor) and the fermionic parton ψ_m (as 8 vector) on the field theory level: χ_Qa=∑_b,mt^m_abϕ_Qbψ_Qm, which applies to both the left- and right-moving modes Q=L,R. One way to make sense of eq: chi on the microscopic level is to consider the bosonic parton ϕ_Qb as a Kondo impurity resting on the boundary of the fermion chain,<cit.> which is treated as a dynamical variable without spacial dependence. There are two types of Kondo impurities: ϕ_Lb and ϕ_Rb. The ϕ_Qb impurity only couples to the χ_Qa fermion and scatters it to the ψ_Qm fermion and vice versa. In this fractionalization scheme, both partons carry the (7)⊂(8) symmetry charge, unlike the fractionalization scheme in the main text where only the bosonic parton carries the (4) symmetry charge. The translation and the chiral symmetry acts on the parton fields as follows:T:ϕ_Rb→-ϕ_Rb,𝒮: ψ_Lm↔ψ_Rm, such that the symmetry action on the physical fermion field in eq: TS sym can be retrieved. Now let us consider putting the fermionic parton ψ_m=(ψ_Lm,ψ_Rm)^⊺ in the same band structure as the physical fermion χ_a=(χ_La,χ_Ra)^⊺, described by ℒ_0^ψ=1/2∑_m=1^8ψ̅_mγ^μ∂_μψ_m, similar to eq: Lchi0. As long as the fermionic parton is gapless, the physical fermion is also gapless, which corresponds to the free fermion fixed point. As the FK interaction is turned on, the following interaction for the fermionic parton will be induced: ℒ_I^ψ=-A(∑_m=1^7ψ̅_mψ_m)^2-B(∑_m=1^7ψ̅_mψ_m)ψ̅_8ψ_8, which contains two types of short-range interactions (A and B terms). The general form of the parton interaction in eq: psi int can be argued on symmetry basis.According to the fractionalization scheme eq: chi, the fermionic parton ψ_m was assigned to the vector representation of (8). So the (7) symmetry group (as a subgroup of (8)) will only rotate seven components of ψ_m and leaving one remaining component invariant. Without loss of generality, we assume ψ_8 to be the (7) invariant component, then the (7) vector can be written as (m=1,⋯,7) Δ^m=(ψ_Lmψ_L8+ψ_Rmψ_R8), in terms of the fermionic parton bilinear form. As shown in eq: FK, the FK interaction H_I∼ - Δ·Δ is just a dot product of (7) vectors, so we expect it to induce a same type of interaction for the fermionic partons-B∑_m=1^7Δ^mΔ^m=-B/2(∑_m=1^7ψ̅_mψ_m)ψ̅_8ψ_8. This gives rise to the B-type of interaction in eq: psi int. As shown in Ref. Fidkowski:2010bf, under the RG flow, A-type of interaction will be generated with A>0 and become relevant. The A-type interaction drives a spontaneous mass generation ⟨∑_m=1^7 ψ̅_mψ_m⟩=M for the first seven ψ_m fermions, which in turn gives rise to the mass BMψ̅_8ψ_8 for the ψ_8 fermion via the B-type interaction. Hence all the fermionic partons are gapped out via the mass generation. The parton mass M is evidently (7) symmetric. It is also invariant under the translation and the chiral symmetries as seen from eq: PSG. So in the presence of the FK interaction, the system will enter the featureless gapped phase via spontaneous mass generation for the fermionic partons.In conclusion, the key point that we learn from this (1+1)D MSG is that the nature of the “symmetric mass” for the physical fermion is actually a bilinear mass for the fermionic parton. The fact that the parton bilinear mass does not break the symmetry is either because the fermionic parton is in a different symmetry representation from the physical fermion (e.g. the (7) symmetry) or because the symmetry charge is carried away by the bosonic parton (e.g. the translation symmetry). These are the key observations that motivate us to propose the (2+1)D SMG theory in the main text. apsrev | http://arxiv.org/abs/1705.09313v2 | {
"authors": [
"Yi-Zhuang You",
"Yin-Chen He",
"Cenke Xu",
"Ashvin Vishwanath"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170525181216",
"title": "Symmetric Fermion Mass Generation as Deconfined Quantum Criticality"
} |
http://arxiv.org/abs/1705.09263v3 | {
"authors": [
"Walter D. Goldberger",
"Siddharth G. Prabhu",
"Jedidiah O. Thompson"
],
"categories": [
"hep-th",
"gr-qc",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170525170957",
"title": "Classical gluon and graviton radiation from the bi-adjoint scalar double copy"
} |
|
Present address: Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA [email protected] Department of Physics, Indian Institute of Technology Kanpur- 208016, India Quantification of the velocity and magnetic field reversals in dynamo remains an interesting challenge.In this paper, using group-theoretic analysis,weclassify the reversing and non-reversing Fourier modes during a dynamo reversal in a Cartesian box.Based on odd-even parities of the wavenumber indices, we categorise the velocity and magnetic Fourier modes into 8 classes each.Then, using the properties of the nonlinear interactions in magnetohydrodynamics, we show that these 16 elements form Klein 16-group Z_2 × Z_2 × Z_2 × Z_2. We demonstrate that field reversals in a class of Taylor-Green dynamo, as well as the reversals in earlier experiments and models, belong to one of the classes predicted by our group-theoretic arguments. Discrete Symmetries in Dynamo Reversals Mahendra K. Verma December 30, 2023 ======================================= § INTRODUCTION In 1919, Larmor proposed that the magnetic field in various astrophysical and geophysical bodies are generated self-inductively by the electric currents and magnetic field by a bootstrap mechanism <cit.>.This mechanism is called dynamo.The generated magnetic field exhibits many interesting phenomena including field reversals. Paleomagnetic records show that the Earth's magnetic field hasreversed its polarity on geological time scales <cit.>.The interval between two reversals is random with an average interval between two consecutive reversals as approximately 200,000 years. On the contrary, the magnetic field of the Sun changes its polarityquasi-periodically approximately every eleven years <cit.>.This phenomena called field reversal is an interesting puzzle, and it has been studied by a large number of researchers.In this paper we will study the symmetry properties of such reversals.Various theoretical modelshave been proposed to describe dynamo mechanism suitable for different situations. Magnetohydrodynamics (MHD), which treats the plasma as fluid, is often used to describe the behaviour of turbulent plasma in the presence of magnetic field. The MHD model however breaks down for collisionless and relativistic limits, and other models are employed for such cases.Dynamo actions in collisionless plasmas have been recently investigated by Rincon et al. <cit.> and Kunz et al. <cit.>. This is pertinent to dynamos in extragalactic plasmas, e.g., in accretion disks, intercluster medium etc. <cit.> In addition, Hall effect becomes important when the ion and electron velocities are sufficiently distinct.Mininni et al. <cit.>, Gómez et al. <cit.>, and Lingam and Bhattacharjee <cit.> have investigated Hall dynamos.There are other possibilities of dynamo action, but we do not list them here due to limited scope of this paper. In this paper, for simplicity, we limit ourselves to MHD dynamos.The equations of magnetohydrodynamics (MHD) satisfy the symmetry properties: u → u andb → -b, where u,b are the velocity and magnetic fields respectively.Note however that such symmetry is not persevered in generalised MHD, such as Hall MHD.From the above symmetry of MHD, one may infer that the magnetic fieldchanges sign after a reversal in MHD, but the velocity field does not.However, researchers observe that b → -b in some experiment, but in some others, only some of the large-scale modes of the b switch sign, and some others do not. In the laboratory experiment involving a Von Karman swirling flow of liquid Sodium (VKS) <cit.>, the magnetic dipolar component D reversesbut the magnetic quadrupolar component Q does not.Petrelis et al. <cit.> and Gissinger et al. <cit.> constructed low-dimensional models whose variables are dipolar and quadrupolar magnetic fields, anddipolar velocity field.Gissinger <cit.> showed that the field reversals in VKS experiment is consistent with the predictions of his low-dimensional models. Earlier, Rikitake <cit.> , Nozieres <cit.> , and Knobloch <cit.> studied magnetic field reversals in disk dynamo model <cit.> and its variations.These models also exhibit chaos.Tobias et al. <cit.> studied chaotically-moduled stellar dynamo. Refer to Moffatt <cit.>, Weiss and Proctor <cit.> and reference therein for further discussions on low-dimensional dynamo models that exhibit reversals of magnetic field. Rayleigh-Bénard convection (RBC) exhibits flow reversals in which the velocity field reverses randomly in time <cit.>. The dynamics of flow reversal has significant similarities with that of dynamo reversals. For the flow reversal in a two-dimensional (2D) box, Chandra and Verma <cit.> and Verma et al. <cit.> constructed group-theoretic arguments to identify theFourier modes that change sign in 2D flow reversals of RBC.They showed that the reversing and non-reversing Fourier modes of 2D RBC form a Klein four-group Z_2 × Z_2.In Appendix A, we generalise the above arguments to three-dimensional (3D) RBC.In this paper we make similar symmetry-based arguments for the dynamo reversals, and classify Fourier modes that change sign in a dynamo reversal.The group consists of 8 elements each of the velocity and magnetic fields.We discuss the details of the group structure in Sec. <ref>.It is important to contrast the reversal dynamicsobserved in box, cylinder, or sphere geometries.For RBC in a box, flow reversals are accompanied by sign changes of some of the Fourier modes <cit.>.In a cylindrical convection, similarreversals have been observed, and they are referred to as cessation-led reversal. In addition, cylindrical convection exhibits rotation-led reversal in which thevertical velocity field changes sign due to the rotation of the large-scale circulation <cit.>.In terms of symmetries, the reversals of flow structures in Cartesian geometry or those in a cessation-led reversal are related to the discrete symmetry.But reversals due to continuous rotation, as in rotation-led reversals in a cylinder, are connected to the continuous symmetry. It is natural to expect that both types of reversals would occur in a spherical dynamo. In the present paper we focus on discrete symmetries of dynamo.Cylindrical and spherical geometries exhibit reversalsconnected to discrete and continuous symmetries.In this paper, to keep the focus on discrete symmetries,we focus on dynamo reversals in a box geometry.Some of the observational works related to dynamo reversals are discussed below. Earth's magnetic field, which is generated by the motion of molten iron inside the Earth,has a dominant dipolar structure. Most of the Earth's past magnetic field data have been measured from the ferromagnet rocks that were formed out of thefrozen magma. Using these measurements, geologists discovered that the Earth's magnetic field has reversed many times in the past. The interval between two consecutive field reversals is randomly distributed, and thefield structure during a reversal is quite complex with possiblemultipolar magnetic-field structure.Some scientists believe that the geomagnetic reversals is a spontaneous process, while others argue it to be triggered by some external sources <cit.>.The solar dynamo too exhibits polarity reversals, but these reversals differ significantly from the geomagnetic reversals.The sunspots, solar wind, and solar flares provide us valuable inputs about the Sun's magnetic field.For example, the poloidal field reverses its direction approximately every eleven years; a field reversal involves interactions among the poloidal and toroidal components <cit.>.Fast supercomputersand sophisticated numerical codes have enabled researchers to simulate and study aforementioned dynamo mechanism inrealistic geometries, e.g. in spherical shells.However the parameters used in simulations are quite far from the realistic values. Field reversals have been reported in several numerical simulations of the geodynamo <cit.> and other 3D simulations of rotating spheres <cit.>.Glatzmaier and Roberts <cit.> ransimulations equivalent to approximately 300000 terrestrial years, and observed field reversals similar to those observed in paleomagnetic records.They reported that the interval distribution between two consecutive reversals is random, and that the magnetic field geometry has a complex structure during a reversal. Based on symmetry arguments, Pétrelis et al. <cit.> proposed a mechanism for dynamo reversals in VKS.They assumed that the magnetic field is decomposed into two parts—a dipolar component of amplitude D, and a quadrupolar component of amplitude Q, and constructed a variable A: A = D + i Q.They wrote the following amplitude equation in powers of A and its complex conjugate A̅underthe constraint that 𝐁→ - 𝐁 (or A → - A):Ȧ = μ A + νA̅ + β_1 A^3 + β_2 A^2 A̅ + β_3 A A̅^2 + β_4 A̅^3.This is up to the lowest order nonlinearity. Here, μ, ν, β_is are complex coefficients that depend on the experimental parameters. Using this model, Pétrelis et al. <cit.> explained various dynamic regimes of the VKS experiment.In a related development, Gissinger et al. <cit.> considereda three-mode model of dynamo reversal. A third mode V representing the large-scale velocity is considered in addition to D and Q. The governing equations are derived based on symmetries, and they areḊ = μ D - V Q,Q̇ = -ν Q + V D, V̇ = Γ - V - Q Dupto quadratic nonlinearities. A nonzero Γ represents the forcing that breaks the rotational symmetry.The models of Pétrelis et al. <cit.> and Gissinger et al. <cit.> invoke rotation and mirror symmetries to construct the nonlinear terms and determine the reversing and non-reversing modes.In this paper we present group-theoretic arguments to determine the reversing and non-reversing modes in a dynamo reversal.Our analysis exploit the nonlinear structure of the equation.We will show that u→ u and b→ - b is a subclass of the possible reversals. Our arguments show that some u,b modes reverse and some others do not.Though our symmetry arguments are similar to those ofPétrelis et al. <cit.> andGissinger et al. <cit.>, yet they are more convenient due to their algebraic structure. Our model also encompasses more modes in contrast to a smaller numberof large-scale modes in the models of Gissinger et al. <cit.>.Krstulovic et al. <cit.> simulatedTaylor-Green dynamo for various boundary conditions. They observed that the dynamo thresholds varies with the boundary conditions. In a box geometry with insulating walls, Krstulovic et al. <cit.> observedan axial dipolar dynamo similar to thatin VKS experiment <cit.>.However Krstulovic et al. <cit.> did not study the dynamo reversals.In this paper we extend Krstulovic et al.'s study so as to include dynamo reversals.We observed interesting reversals for the insulating boundary condition; here the dipolar mode does not flip, but higher Fourier modes flip.We will show that the set of reversing and non-reversing modes belong to one of the solutions of the group-theoretic model.Kutzner and Christensen <cit.> performed direct numerical simulations (DNS) of MHD equations, and observed transitions between dipolar and multipolar regime accompanied by reversals of the dipolar field. Oruba and Dormy <cit.> showed that such transitions from the static dipolar to the reversing multipolar dynamo are due to balance between the inertial, viscous and Coriolis forces.These investigations raise interesting question on the reversing and non-reversing modes in a dynamo.The general symmetry classes of our group-theoretic arguments would be useful for such analysis.The outline of the paper is as follows: in Sec. <ref>, we discuss thegoverning equations and the boundary conditions of the system. In Sec. <ref>, we describe the symmetries of dynamo reversals.We extend these arguments to magneto-convection in Sec. <ref>. In Sec. <ref> we show that the reversing and non-reversing modes in a dynamo reversals observed in a DNS belong to one of the classes of group-theoretic model. In Sec. <ref> we discuss the symmetry classes of some other dynamo reversals.We conclude in the Sec. <ref>.§ EQUATIONS AND METHODThe governing equations of a dynamo are∂𝐮/∂ t + (𝐮·∇)𝐮=- ∇ P + (𝐣×𝐛) + ν∇^2 𝐮 + 𝐟,∂𝐛/∂ t + (𝐮·∇)𝐛= (𝐛·∇)𝐮 + η∇^2 𝐛,∇·𝐮= 0,∇·𝐛= 0 .where 𝐮 is the velocity field, 𝐛 is the magnetic field, P is the pressure field, 𝐟 is the external mechanical forcing, 𝐣=(∇×𝐛)/μ_0 is the current density, ν is the kinematic viscosity, and η is the magnetic diffusivity. We consider the flow to be incompressible [see Eq. (<ref>)], and set the fluid density ρ to unity.Two important parameters used in a dynamo literature are the Reynolds number Re and magnetic Reynolds number Rm, which are defined asRe=UL/ν, Rm=U L/η,where U = √(2 E_u) is the root-mean-square velocity (E_u = the total kinetic energy), and L is the characteristic length scale of the flow, which is defined asL= 2π∫ k^-1 E_u(k) dk/∫ E_u(k) dk.Here the one-dimensional kinetic energy spectrum E_u(k) is defined as the energy contents of a shell of radius k and unit width:E_u(k) = ∑_k-1 < |𝐤^'| ≤ k1/2 |𝐮̂(𝐤^')|^2.The one-dimensional magnetic energy spectrum is defined similarly.One other important dimensionless parameter for dynamo ismagnetic Prandlt number, which is defined asPm = ν/η = Re/Rm. For the analysis ofthe large-scale structures and dynamo reversals, it is convenient to work in the Fourier space with Fourier basis function exp(i k· r):u=∑_k_x,k_y,k_zû (k_x,k_y,k_z) exp(ik · x) , b=∑_k_x,k_y,k_zb̂ (k_x,k_y,k_z) exp(ik · x),where k_x,k_y,k_z are integers for a (2π)^3 box; they take both positive and negative values.In this representation,the MHD equations ared/dtû_m(𝐤) =-i k_n ∑_𝐤=𝐩+𝐪û_n(𝐪) û_m(𝐩) + i k_n ∑_𝐤=𝐩+𝐪b̂_n(𝐪) b̂_m(𝐩) - ν k^2 û_m(𝐤) - i k_m p̂(𝐤) + f̂_m(k), d/dtb̂_m(𝐤) =-i k_n ∑_𝐤=𝐩+𝐪b̂_n(𝐪) û_m(𝐩) + i k_n ∑_𝐤=𝐩+𝐪û_n(𝐪) b̂_m(𝐩) - η k^2 b̂_m(𝐤), k_mû_m(𝐤) =0 , k_mb̂_m(𝐤) =0.where û_m(𝐤), b̂_m(𝐤), f̂(𝐤), and p̂(𝐤) are the Fourier transforms of the velocity, magnetic, external force, and pressure fields respectively. We employ Taylor-Green (TG) forcing: 𝐟 = F_0[sin(k_0 x)cos(k_0 y)cos(k_0 z); - cos(k_0 x)sin(k_0 y) cos(k_0 z); 0 ]where F_0 is the forcing amplitude, and k_0 is the wavenumber of that forcing.We choose k_0=1.For the velocity field we employ the free-slip or stress-free boundary condition at all the six sides of thebox:u_⊥ = 0;∂ u_∥/∂ n=0, where n̂ is the normal to the surface, and u_⊥ and u_∥ are respectively the velocity components normal and parallel to the wall. For example, if we consider the boundary condition at the wall at z=0 (xy plane), the normal vector n̂ will be the -ẑ vector. So, in this case, u_⊥ = -u_z = 0,∂ u_x/∂ z = 0, and ∂ u_y/∂ z = 0. For the magnetic field, we employ the insulating boundary condition at all the walls <cit.>:b_∥ = 0;∂ b_⊥/∂ n=0, where b_⊥ and b_∥ are respectively the components of the magnetic field, normal and parallel to the wall. We call this insulating wall because the current𝐣 = (∇×𝐛)/μ_0 on the surface is zero.The aforementioned boundary conditions are satisfied forthe following basis functions for u and b:u_x=∑_k_x,k_y,k_z8 û̂_x(k_x,k_y,k_z) sin(k_xx) cos(k_yy) cos(k_zz), u_y=∑_k_x,k_y,k_z8 û̂_y(k_x,k_y,k_z) cos(k_xx) sin(k_yy) cos(k_zz), u_z=∑_k_x,k_y,k_z8 û̂_z(k_x,k_y,k_z) cos(k_xx) cos(k_yy)sin(k_zz) , b_x=∑_k_x,k_y,k_z8 û̂_x(k_x,k_y,k_z) cos(k_xx) sin(k_yy) sin(k_zz), b_y=∑_k_x,k_y,k_z8 b̂̂̂_y(k_x,k_y,k_z) sin(k_xx) cos(k_yy) sin(k_zz), b_z=∑_k_x,k_y,k_z8 b̂̂̂_z(k_x,k_y,k_z) sin(k_xx) sin(k_yy)cos(k_zz) ,where, in a π^3 box,k_x, k_y, k_z are positive integers including zero, and û̂, b̂̂̂ represent the basis functions for the free-slip and insulating boundary conditions. We choose the above basis functions for our simulation.We refer to the above as free-slip, insulating basis function, for which we follow the conventions and definitions of FFTW <cit.>.Using 2 i sin( k · r) = exp( k · r) -exp(- k · r) and 2cos( k · r =exp( k · r) -exp(- k · r), we can relate û̂_i(k_x,k_y,k_z) and b̂̂̂_i(k_x,k_y,k_z) with û (± k_x, ± k_y,± k_z) and b̂ (± k_x, ± k_y,± k_z) of Eqs. (<ref>,<ref>). For example, û_x(-k_x,-k_y,-k_z) = - i û̂_x(k_x,k_y,k_z), û_x(-k_x,k_y,-k_z) =û̂_x(k_x,k_y,k_z), etc. These properties enable us to use 1/8th Fourier modes (of Eqs. (<ref>,<ref>)) for a pseudo-spectral simulation. However, in our simulation, we impose the above condition in each time step, and time-step all the Fourier modes, i.e., û (± k_x, ± k_y,± k_z) and b̂ (± k_x, ± k_y,± k_z) of Eqs. (<ref>,<ref>).The Fourier decomposition of the MHD equations yield a set of coupled nonlinear ordinary differential equations (ODEs) given by Eqs. (<ref>,<ref>). These equations are often solved numerically using pseudo-spectral method, as done in this paper (see Sec. V).It is also customary to truncate the Fourier expansion drastically and focus only on a limited set of modes. Hence we obtain a small set ofnonlinear ODEs that can be analysed using the tools of nonlinear dynamics.The dimension of the system, and consequently, its complexity will depend on the order of truncation. Quantities like Lyapunov exponents can be used to study properties of such systems, e.g., the transition between deterministic and chaotic behaviour. Following the above procedure, Verma et al. <cit.> constructed a truncated six-model using the Fouier modes u(1,0,1),u(0,1,1),u(1,1,2),b(1,0,1),b(0,1,1),b(1,1,2). These Fourier modes are part of an interacting triad.The above model exhibits dynamo transition, but no chaos.In the next section, we discuss the symmetries of the MHD flows; these symmetries provide valuable insightsinto the dynamo reversals. § SYMMETRIES OF THE MHD EQUATIONS AND PARTICIPATING MODESThe structure of the MHD equations <cit.> reveal that the equations are invariant under the transformation u → u andb → -b.However, in several dynamo simulations and models, only some modes of the velocity and magnetic fields reverse during a dynamo reversal, e.g., in Gissinger <cit.>, the dipolar component of the magnetic field reverses, but not the quadrupolar component. The rules for such reversals can be derived using the symmetry properties of the MHD equations in the Fourier space.The arguments are somewhat simpler for the velocity field only, which appears in RBC.In Appendix A, we discuss the symmetry properties for RBC.For the basis functions of Eqs. (<ref>-<ref>), we classify the Fourier modes according to the parity of the modes (k = (k_x, k_y, k_z)).For the same, we divide each Fourier component, k_x, k_y, k_z, according to their parities—even, represented by e, and odd, represented by o. Let us denote the parity function by P. To illustrate, P(3) = o, but P(4) = e.Thus, for MHD, the Fourier modes of the velocity field (under the free-slip boundary condition) is classified into eight classes: E= (eee), odd O=(ooo), and mixed modes—M_1=(eoo), M_2 =(oeo), M_3=(ooe), M_4=(eeo), M_5=(oee), M_6= (eoe). The corresponding classes for the magnetic Fourier modes are labeled as E, O, M_1, M_2, M_3, M_4, M_5, and M_6 respectively.To illustrate, û_x(1,1,1) ∈ O , û_x(2,2,2) ∈ E, and b̂_x(2,1,1) ∈M_1. In the following discussion, we will derive a group structure for the above modes that will help us identify the reversing and nonreversing Fourier modes. | http://arxiv.org/abs/1705.09630v1 | {
"authors": [
"Riddhi Bandyopadhyay",
"Mahendra K. Verma"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170526155847",
"title": "Discrete Symmetries in Dynamo Reversals"
} |
[email protected] [email protected] [email protected] of Physics, Florida State University, Tallahassee, Florida 32306-4350, USA We present phase diagrams, free-energy landscapes, and order-parameter distributions for a model spin-crossover material with a two-step transition between the high-spin and low-spin states (a square-lattice Ising model with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions) [P. A. Rikvold et al., Phys. Rev. B 93, 064109 (2016)]. The results are obtained by a recently introduced, macroscopically constrained Wang-Landau Monte Carlo simulation method [C. H. Chan, G. Brown, and P. A. Rikvold, Phys. Rev. E 95, 053302 (2017)]. The method's computational efficiency enables calculation of thermodynamic quantities for a wide range of temperatures, applied fields, and long-range interaction strengths. For long-range interactions of intermediate strength, tricritical points in the phase diagrams are replaced by pairs of critical end points and mean-field critical points that give rise to horn-shaped regions of metastability. The corresponding free-energy landscapes offer insights into the nature of asymmetric, multiple hysteresis loops that have been experimentally observed in spin-crossover materials characterized by competing short-range interactions and long-range elastic interactions. Phase diagrams and free-energy landscapes for model spin-crossover materials with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions P. A. Rikvold December 30, 2023 ========================================================================================================================================================================§ INTRODUCTIONSpin-crossover (SC) materials are molecular crystals in which the individual molecules contain transition metal ions that can exist in two different spin states: a low-spin ground state (LS) and a high-spin excited state (HS). Molecules in the HS state have larger volume and higher effective degeneracy than those in the LS state <cit.>. Due to its higher degeneracy, crystals of such molecules can be brought into a majority excited HS state by increasing temperature, changing pressure or magnetic field, electrochemical stimuli, or exposure to light <cit.>. The size difference between the HS and LS molecules causes local elastic distortions that lead to effective long-range elastic interactions mediated by the macroscopic strain field <cit.>. In addition to such long-range interactions, the materials will also typically have local interactions caused by, e.g., quantum-mechanical exchange or geometric restrictions. These intermolecular interactions may cause first-order phase transitions that can render the HS state metastable and lead to hysteresis when exposed to time-varying fields <cit.>. In the case of optical excitation into the metastable phase, this phenomenon is known as light-induced excited spin trapping (LIESST) <cit.>. The metastable properties in combination with the SC materials' sensitivity to a wide range of external stimuli make them promising candidates for applications such as switches, displays, memory devices, sensors, and actuators <cit.>.In the SC literature, the phase transitions caused by the short-range and long-range interactions are often discussed using an Ising-like pseudospin formulation, in which the HS state is represented as s=+1 and the LS state as s=-1. This is the representation we will use in this paper. It has the advantage of a high degree of symmetry, and it enables easy reference to studies of other Ising-like models. To minimize the strain energy, the elastic long-range interaction favors different molecules being in the same state (LS-LS or HS-HS). In this pseudospin language it is therefore called ferromagnetic-like, or simply ferromagnetic. The short-range interactions depend on the particular material and may either be ferromagnetic-like, or they may favor neighboring molecules in opposite states (LS-HS), which is analogously called antiferromagnetic-like, or simply antiferromagnetic. We emphasize that this nomenclature only represents an analogy and does not imply a magnetic origin of the interactions. In the remainder of this paper, we will use the simplified terms, ferromagnetic and antiferromagnetic, for interactions that favor uniform and checkerboard spin-state arrangements, respectively.If the short-range interaction is ferromagnetic, it has been found that adding even a very weak long-range interaction causes the universality class of the critical point to change from the Ising class to the mean-field class <cit.>. On the other hand, if the short-range interaction is antiferromagnetic, the critical line will terminate at a certain point, with the appearance of metastable regions in the phase diagram, bounded by sharp spinodal lines <cit.>. Then, with sufficiently strong long-range interaction, new mean-field critical points emerge in the phase diagrams – a phenomenon which is not predicted by simple Bragg-Williams mean-field theory <cit.>. These new mean-field critical points also become the end points for the spinodal lines bounding the metastable regions.In some SC materials, the transition between the LS and HS phases proceeds as a two-step transition via an intermediate phase<cit.>, giving rise to complex, asymmetrical hysteresis loops. In the case of Fe(II)[2-picolylamine]_3Cl_2 ·Ethanol <cit.>, x-ray diffraction has revealed an intermediate phase, characterized by long-range order on two interpenetrating sublattices with nearest-neighbor molecules in different states (HS-LS) <cit.>. Several of these experimental results were recently reviewed <cit.>. This situation can be modeled by an Ising-like model with antiferromagnetic nearest-neighbor interactions. Various mean-field approximations to this model have been considered, both without <cit.> and with <cit.> a long-range ferromagnetic term.Recently, Rikvold et al. used standard importance-sampling Monte Carlo (MC) simulations to obtain phase diagrams and hysteresis curves for such an Ising model with nearest-neighbor antiferromagnetic interactions and ferromagnetic long-range interactions approximated by a mean-field equivalent-neighbor (Husimi-Temperley) term <cit.>. (See Hamiltonian in Sec. <ref>.) To locate the various transition lines in the phase diagram, this method requires separate simulations for different values of temperature, field, and long-range interaction strength. This procedure is very computationally intensive, and phase diagrams could therefore only be drawn for three different interaction strengths. In the present paper, weprovide detailed phase diagrams for this system with a range of different long-range interaction strengths, from quite weak to quite strong. In addition to phase diagrams, we also obtain free-energy landscapes and order-parameter probability densities in terms of the model's two order parameters, magnetization (M) and staggered magnetization (M_s). To obtain these results with a reasonably modest computational effort, we use a recently proposed, macroscopically constrained Wang-Landau (WL) MC algorithm <cit.>. With this method,a simple analytic transformation of the system energy E enables us to extract results for any combination of temperature, applied field, and long-range interaction strength from one single, high-precision simulation of the joint density of states (DOS), g(E,M,M_s), for a simple square-lattice Ising antiferromagnet in zero field. The details of how to use the algorithm to calculate the joint DOS, and how to extract from it free-energy landscapes, order-parameter probability densities, and phase diagrams are given in our recent papers, Ref. <cit.>. Here, we concentrate on the physical aspects of this model SC material and, in particular, their dependence on the long-range interaction strength. In the process, we also obtain improved estimates for the positions and shapes of the first-order coexistence lines in the phase diagrams.Studies of Ising models with long-range interactions have a long history. Some notable examples are work on Ising models with weak long-range interactions by Penrose, Lebowitz, and Hemmer<cit.>, and with long-range lattice coupling by Oitmaa and Barber <cit.>. Herrero studied small-world networks with both ferromagnetic <cit.> and antiferromagnetic interactions <cit.>. Hasnaoui and Piekarewicz <cit.> recently used an Ising model with Coulomb long-range interaction to simulate nuclear pasta in neutron stars.It should also be mentioned that the Ising model with long-range interactions decaying as r^-(d+σ) with d=1,2,3 and 0<σ<d/2 was studied by Luijten and Blöte <cit.>, and the effect of long-range interactions on phase transitions in short-range interacting systems were studied by Capel et al. <cit.>.The remainder of this paper is organized as follows. In Sec. <ref> we present the Ising-like model Hamiltonian and its interpretation as a model for SC materials. In Sec. <ref> we briefly discuss the macroscopically constrained WL algorithm and present the analytic energy transformation that enables us to extract data for arbitrary model parameters from a single simulated joint DOS. We also show how constrained partition functions are obtained from the joint densities of states, and how the partition functions lead to free-energy landscapes and order-parameter probability densities. Sec. <ref> contains our main results: phase diagrams, as well as probability densities and free-energy landscapes at selected phase points. All these are obtained for several values of the long-range interaction strength, ranging from quite weak to quite strong, and producing a number of topologically different phase diagrams. Section <ref> contains a brief summary and conclusions. Details of our estimates of finite-size and statistical errors are given in Appendix <ref>.§ 2D ISING-ASFL MODEL To approximate a SC material with antiferromagnetic-like nearest-neighbor interactions and ferromagnetic-like elastic long-range interactions, we here employ the model introduced by S. Miyashita and first used in Refs. <cit.>. This is a L × L square-lattice nearest-neighbor Ising antiferromagnet with ferromagnetic equivalent-neighbor (aka Husimi-Temperley) interactions. It is defined by the Hamiltonian,ℋ = J ∑_⟨ i,j ⟩s_is_j-HM -A/2L^2M^2 ,with J>0. We name it the two dimensional Ising Antiferromagnetic Short-range and Ferromagnetic Long-range (2D Ising-ASFL) model. The first two terms constitute the Wajnflasz-Pick Ising-like model <cit.>, in which the pseudo-spin variable s_i denotes the two spin states at site i (-1 for LS and +1 for HS), and M=∑_is_i is the pseudomagnetization. The effective field term,H =1/2 (k_ B T ln r - D) ,contains D > 0, which is the energy difference between the HS and LS states, and r, which is the ratio between the HS and LS degeneracies. T is the absolute temperature, and k_ B is Boltzmann's constant. (Changing the temperature in the physical SC system therefore corresponds to a combined change in temperature and effective field in this pseudospin model. See Figs. 5(a) and 8 of Ref. <cit.>.)The last term in Eq. (<ref>) approximates the elastic long-range interactions of the SC material as in Refs. <cit.>. Since it lowers the energy of more uniform spin-state configurations (mostly +1 or mostly -1) in a quadratic fashion, it is a ferromagnetic term. Throughout the paper, temperature (T), energy (E), magnetic field (H), and long-range interaction strength (A), will be expressed in dimensionless units (|J|=k_B=1). The order parameters of this model are magnetization (M) and staggered magnetization (M_s). They can be normalized as m=M/L^2 and m_s=M_s/L^2. If we break the two-dimensional square lattice into two sublattices (A and B), like the black and white squares on a chessboard, m and m_s can be expressed in terms of the normalized magnetizations (m_A, m_B) of these two sublattices asm=(m_A+m_B)/2m_s =(m_A-m_B)/2 .The usual order parameter for SC materials is the proportion of HS molecules, n_ HS, which is related to the pseudospin variables asn_ HS = ( m + 1 )/2.The equilibrium (stable) and metastable phases at zero temperature were obtained from the Hamiltonian by simple ground-state calculations in <cit.>. We briefly repeat the results here for convenient reference, also introducing the following short-hand notation for the low-temperature ordered phases:antiferromagnetic (which is doubly degenerate), is called AFM;ferromagnetic with majority of s_i = +1, is called FM+;and ferromagnetic with majority of s_i = -1, is called FM-. A < 8: AFM is stable for -4 + A/2 < H < 4 - A/2, metastable against transition to FM+ for 4 - A/2 < H < 4, andmetastable against transition to FM- for -4 < H < -4 + A/2. FM+ is stable for H > 4 - A/2, and metastable for transition to AFM or FM- for 4 - A < H < 4 - A/2. FM- is stable for H < -4 + A/2, and metastable for transition to AFM or FM+ for -4 + A/2 < H < -4 + A.A > 8: AFM is never the stable ground state, but it is metastable for -4 < H < 4. FM+ is stable for H > 0 and metastable for 4 - A < H < 0. FM- is stable for H < 0 and metastable for 0 < H < -4 + A.§ METHOD §.§ Obtaining joint density of states The resultspresented in this paper are all based on the joint DOS, g(E,M,M_s), determined once for H=A=0, which corresponds to a simple square-lattice Ising antiferromagnet. Using this, the joint DOS for any arbitrary value of (H,A) can be obtained byg(E(H,A),M,M_s) = g(E(0,0),M,M_s)whereE(H,A) = E(0,0)-HM- AM^2/2L^2 .Note that this is an alternative, but equivalent way to express the content of Eq. (10) in Ref. <cit.>. This result is based on the fact that all the microstates are equally shifted in energy when a field-like parameter couples to a function of the global property M, as shown inEq. (<ref>). With the joint DOS, all thermodynamic quantities can be calculated, as demonstrated in <cit.>. From g(E,M,M_s) at different (H,A), we can obtain g(E,M) and g(E,M_s), as shown in Ref. <cit.>.To obtain an accurateg(E,M,M_s) at H=A=0, the macroscopically constrained WL method is used <cit.>. With the help of simple combinatorial calculations in the (M,M_s) space, the method converts what would otherwise be a time-consuming multi-dimensional random walk in the (E,M,M_s) space into many independent, one-dimensional random walks in E, each constrained to a fixed value of (M,M_s). Through further, symmetry based simplifications <cit.>, the method can obtain an accurate estimate of g(E,M,M_s) in a relatively short time.As the details of how to arrive at these results have already been presented in <cit.>, here we simply focus on the physics of the model SC material as A is changed. All the phase diagrams, free-energy landscapes, and probability densities shown in Sec. <ref> areobtained with L=32. §.§ From joint density of states to thermodynamic quantitiesWe define the constrained partition function of any macrostate (m,m_s) asZ_m,m_s=∑_E g(E,m,m_s) e^-E/T .The overall partition function of the system is thenZ_all=∑_m,m_s Z_m,m_s .The joint probability of finding the system in a macrostate (m,m_s) isP(m,m_s)Δ m Δ m_s=Z_m,m_s/Z_all ,where Δ m, Δ m_s are the order-parameter step sizes, both chosen to be the same value, around 0.03. The free energy of macrostate (m,m_s) isF(m,m_s)= -T ln Z_m,m_s .We will plot these quantities in terms of (m_A,m_B) which have a one-to-one relation with (m,m_s)(see Eqs. (<ref>) and (<ref>)). Summing over the contributions of the joint probability (Eq. (<ref>)) in one direction, we obtain the marginal probability densities asP(m)Δ m= ∑_m_s Z_m,m_s/Z_all P(m_s)Δ m_s = ∑_mZ_m,m_s/Z_all .With these densities, we can calculate the expectation values of the order parameters and other quantities. We can express the free energy in terms of one order parameter asF(m)=-T ln∑_m_s Z_m,m_s F(m_s) =-T ln∑_mZ_m,m_s . The presence of the long-range interaction induces metastable phase regions in the phase diagrams.A very important point is thatwhen we consider values of (T,H,A) lying in those regions, the stable phase will be the phase that has larger total area in the marginal probability density, rather than the phase that shows the higher peak. Systems lying on the coexistence line between two phases will have equal areas in the marginal probability density.In a free-energy contour plot or joint probability density plot, against m and m_s (or against m_A and m_B), the region around (m,m_s)=(1,0) [or (m_A,m_B)=(1,1)]corresponds to the FM+ phase. Similarly, the region around (m,m_s)=(-1,0) [or (m_A,m_B)=(-1,-1)] corresponds to the FM- phase. The region around (m,m_s)=(0,1) [or (m_A,m_B)=(1,-1)] corresponds to the AFM+ phase, and the region around (m,m_s)=(0,-1) [or (m_A,m_B)=(-1,1)] corresponds to the AFM- phase. Finally, the region around (m,m_s)=(0,0) [or (m_A,m_B)=(0,0)] corresponds to the disordered phase. However, these are just the most extreme cases. Some AFM phases have significant ferromagnetic properties, and some FM phases may be quite disordered.In our model, for a particular(T,H,A) triple, if the system can exist as a disordered phase, it cannot exist as an AFM phase, and vice versa. However it may happen that a disordered phase shows strong AFM properties. Changing (T,H,A) may let the system change from one phase to another through a continuous phase transition, as it crosses the critical line between the two phases. In the Ising-ASFL model, a critical line only exists between the disordered phase and the AFM phase. The phase boundary between the ferromagnetic phase and the disordered phase is a coexistence line, and it ends with a mean-field critical point for sufficiently strong long-range interaction A. This critical point is located where the two spinodal lines meet. The expectation values of the two order parameters can be obtained easily as⟨ m ⟩ = ∑_m m P(m) Δ m⟨ m_s⟩ = ∑_m_s m_s P(m_s) Δ m_s .As the two AFM phases always exist in pairs and the probability of finding the system in both are the same, ⟨ m_s⟩=0.As ⟨ m_s⟩=0, we define the corresponding fourth-order Binder cumulant as <cit.>, u_m_s=1-⟨ m_s^4⟩/3 ⟨ m_s^2⟩^2 .Here we only define the cumulant for the order parameter m_s, as only the critical line will be located by the cumulant. When we take the ensemble average, we have to exclude all the phase points that belong to the metastable FM+ or FM- phase. That is, when we look at F(m), if we find more than one minimum (i.e. more than one phase are found), we neglect the states that have values of |m| greater than the separating value of m. The critical line in this model is commonly accepted to be in the Ising universality class <cit.>, which (assuming isotropy, periodic boundary conditions, and a square shape as in the present study) has a cumulant fixed-point value of 0.6106924(16) <cit.>. We therefore locate the critical line by finding the phase point within a temperature range where the cumulant is close to 0.61, and does not deviate from 0.61 by more than 0.01. The resulting critical line for A=0 is included in Fig. <ref> together with the analytically approximated critical line for the pure square-lattice Ising antiferromagnet in the thermodynamic limit from Ref. <cit.>. Withinthe resolution of this figure, our L=32 datacoincide with this highly accurate approximation.The variance of the order-parameter m, which is proportional to the susceptibility times the temperature,var(m)=χ_mT = L^2( ⟨ m^2⟩ - ⟨ m ⟩^2 ) ,is considered as we use its maxima to separate the FM± phases from the disordered and AFM phases. All the coexistence lines that we show are located by using this quantity. Note that this quantity is very difficult to measure through importance-sampling MC, while our approach can directly calculate it using g(E,M,M_s). Further details on the method are given in Ref. <cit.>. In next section, we consider the phase diagrams for different values of A and study selected phase points. These are the main results of the present paper. Notice that all the phase diagrams are symmetric about the T axis, with an exchange between FM+ and FM-. For A=0, the model reduces to the standard square-lattice antiferromagnetic Ising model <cit.>.§ PHASE DIAGRAMS §.§ Weak long-range interaction, A=1,4 It is reasonable to assume that adding a ferromagnetic long-range interaction A to the pure antiferromagnet must favor the appearance of the ferromagnetic phases, and thus push the critical line towards lower values of |H|. Figure <ref> supports this assumption. Moreover, the critical lines also terminate at lower |H| and higher T for larger A. The phase diagrams in Fig. <ref> show that the critical lines end with the appearance of a metastable region in the phase diagram, and that the metastable region grows as A increases. All phase diagrams shown in this paper are symmetric under simultaneous reversal of H and m. Error bars including statistical and finite-size errors are included with every data point in this and all subsequent phase diagrams. With the exception of Fig. <ref>, they are everywhere smaller than the symbol size. A discussion of how the errors were estimated is found in Appendix <ref>. Introducing the long-range interaction A with the M^2 term makes it much weaker than the HM term for small M, so that the long-range interaction effect is negligible when H and A are small, and so it does not significantly affect the critical temperature near H=0. On the other hand, when we increase H, the M^2 term will eventually be larger than the M term, and finally causes a local free-energy minimum to show up in the FM+ region, corresponding to a metastable FM+ phase region in the phase diagram (Figs. <ref> (a) and (b)). A new FM+ peak also appears in thejoint probability density (P(m_A,m_B)) and marginal probability densities (P(m) and P(m_s)). One peak may be much smaller than the other, such that it may not be easy to discover the presence of metastability through looking at the probability density (Figs. <ref> (b) and (d)). Notice that although one phase may have much smaller probability density than the other, the lifetimes for these metastable phasesincrease exponentially with system volume, e^cL^2 for a two-dimensional system, so that they are still macroscopic, and thus cannot be neglected <cit.>.The AFM and FM+ phases are separated by the coexistence line in the metastable region, and we observe that when T is low, the coexistence line is a practically straight line at constant H in the phase diagram. Note that this result is different from Rikvold et al.'s former result <cit.> for A=4, which indicates a reentrant behavior of the coexistence line at low T. This discrepancy is probably due to incomplete ergodicity in the importance-sampling MC with mixed initial conditions used in Ref. <cit.>.For any point lying on that straight vertical segment of the coexistence line, as in Fig. <ref> (a)-(c), the coexisting AFM phases and the FM+ phase are located at their extreme locations, i.e., m=+1, m_s=±1.Increasing T bends the coexistence line toward lower |H| values. Simultaneously, the AFM phases and the FM+ phase move away from the extreme positions and towards each other, as shown in Fig. <ref> (d)-(f). The coexistence line finally joins the critical line at the tricritical point, where the two AFM phases and the FM+ phase become indistinguishable at the continuous phase transition point. Figure <ref> (g)-(i) represent a point lying on the coexistence line, below the tricritical point. We see from the joint probability density in Fig. <ref> (g) that the ferromagnetic phase and the AFM phases are coalescing. However, the marginal probability along the m axis in Fig. <ref> (h) still has two peaks. We therefore regard the system as in AFM/FM+ coexistence, with this small system fluctuating easily between the two phases. Extrapolation of the end points of the two spinodal lines gives the merging temperature, which corresponds to the tricritical point.When the two spinodal lines merge, the distance between them (Δ H) varies against temperature as <cit.>(Δ H)^2/3∝ T_x-T ,where T_x represents the tricritical or critical temperature, where the coexistence line ends. After obtaining the tricritical temperature, we can estimate the tricritical field as the average of the extrapolation points of the two spinodal lines.Figure <ref> (j)-(l) show data at the tricritical point for A=4, where the AFM phases and the FM+ peak finally join together into one phase.§.§ Medium long-range interaction, A=6,7,8 As mentioned above for small A, moving along the coexistence line toward the critical line, one approaches a tricritical point, where the two AFM phases and the FM phase become indistinguishable. Below the tricritical temperature, the three phases are distinct. Then it is reasonable to expect that, if A is big enough, the two AFM phases may combine into one disordered phase at a lower T than the one where they further combine with the FM phase. In this scenario, we will find that the critical line, which represents the AFM/disordered phase transition, intersects the coexistence line at a critical end-point, and new metastable regions (horn regions) emerge in the phase diagram as shown for A=6,7, and 8 in Fig. <ref>.Figurte <ref> is a closer look at the horn region for A=7. The coexistence line separates the FM phase from the AFM phases at low T. After passing through the critical end-point, it separates the FM phase from the disordered phase. At a higher T, it ends in a mean-field critical point, where the disordered and FM phases become indistinguishable.Figure <ref> shows the case near the critical end-point. As this point is the intersection of the critical line and the coexistence line, it has properties of both lines. Since it is on the coexistence line, the combined AFM/disorderedphase is equally probable as the FM+ phase, as shown in (c) and (e). Since it is on the critical line, the AFM peaks are connected through the middle disordered region as it corresponds to a continuous phase transition between the AFM phases and the disordered phase (shown in (b)). For the marginal probability density function P(m_s), if we remove the contribution from the FM+ phase as shown in the inset, the height ratio between a AFM peak to the central point in the middle between the two peaks is around 26/1, which is close to the established value of about 22/1 <cit.>.Figure <ref> shows a point close to the mean-field critical point at (H,T)=(0.566,2.63) for A=7, where we see that the two peaks in P(m) have coalesced into one single peak. We note that the position of the critical point found here is consistent with the one found in Ref. <cit.> by importance-sampling MC with system sizes up to L=1024, H=0.561(1) and T=2.61(1).Figure <ref> shows results as we move along the coexistence line to a point near the mean-field critical point. The disordered phase peak gradually contracts to m_s=0 as the AFM fluctuations weaken (refer to the first row of the figure), and the FM+ peak slowly merges with the disordered phase peak until only one peak is left in the marginal probability along the m direction (refer to the second row of the figure).We see that the two peaks in the marginal probability density, P(m), along the FM axis, which correspond to two different phases, become less sharp and merge. Note that the joint probability density in (g) seems to show only one peak, but after summing up all the contributions from different m_s, the marginal probability density in (h) shows two peaks, and we still regard them as two phases even though they are strongly connected by fluctuations.Figure <ref> shows the results observed at four points that are equidistant from the coexistence line, but lie in four different phase regions, with the critical end-point nearly at the center, as shown by the four red dots in Fig. <ref>. Parts (a)-(b) show a point lying in region III, which has stable disordered phase and metastable FM+ phase;(c)-(d) show a point lying in region IV, which has stable AFM phase and metastable FM+ phase;(e)-(f) show a point lying in region V, which has stable FM+ phase and metastable disordered phase; and(g)-(h) show a point lying in region VI, which has stable FM+ phase and metastable AFM phase.The phase diagram for A=7 is well suited for comparison with a number of experimental results for SC materials that show asymmetric, two-step thermal hysteresis loops <cit.>. Such a two-step loop, obtained directly from the joint probability density, P(m,m_s), along a path between (H,T) = (1.0,2.5) and (-1.5,1.75) in Fig. <ref> is shown in Fig. <ref>. This path corresponds to the parameters ln r=20/3 and D=44/3 in Eq. (<ref>). The narrow high-temperature loop corresponds to the crossings of the spinodal lines in the horn region, while the wide low-temperature loop corresponds to crossings of the spinodals in the negative-H region. In order to calculate these hysteresis loops, at each point along the hysteresis path we first located the local maximum in F(m) that separates the two phases. Then, ⟨ m ⟩ and⟨ |m_s| ⟩ were obtained by summing over P(m,m_s) as described in Sec. <ref>. Although we do not show other examples of hysteresis loops here, we emphasize that our macroscopically constrained WL method enables the calculation of such loops for any value of A and any choice of hysteresis path, solely based on the DOS data for the pure Ising antiferromagnet, without any further MC simulations. The hysteresis loop shown here is fully consistent with the one obtained by importance-sampling MC simulations for the same parameters in Ref. <cit.> <cit.>. The only significant differences are the slopes of the ⟨ |m_s| ⟩ curve where the path crosses the critical line, which in both cases are due to finite-size effects. On the other hand, finite-size effects in the positions of the spinodals are negligible, as discussed in Appendix <ref>. The phase diagrams for A=6,7,8 in Fig. <ref>show several additional, noteworthy features. First, the phase diagrams shown are symmetric about the T axis, with an exchange between FM+ and FM-. This is because the FM+ spinodal line is just touching the T axis at T=0 for A=4 (Fig. <ref>) (c). Further increases of A beyond 4 will make a FM- spinodal line show up in the positive H region. Thus, the strong AM^2/(2L^2) causes a FM- metastable region to appear in the positive H field region. Figure <ref>(c)-(d) illustrate the case of a point lying on the coexistence line between the FM+ phase and AFM phases, inside the FM- metastable region. The small drop in the free energy in (d) near m=-1 indicates the metastable FM- phase. Figure <ref>(a)-(b) illustrate a point at H=0 and at a low T, where both FM phases are metastable.Second, observe that the coexistence lines turn toward stronger |H| when approaching the mean-field critical points (Fig. <ref>). This isbecause the disordered phase is more favorable than the FM phases at high T, so a stronger |H| field is required to balance this effect.Third, when A increases, the mean-field critical temperature also increases, which makes the area of the horn region increase. This is because the ferromagnetic effects increase with A according to the Hamiltonian (<ref>), so a stronger disordering effect (higher temperature) is required to balance it. Fourth, the coexistence line moves toward lower |H| as A increases, which makes the stable AFM region shrink and the stable FM regions grow. This is because strong -AM^2/(2L^2) stabilizes the ferromagnetic phases at lower |H|. The coexistence line for A=8 is at H=0 for low T (Fig. <ref>(c)). In that case, the two AFM phases and the two FM phases are equally probable as shown in Fig. <ref>(a). When T increases to a high enough value, disorder effects start to show up, making |m_s| decrease from 1 (Fig. <ref>(c)). At low |H| and high T, the disordered phase is preferred over the ferromagnetic phase. This effect starts to show up before reaching the critical temperature, making the coexistence line turn away from H=0before it crosses the critical line, as shown in Fig. <ref>(a).Fifth, the FM- spinodal line continues moving toward higher T when A increases as the ferromagnetic phase is getting stronger. At A=8, the FM- spinodal line has moved above the critical line. This produces a region (Fig. <ref>(a)) that is stable in the FM+ phase, and metastable in both the FM- and disordered phases (region VIII), and another region that is stable in the disordered phase, and metastable in both FM± phases (region VII).Observe from Figs. <ref> and <ref>(a) that the coexistence line makes a relatively large bend at the critical end-point. This is because after passing through this point, the AFM phase changes to the disordered phase, which is favored at high temperature, making the coexistence line have a smaller slope. Therefore, a relatively large bend in the coexistence line is found at the critical end-point. This agrees with the previously observed result that d^2H/dT^2 along the coexistence line reaches a maximum at the critical end-point <cit.>. Note that the location of the coexistence lines given byRikvold et al.'s Ref. <cit.> is different from the current result for A=7. The former result may be due to incomplete ergodic sampling by the mixed-start importance-sampling MC method used in that work to locate the coexistence lines. This might also affect experimental attempts to accurately detect phase coexistence. Further analysis of the discrepancy between the importance-sampling MC using the mixed-start method and the present method in locating coexistence lines is in progress <cit.>.§.§ Transitional long-range interaction strength A=8.1,9From the ground-state analysis in Ref. <cit.>, A=8 is the dividing line for the stable phase at T=0. For A>8, the stable phase at T=0, H>0 can only be the FM+ phase. Figures <ref>(a)-(d) show that increasing A from 8 to 8.1 makes the FM phases overtake the AFM phases and become the stable phases below the critical line. The Bragg-Williams mean-field approximation <cit.> also suggests that phase diagrams having A>8 belong to the same group (large long-range interaction group) and possess the same nature <cit.>. While Ref. <cit.> has already pointed out that the Bragg-Williams mean-field approximation fails in predicting the existence of the horn regions (Figs. <ref> and <ref>), here we find that the existence of the horn region induces a range of transitional long-range interaction strengths, between the medium long-range interaction and the strong long-range interaction. A=8.1 (Fig. <ref>(b))and A=9 (Fig. <ref>(c)) belong to this range.In this transitional range of A, we notice several things. First, the coexistence lines still exist, but the FM phases have pushed them to meet the T axis at high temperatures, and this intercept temperature increases with A (Fig. <ref>). Second, while for A=8 and when T is low, the AFM phases and the FM± phases are equally stable along the T axis (Fig. <ref>(a)). IncreasingA makes the FM± phases overtake the AFM phases along the T axis. Figure <ref> demonstrates this by comparing four points on the T axis for A=8 and 8.1. Third, the FM phases push the two spinodal lines originating from the mean-field critical point toward H=0. As a result, at around A=9 (Figs. <ref>(c) and <ref>(a)), the disordered spinodal lines nearly touch the T axis before the two mean-field critical points from the ± H side of the phase diagram coalesce at even higher A.While Fig. <ref>(h) shows a point close to the coexistence line for A=8.1, which has the disordered phase spread to the two AFM corners without connecting to the two FM± peaks, Fig. <ref> showsa point close to the coexistence line for A=9, which has the disordered phase connected to thetwo FM± peaks. The connecting bridge should disappear and the three peaks should become sharper, as the system size is increased. §.§ Strong long-range interaction, A=9.5,11 When the long-range interaction is sufficiently strong, the two mean-field critical points in the ± Hhorn regions will coalesce into one critical point as shown in the phase diagrams for A=9.5 and 11 in Fig. <ref>(b)-(c). Above this mean-field critical temperature, the system is in a disordered phase. If we increase H, the system undergoes a continuous crossover from the disordered phase to the FM+ phase, but there is no sharp transition point. The combined mean-field critical point is also the end point of the FM± spinodal lines.When H>0 and T is below the FM- spinodal line (region IX in Fig. <ref>), the marginal probability density P(m) has two peaks, and the system has a stable disordered/FM+ phase and a metastable disordered/FM- phase(Fig. <ref>). As there is a continuous crossover between the disordered phase and the FM+ phase above T_c, it is natural that near the mean-field critical point, the marginal probability density has a large peak at a value of |m| that is smaller than 0.5. Moreover, the metastable phase can show very strong disordered properties, so we consider the metastable phase below the FM- spinodal line to be a disordered/FM- phase. The topology of the phase diagrams for A=9.5 and 11 is the same as found for A=10 in Ref. <cit.>. Fig. <ref>(a)-(b) show probability densities at the coalesced mean-field critical point. It is found by extrapolation of the FM- spinodal line and Eq. (<ref>).Note that, as the critical point is in the mean-field universality class, at T=6, which is below the critical point for A=11 as shown in Fig. <ref>(c)-(d), we regard it as having stable FM± phases, connected by fluctuations resembling the disordered phase. However, we do not regard the system as having a metastable disordered phase. The fluctuation connection has disappeared at around T=5.5. At T=2.5 as shown in Fig. <ref>(a)-(b), the system is close to the AFM and disordered spinodal line, the free energy in (b) shows a flat maximum around m=0. Figure <ref>(c)-(d) shows the case at T=2.35 for A=11, which is a point in region X in the phase diagram of Fig. <ref>(c). The free-energy contourand the free-energy drop near m=0 indicate the existence of the metastable disordered phase.Further reduction in T below the critical line brings the system to the stable FM+ phase with metastable AFM phases, i.e. region XI in the phase diagram of Fig. <ref>(c), as shown in Fig. <ref>(e)-(f) for T=2.2. The free-energy contour, and the drop in free energy near m=0, indicate the existence of the metastable AFM phases. As A increases (Fig. <ref>), the disordered/AFM spinodal line merges with the critical line. We expect region X, the disordered metastable phase region, to disappear when A becomes very large. § SUMMARY AND CONCLUSIONIn this paper we have presented detailed phase diagrams, free-energy landscapes, and order-parameter distributions for a model SC material with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions <cit.>, covering a wide range of temperatures T, fields H, and long-range interaction strengths A. This was accomplished with a relatively modest computational effort by a recently developed, Macroscopically Constrained WL method for systems with multiple order parameters <cit.>. The method produces DOS for given values of the system energy E, magnetization m, and staggered magnetization m_s for a square-lattice Ising antiferromagnet (i.e., A=0) in zero field. The DOS for arbitrary values of H and A are then found by a simple transformation of E [Eq. (<ref>)], without the need for additional simulations. From the transformed DOS, we obtain free-energy landscapes and (H,T) phase diagrams, including metastable regions important to applications of SC materials <cit.>. Topologically different phase diagrams are obtained, depending on the strength of A. For A=0, the numerically well-known phase diagram for the square-lattice antiferromagnet is recovered (Fig. <ref>).For weak long-range interactions, 0 < A ≲ 4, the high-temperature critical line terminates in a tricritical point at a nonzero temperature, from which sharp spinodal lines marking the extent of metastable phase regions extend to T=0 (Fig. <ref>). In this parameter range, the phase diagram is topologically identical to what is predicted by a simple Bragg-Williams mean-field approximation as discussed in Ref. <cit.>.At a value of A between 4 and 6 (which we have not attempted to determine accurately), the tricritical point decomposes into a critical end-point and a mean-field critical point at a higher temperature. The resulting horn structure of the phase diagram, which is not seen in simple Bragg-Williams mean-field calculations, is illustrated in Fig. <ref> for the intermediate interaction strengths, A=6, 7, and 8. The phase diagram obtained for A=7 (Fig. <ref>) is in excellent agreement with that obtained by computationally intensive importance-sampling MC simulations in Ref. <cit.>. The only clear difference is the shape of the AFM/FM coexistence lines. A detailed investigation of this issue is in progress <cit.>. (Very recently, horn regions and asymmetric, two-step hysteresis loops, analogous to those seen in the model studied here, have also been observed for a model with antiferromagnetic-like nearest-neighbor interactions and genuine elastic interactions <cit.>.) The horn structure gives rise to asymmetric, two-step hysteresis loops (see example in Fig. <ref>) that are similar to experimental observations in several different SC materials <cit.>.For A>8, the AFM phase is no longer a possible ground state of the model. In the transitional region, 8 < A ≲ 9, the horn region shrinks as shown in Fig. <ref>, until the two mean-field critical points coalesce into a single critical point at H=0 for a value of A somewhere between 9 and 9.5. (This value we also have not attempted to determine accurately.) To our knowledge, this regime of transitional interaction strengths has not been investigated before. Phase diagrams for the strong-interaction case, represented by A=9.5 and 11, are shown in Fig. <ref>. These are topologically identical to the one shown for A=10 in Ref. <cit.>. We believe our results can contribute to the interpretation of the fascinating phase diagrams and hysteresis loops observed in many SC materials and other systems with competing short- and long-range interactions.§ ACKNOWLEDGMENTSThe Ising-ASFL model was first proposed by Seiji Miyashita, and we thank him for useful discussions. The simulations were performed at the Florida State University High Performance Computing Center. This work was supported in part by U.S. National Science Foundation grant No. DMR-1104829.§ FINITE-SIZE EFFECTS AND ERROR ESTIMATES The questions of finite system sizes and error estimates are intimately connected, and it is reasonable to ask whether the system size of L=32 that we use here is sufficient to ensure reliable results. The fourth-order Binder cumulant presumably leads to cancellation of leading corrections to scaling <cit.> and is a remarkably accurate method to locate critical points. The most general way to utilize the method is to look for the crossings between plots of cumulant vs temperature or field for different system sizes. However, the model studied here fulfills all the symmetry requirements to yield a fixed-point value of 0.6106924(16) <cit.>. As a consequence, it is possible to obtain good estimates of critical points as the phase points where the cumulant is near this value for a single system size, as we have done here. This is demonstrated in Fig. <ref>, where we compare critical lines obtained here using the macroscopically constrained WL method with L=32, with those obtained in Ref. <cit.> by importance-sampling MC using the standard method of cumulant crossings for L ≤ 1024. The differences are indeed very small, and although they are included as error bars in all the phase diagrams shown in this paper, they only exceed the symbol size in the lower right quadrant of the enlarged view of the horn region for A=7, shown in Fig. <ref>. 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Rikvold, “Nontrivial phase diagram for an elastic interaction model of spin crossover materials with antiferromagnetic-like short-range interactions," Phys. Rev. B 96, 144425 (2017). | http://arxiv.org/abs/1705.09465v2 | {
"authors": [
"Chor-Hoi Chan",
"Gregory Brown",
"Per Arne Rikvold"
],
"categories": [
"cond-mat.mtrl-sci",
"cond-mat.other",
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170526074439",
"title": "Phase diagrams and free-energy landscapes for model spin-crossover materials with antiferromagnetic-like nearest-neighbor and ferromagnetic-like long-range interactions"
} |
Dynamic analysis in Greenberg's traffic model Han Zhao^†The first two authors contributed equally to this work. [email protected] Shanghang Zhang^* [email protected] Guanhang Wu^♮ [email protected] JoãoP. Costeira^♭ [email protected] José M. F.Moura^ [email protected] Geoffrey J. Gordon^† [email protected]^†Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, USA^Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA^♮Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, USA^♭Department of Electrical and Computer Engineering, Instituto Superior Técnico, Lisbon, Portugal Received; accepted =======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Oscar A. Rosas–JaimesFacultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla,Prol. 24 Sur S/N Ciudad Universitaria, San Manuel, 72570 Puebla, México.e-mail: [email protected] A. Quezada–TéllezDepartamento de Matemáticas Aplicadas y Sistemas, UAM-Unidad Cuajimalpa,Avenida Constituyentes 1054, 11950 Ciudad de México, México.e-mail: [email protected] Fernández–AnayaDepartmento de Física y Matemáticas, Universidad Iberoamericana,Prol. Paseo de la Reforma 880, Álvaro Obregón, Lomas de Sta. Fe, Cd. de México,e-mail: [email protected] Based on the classical traffic model by Greenberg, a linear differential equation, we analyze it by means of varying the critical velocity v_o that appears in it as a parameter. In order to make such analysis we have obtained a solution for such a model and discretized it, obtaining related expressions for density k, flow q and velocity v to be treated as paired functions to obtained maps in phase-planes in which it is possible to observe distinct behaviors which span from monotonic and oscillatory stable trajectories, limit cycles of distinct periodicity, and chaotic ones. These behaviors are analyzed from a dynamical approach and then ilustrated with simulations performed in each case. As it is shown in this paper, these analyses are similar to those carried out in similar though simpler expressions (i.e. logistic-type functions), but taking in this case a new and direct approach through a nonlinear expression not used before to perform studies like these presented in this document, with a deep detail in the manner in which traffic variables are involved qualitatively and quantitatively.Keywords: Discrete Greenberg Model, Traffic Fundamental Diagram, Dynamical Analysis, Chaos.§ INTRODUCTION Traffic of vehicles has a presence in every aspect of persons and products mobility that it has become a phenomenon by itself. The physics of traffic can be modeled by many approaches. Due to its probabilistic aspects, it can be seen as a set of stochastic models (see <cit.>), but it is undeniable that traffic observe rules and relations showing a deterministic character (see <cit.>). This ambiguous nature leads to adapt the best scheme depending on the planning, research or design needs.Many traffic specialists prefer to follow more practical divisions that depend on the involved measures and variables. In this way, traffic can be explained by its individual units, the vehicles or small groups of them, through their positions, gaps, velocities or accelerations, obtaining microscopic models (see <cit.>). These type of models have the advantage of focusing in fine details, making possible to distinguish among different classes of vehicles or even to simulate driving styles (see <cit.>). However, these advantages are at the same time handicaps in other circumstances, due to they are difficult to analyze under a global perspective.Most of the times, cumulative approaches are used in road networks because traffic needs to be emulated or analyzed as a continuum, using aggregated variables which are modeled, simulated, analyzed and controlled macroscopically as a stream of vehicles (see <cit.>, <cit.> and <cit.>). This point of view allows to manage a large set of road networks today, becoming the main reason of its extensive use by many professionals, from its first developments to nowadays (see for example <cit.>).A first aspect of the nature of the traffic flow, independently of the type of model being used, is that it is nonlinear. As a consequence, all models related to traffic flow can exhibit a distinguishable behavior, ranging from stable and monotonous conditions to cyclic behaviours, which in turn can go from free-flow conditions to congested regimes, in the effort of emulating real situations (see <cit.>).Through the distinct types of models it is possible to perform analysis in order to help understand and predict those dynamical behaviours observed in roads. However, it is well known that there is no model able to describe the whole range of possibilities that can be observed in any traffic scenario. Some of such models are better than others for a certain kind of analysis, while others have a better fit to necessities of different understanding.In Section <ref> a brief view related to traffic models is developed, but it focuses on those models that have a macroscopic approach and their variables. These models have a useful and graphical tool that helps in understanding their theoretical background, known as fundamental diagrams, which represents basic relationships between pairs of common macroscopic variables. This document is devoted to the analysis of a specific behavior of traffic through three possible fundamental diagrams, paying attention to some trajectories that can be observed when an adjustable parameter is modified.In this same section a very well-known classical model proposed by <cit.> is used in this manuscript to generate a discrete version to perform analysis based on the fundamental diagrams generated directly from it. Fundamental diagrams have very well identified generic forms, but their exact shapes depend on the traffic model over which they are calculated. Even though it has been shown that Greenberg's model is not well suited for free-flow values of velocity, it provides a sufficiently and easy approach to emulate average values of velocity v, density k and flow q of a stream of vehicles moving in a road.Section <ref> presents some mathematical concepts and definitions, as well as some propositions that will be useful for the dynamical analyses applied to this discretized model, while Section <ref> presents some graphical results from numerical simulations obtained by varying a parameter that is capable to modify the scale of these fundamental diagrams, which is equivalent to the modification of the system and its conditions, as when the capacity permits smaller or bigger volume of vehicles, or the environmental conditions allow or restrict the driving possibilities. it is possible to identify points and ranges that are related to those analysis and their respective plotting representations, from stable fixed points to chaotic orbits.Many documents (see <cit.>, <cit.> and <cit.>) present analyses of chaotic trajectories on traffic model systems, but they are related to microscopic approaches and not to macroscopic ones. <cit.> develop an analysis over Greenshield's model, even though they adapt that model to a logistic-type mapping only for a flow–density relation. In <cit.> a polynomial approach is suggested for the calculation of a flow–density fundamental diagram, and then it is used to perform a discrete dynamical analysis. In the present case of this article the velocity–density and the velocity–flow fundamental diagrams are also included, and they are not adapted versions of a fundamental diagram model, but they are directly derived from the solution of Greenberg's model and other basic expressions in traffic theory. This has resulted in a new discrete nonlinear expression not used before to achieve stable, cyclic and chaotic trajectories.In Section <ref>, further analysis is provided through measuring the chaos character, such as the divergence speed of the nearby trajectories using Lyapunov exponents, which are directly related with the nonlinear nature of the model used, being at the same time a qualitative indicator of stable or chaotic behavior of the system.At the end, some concluding remarks are written for the main results and consequences obtained. § TRAFFIC MODELS§.§ Fundamental DiagramsTraffic models arise mainly from a necessity of understanding phenomena that have strong implications in economic and social aspects of modern life, specially when the physics of such phenomena has to do with an increasing frequency congestion (see <cit.>). First attempts to produce them began with statistical approaches as well as analogies with other physical phenomena (see <cit.>, <cit.> and <cit.>), with the intention they match up with real data, which seem to follow curves that began to be called fundamental diagrams.Figure <ref> shows the flow-density map, the velocity-density map and the velocity-flow map for a linear velocity-density relation model, from where the other two relationships can be obtained (see <cit.>). These functions are idealizations of the true points that represent the values of the traffic macroscopic variables, and their utility lies in that they are the best comprehensive approaches for any set of traffic macroscopic plots.In a macroscopic approach to traffic flow, velocity v is referred as the velocity of the wave front of vehicles, many times considered as the maximum velocity reached by the average of the cars. When congestion appears, it can be more valuable to know the backward front of congestion velocity w, a quantity that measures a congestion wave front that moves in the opposite direction to traffic flow (see <cit.>). However, few works have taken advantage of this quantity (see for example <cit.>).Flows q are quantities related to a set of vehicles moving with respect to time. It can be a number of cars trespassing a point in a period of time or a set of cars traversing a section of a way in a time interval (see <cit.>).Density k is defined as the number of vehicles occupying a section of a lane or stretch of a road. Direct measure of density can be get by air photographs, videorecorded images or by in situ observations, limiting a length of a way and counting the vehicles present on it in a moment of time (see <cit.>).These three quantities are simplistic related throughq=vkand relations between pairs of them are represented in Figure <ref>. The form of these curves is rather descriptive and at the same time idealized. It depends on particular cases of particular roads and their conditions. Data sets of each of them depict a complete and continue function but it is little probable to find the whole range of values of each variable in a measuring location. Data obtained in real life have multiple discontinuities in which many parts of these curves are not present (see <cit.>).Nevertheless these curves illustrates several significative points. Note that null flow occurs in two different conditions: * When there are no cars in the road, density and flow are zero. Velocity is theoretical and it will be that of the first driver appearing, supposedly a high value. This velocity is represented in the fundamental diagram as v_f and it is named free-flow velocity.* When density becomes so high that all vehicles are forced to stop, flow is zero again, due to there is no movement. The density in this situation is known as jam density and it is referred to as k_j. Between these two extremes there are many conditions of vehicular flow. As density increments from zero, flow does the same, due to there are more cars on the road, but as this happens velocity declines, because of the growing interaction among vehicles. This decrement in velocity is imperceptible when densities and flows are low.Velocity decreases significatively a little before reaching the maximum flow. This condition is showed in the diagrams of Figure <ref> as the critic velocity or optimum velocity v_o, the optimum density k_o and the maximum flow q_max.Slope of a straight line drawn from the origin of the velocity-flow diagram towards any point in the curve represents density. Likewise, a straight line from the origin of the density-flow diagram to any point over the curve represents velocity v. These slopes can be calculated from Equation (<ref>). Another important slope is that corresponding with the backward front velocity of congestion w, not depicted in the figure, but that would be plotted from the jam density extreme and with a negative slope.Notice that the three diagrams showed are redundant, because once established a relation between two of the variables, the another stays defined. Each of these relations (and their respective fundamental diagram) has its own field of application. For example, the velocity-density diagram is the base for vehicular flow models, because for a density value corresponds only a velocity value. The flow-density relationship is the point of departure for traffic control, due to it is possible to identify easily those regions where the traffic can be consider free, congested or in transition. The velocity-flow relationship is useful because it depicts regions of these values that can be related directly with levels of service in the roads (see <cit.> and <cit.>).As can be observed in Figure <ref>, any flow value distinct of the maximum can occur in two different conditions, one with low density and high velocity, and another with high density but low velocity. The portion of the curves for this last case represents the congested situation, with sudden changes in the traffic, some of them periodic, some of them chaotic, as will be seen later. §.§ Greenberg's ModelEquation (<ref>) relates density, velocity and flow, but it tells little about the way in which the corresponding data matches with the model used, and several fitness schemes has been proposed. Unfortunately, traffic flow data seem to be quite complex, and no model is able to achieve a perfect fit. Some models are better than others for different traffic flow regimes.Greenberg's model is a well stablished expression for macroscopic traffic, the result of observing velocity-density data sets for tunnels, specially those data that describe congestion (see <cit.>). This author correlated this information with the hydrodynamic analogy (given by the set of works by <cit.>, <cit.> and <cit.>) due to the equation of motion of a one-dimensional fluiddv/dt=-v_o^2/k∂ k/∂ xwhere x is the distance coordinate along the road, t is time, the parameter v_o is the optimum or critical velocity, and v and k are the already-known variables for velocity and density.Due to velocity is a function of distance and time, and by the properties of the total derivative, then Equation (<ref>) can be written in the formdv/dk∂ k/∂ t + vdv/dk∂ k/∂ x + v_o^2/k∂ k/∂ x = 0 which can be divided by dv/dk to obtain∂ k/∂ t + ( v + v_o^2/kdk/dv) ∂ k/∂ x = 0 Greenberg then uses the mass conservation expression∂ k/∂ t+∂ q/∂ x=0 which, by means of Equation (<ref>), can be written as∂ k/∂ t + v∂ k/∂ x + kdv/dk∂ k/∂ x = 0 In this way, it is possible to describe the behavior of vehicles using Equations (<ref>) and (<ref>).As this pair of equations constitutes a system, a non-trivial solution is achieved by this author asdv/dk = - v_o/k Differential equation (<ref>) follows the trajectory given byv = v_o ln( k_j/k) which functional plot is shown in Figure <ref> in the portion that corresponds to the velocity-density fundamental diagram where, as can be seen, infinity values for free-velocity are calculated. Even though this fact can be seen as a drawback for this model, Greenberg was able to show that for regions going away from those free-velocity values the model adjusts quite well.Combining equations (<ref>) and (<ref>) results in a relationship for density k and flow qq = v_ok ln( k_j/k)which is depicted in Figure <ref> in the respective portion of the flow-density fundamental diagram.The corresponding plot for the velocity-flow relationship is easier to get through a set of values obtained from flow and density data through the Equation (<ref>) in the formk=q/vThis one is represented in its respective portion of Figure <ref>. This figure gives a deeper detail of the relationship of the variables density k, flow q and velocity v in comparison to Figure 1. In this way, an improved accuracy for the fundamental diagrams used for the analysis is shown by Figure 2. <cit.> calculated and analyzed parameter values such as the maximum flow q_max, the optimum velocity v_o and the jam density k_j. These last two expressions are obtained throughout direct observations of graphical representations of the data.Real collections of traffic data show big variance and dispersion and, as has been mentioned earlier, it is difficult for any model to achieve a satisfactory fit to them. <cit.> commented on this fact, pointing out that his model represents a very good average of any set of traffic data for a wide range of road conditions, what has been confirmed by latter measurements (see <cit.>).Let the set of equations (<ref>), (<ref>) and (<ref>) be the functions which describe the relationship between each pair of the macroscopic variables of traffic. In order to generalize a result, let the maximum density k_j=1. with this assumption and from the flow-density function, it is possible to notice that the maximum flow q_max occurs when k_j/k = e, and the optimum density is then k_o=e^-1.With these values, a normalization can be established for this set of functions, allowing to lead interesting analyses about some generalized properties of those expressions related to the general shape of the fundamental diagrams obtained directly by the Greenberg's model.These analyses will be conducted through the optimum velocity v_o, a parameter that can be used to adjust these fundamental diagrams to different type of traffic data, with the effect of enlarging or shrinking their shapes. Figure <ref> shows the aspect obtained by the fundamental diagrams for five distinct values of v_o. Physically, this is equivalent to modify the road conditions for the model, since for a bigger value of v_o corresponds a bigger capacity or some other enhanced characteristic for vehicles displacement.The parameter v_o is left constant for each specific process and modifies the functions (<ref>) and (<ref>) of variable density k. These functions, along with that of Equation (<ref>) preserve the shape of the respective fundamental diagrams as seen in Figure <ref>. By the normalization made, it is possible to see that k ∈[ 0 1 ] and q ∈[ 0 1 ].We want to treat these fundamental diagrams in a similar manner as is done in the paper by <cit.>. In that paper, only the flow–density fundamental diagram is tested by means of a polynomial expression. In this present work we are spanning such treatment to those three fundamental diagrams already mentioned, using Greenberg's model. In consequence, each of these three fundamental diagrams are supposed to achieve an independent and parallel iterative process of the form k_i+1 = Q( k_i ), where Q( k_i ) is the particular function (<ref>) or (<ref>). Q_v(k_i + 1) = v_o ln( k_j/k(i)), i=1,2,3,... Q_q(k_i+1) = v_o k(i) ln( k_j/k(i)), i=1,2,3,... That is to say, starting from an initial condition, these expressions will generate a new value for Q_v(k_i+1) and Q_q(k_i+1) from previous inputs Q_v(k_i) and Q_q(k_i), constructing sequences { Q_v(k_n) } = {Q_v(k_0)Q_v(k_1), ⋯ , Q_v(k_i), ⋯} and { Q_q(k_n) } = {Q_q(k_0)Q_q(k_1), ⋯ , Q_q(k_i), ⋯} in an iterative way, and all this values will create a set that can be registered and plotted. These processes have been performed in other works over logistic-type functions (see, for example <cit.>, <cit.>, <cit.> and <cit.>).As it was mentioned earlier, one of the main purposes of this work is to fit the shapes obtained from Greenberg's model to calculate the same iterative processes that appear in those published papers, where such iterations are done over one-dimension functions of the form X_i+1 = F(X_i). Even though Equations (<ref>) and (<ref>) are evidently two-dimension functions, due to the normalization properties, the fundamental diagrams obtained can follow the same processes.Physically, this would be the case of a closed traffic network composed by arcs and nodes, all included in a control volume, but only a point of measuring is taken into account to know the state, and interactions and geometry of this network will be reflected in that sole point.In order to complete our three-variable scheme, density k is calculated in a parallel form byk(i+1) = Q_q(k_i)/Q_v(k_i)§ MATHEMATICAL CONCEPTSIn this section, some useful definitions, theorems and lemmas are presented, which will be used for the analyses that will be developed through the rest of this article. §.§ Fixed Points in Greenberg's Discrete Model (<cit.>) For Ω_α⊆ℝ^n and Λ⊆ℝ^m which α∈ℤ, consider a vector function 𝐟_α: Ω_α×Λ→Ω_α wich is C^r(r≥ 1)-continuous, and there is a discrete equation in the form of 𝐤_i+1=𝐟_α(𝐤_i,𝐩_α) for 𝐤_i,𝐤_i+1∈Ω_α, i ∈ℤ and 𝐩_α∈Λ. With an initial condition of 𝐤_i=𝐤_0, the solution of equation (<ref>) is given by 𝐤_i=𝐟_α(𝐟_α(⋯(𝐟_α(𝐤_0, 𝐩_α)))) for 𝐤_i∈Ω_α, i ∈ℤ and 𝐩∈Λ. * The difference equation with the initial condition is called a discrete dynamical system. * The vector function f_α(k_i, p_α) is called a discrete vector field on domain Ω_α. * the solution k_i for all i ∈ℤ on domain Ω_α is called the trajectory, phase curve or orbit of discrete dynamical system, which is defined as Γ = { k_i | k_i+1 = 𝐟_α(𝐤_i,𝐩_α)k ∈ℤ𝐩_α∈Λ}⊆∪_αΩ_α (<cit.>) Consider a discrete, nonlinear dynamical system 𝐤_i+1=𝐟 (𝐤_i,𝐩). A point 𝐤^*_i∈Ω_α is called a fixed point or a period-1 solution of a discrete nonlinear system 𝐤_i+1=𝐟(𝐤_i, 𝐩) under map P_i if for 𝐤_i+1=𝐤_i=𝐤^*_i 𝐤^*_i=𝐟(𝐤^*_i,𝐩) The linearized system of the nonlinear discrete system 𝐤_i+1=𝐟(𝐤_i,𝐩) at the fixed point 𝐤^*_i is given by 𝐲_i+1=DP(𝐤^*_i,𝐩)𝐲_i=D𝐟(𝐤^*_i, 𝐩)𝐲_i where, 𝐲_i=𝐤_i - 𝐤^*_i and𝐲_i+1=𝐤_i+1 - 𝐤^*_i+1By Definition <ref> and if 𝐤_j=1 as mentioned in Section <ref>, a fixed point in Greenberg's model is𝐤^*_i = e^-1/v_o Derivation of (<ref>) givesd/dk_i[v_o 𝐤_i ln(𝐤_j/𝐤_i)] = -v_o + v_o(ln𝐤_j - ln𝐤_i) Evaluating this derivative with k_j=1d/dk_i[v_o 𝐤_i ln(𝐤_j/𝐤_i)] |_𝐤^*_i = -v_o(1-1/v_o) = -v_o + 1 (<cit.>)Consider a discrete nonlinear dynamical system 𝐤_i+1 = 𝐟 (𝐤_i,𝐩) with a fixed point 𝐤^*_i. The corresponding solution is given by 𝐤_i+j = 𝐟(𝐤_i+j-1,𝐩) with j∈ℤ. Suppose there is a neighborhood of the fixed point 𝐤^*_i(i.e., U_i(𝐤^*_i)⊂Ω_α), and 𝐟(𝐤_i,𝐩) is C^r(r≥1)-continuous in U_i(𝐤^*_i). The linearized system is 𝐲_i+j+1 = D𝐟(𝐤^*_i,𝐩)𝐲_i+j(𝐲_i+j=𝐤_i+j - 𝐤^*_i) in U_i(𝐤^*_i). The matrix D𝐟(𝐤^*_i,𝐩) possesses n eigenvalues λ_q(q=1,2,...,n). * The fixed point 𝐤^*_i is called a hyperbolic point if |λ_q|≠ 1 (q=1,2,...,n). * The fixed point 𝐤^*_i is called a sink if|λ_q| < 1(q=1,2,...,n). * The fixed point 𝐤^*_i is called a source if |λ_q| > 1(q=1,2,...,n). * The fixed point 𝐤^*_i is called a center if |λ_q| = 1(q=1,2,...,n) with distinct eigenvalues. We apply these definitions to the particular case of the fixed point found for Greenberg's model. Notice that v_0 is always a non-negative value. Therefore: * For|-v_o + 1| = 1 only when v_0 = 0. Except for such a value 𝐤^*_i is a hyperbolic point. * For v_0∈(0,2), we have |-v_o + 1| < 1, then 𝐤^*_i is a sink. * For v_0∈ (-∞,0) ∪ (2,∞), it is obtained |-v_o + 1| > 1 and then 𝐤^*_i is a source. * For values v_0=0 and v_0=2, |-v_o + 1| = 1 and in such cases a pair of centers result, a hyperbolic and a non-hyperbolic. §.§ Bifurcation in Greenberg's Discrete Model <cit.> Consider a 1-D map P: 𝐤_i→𝐤_i+1 with 𝐤_i+1=𝐟 (𝐤_i,𝐩) where 𝐩 is a parameter vector. To determine the period-1 solution (fixed point) of Equation <ref>, substitution of k_i+1=k_i into Equation <ref> yields the periodic solution k_i=k^*_i. The bifurcation of the period-1 solution is presented. * Period-doubling bifurcation dk_i+1/dk_i=df(k_i,𝐩)/dk_i|_𝐤_i=𝐤^*_i=-1 As it will be depicted in Section <ref>, Greenberg's model (<ref>)–(<ref>) present n-period cycles in its trajectories and bifurcations in corresponding mappings. Taking n=1 in (<ref>) a period-doubling bifurcation presents when v_0 = 2, through Definitions <ref> and <ref>. df(k_i,𝐩)/dk_i|_𝐤_i=𝐤^*_i=-v_o + 1=-1 And thenv_0 = 2§.§ Stability analysis for Greenberg's Discrete Dynamic SystemIn this last subsection, we include the fundamentals of stability for a nonlinear discrete dynamical system (see <cit.>). Consider a nonlinear discrete-time system of the form𝐤_i+1 = 𝐟(i,𝐤_i), i ≥ i_0 where 𝐟: ℤ_+×ℝ^n→ℝ^n is a given function such that 𝐟(i,0) = 0, for all i ∈ℤ_+ (i.e., ξ = 0 is an equilibrium of the system (<ref>)).It is clear that for i_0∈ℤ_+ and 𝐤_0∈ℝ^n, (<ref>) has a unique solution, denoted by 𝐤_i(,i_0,𝐤_0) satisfaying the initial condition 𝐤_i=0 = 𝐤_0(<cit.>) The zero solution of (<ref>) is exponentially stable if there exist M ≥ 0 and β∈ [0,1) such that: ∀ i,i_0∈ℤ_+, i ≥ i_0; ∀𝐤_0∈ℝ^n: ∥𝐤(i,i_0,𝐤_0)∥⩽ Mβ^i-i_0∥𝐤_0∥A simple sufficient condition for exponential stability to (<ref>) is given by the following lemma <ref>.(<cit.>) Suppose there exists A ∈ℝ^n × n_+ such that |𝐟(i,𝐤_i) |≤ A |𝐤_i|,∀ i ∈ℤ_+, ∀𝐤_i∈ℝ^n. If ρ(A) < 1 then the zero solution of (<ref>) is exponentially stable.Direct application to (<ref>) is given in the following theorem. Consider the scalar system 𝐤_i+1 = v_0𝐤_iln(𝐤_j𝐤_i). If | v_0| < 1 and |𝐤_i| < 1, ∀ i ∈ℤ_+then the system is exponentially stable. From equation (<ref>) 𝐤_i+1 = v_0𝐤_iln (𝐤_j/𝐤_i) let 𝐤_j = 1 𝐤_i+1 = v_0𝐤_iln𝐤_i and taking absolute value |𝐤_i+1| = | v_0||𝐤_i|| ln𝐤_i| we obtain |𝐤_i+1|/|𝐤_i|= | v_0|| ln𝐤_i| < | v_0||𝐤_i| and rewriting |𝐤_i+1| < | v_0||𝐤_i|^2 by hypothesis |𝐤_i| < 1, ∀ i ∈ℤ_+, in consequence |𝐤_i+1| <| v_0||𝐤_i|^2≤| v_0||𝐤_i| |𝐤_i+1| < | v_0||𝐤_i| therefore by Lemma <ref> if | v_0| < 1, then the system is exponentially stable. These results about stability can be extended to the other two fundamental diagrams due to their relationships by means of expressions such as (<ref>) and (<ref>). On the other hand, the implications in traffic situations from this theorem are directly related with decreasing or increasing amounts of traffic density k and flow q, as will be better explained by the simulations performed and depicted in Section <ref>.The nonlinear nature of these equations leads to a quasi–periodic behaviour which reaches chaotic dynamics, whose trajectories are characterized by an exponential divergence of initially close points (as explained by <cit.>). Instead of developing a stability analysis for such situations, we opt to take care about other type of measuring. Taking the case of one-dimensional discrete maps of an interval k_n+1=f(k_n),x ∈ [0,1] The so-called Lyapunov exponent is a measure of the divergence of two orbits starting with slightly different initial conditions k_0 and k_0+Δ k_0. The distance after n iterations Δ k_n = | f^n(k_0+Δ k_0)-f^n(k_0) |increases exponentially for large n for a chaotic orbit according toΔ k_nΔ k_0e^λ_L It is possible to relate the Lyapunov exponent analytically to the average stretching along the orbit k_0,k_1=f(k_0),k_2=f(f(k_0)),…,k_n=f^n(k_0)=f(f(f(…(k_0)…)). Through proper mathematical treament, it is possible to obtain: lnΔ k_n/Δ k_0 = ln|f^n(k_0+Δ k_0)-f^n(k_0)/Δ k_0|ln|df^n(k)/dk| = ln∏_j=0^n-1| f'(k_j) |or equivalently lnΔ k_n/Δ k_0 = ∑_j=0^n-1ln| f'(k_j) |and finally λ_L = lim_n→∞1/nlnΔ k_n/Δ k_0 = lim_n→∞1/n∑_j=0^n-1ln| f'(k_j) |where the logarithm of the linearized map is averaged over the orbit as k_0, k_1, …, k_n-1. Negative values of the Lyapunov exponent indicate stability, and positive values chaotic evolution, where λ_L measures the speed of exponential divergence of neighboring trajectories. At critical bifurcation points the Lyapunov exponent is zero (see <cit.>).§ NUMERICAL SIMULATIONS As it has been shown through this document, discrete iterative equations have been defined for pairs of variables depicting fundamental diagrams density versus flow (k vs q), density versus velocity (k vs v) and flow versus velocity (q vs v). These expressions are useful to perform simulations which plottings show interesting trajectories as parameter v_0 is varied. Each simulation is a set of 300 iterations.If k = 0.25 is an initial condition with v_o= 0.25, a first series of iterations can be executed (Figure <ref>). Its possible to follow the trajectory of the iterations. The q = k line is drawn over the flow-density plot, which is the most common depicted in literature. It is helpful in order to visualize the form in which iterations develop, represented by vertical and horizontal lines that intersect it from values calculated.In that way, it can be noticed that starting from the initial condition the final value stops in a point near q = k = 0.0183, indicating that the final state stabilizes for any future time (iteration) in a free-flow value, like a transient of a road being emptied. The initial condition chosen, as well as the others that will be shown in the rest of this work, has been selected due to it exhibits a rapid transient, which permits a better view of the iterations behavior.We are including the other two fundamental diagrams with their respective iterations carried on each value of v_o for the respective initial condition. The plots that correspond with velocity–density and velocity–flow diagrams show the convergence to v = 1 and confirm the stable point for k and q as calculated in Section <ref>. These plots do not repeat the same idea of including a 45-degree line to yield the iterations, but it is possible to watch that the iterations follow not only the respective fundamental diagram profile, but they follow the profile of the missing variable diagram, i.e. if the iterations are over the velocity-density fundamental diagram, iterations also touch the velocity-flow fundamental diagram profile. This is a consequence of the relation among the three macroscopic variables.If the initial condition is changed, the transient will follow a distinct trajectory, but it will reach the same final value, that is to say, the steady state of both situations are equal. It is said that the point q=k=0.0183 (and v=1, simultaneously) is an attractor for the trajectories of the system (as described by <cit.>, for example).As the parameter v_o is varied, a set of final values result, which can be plotted as in Figure <ref>, which is called a bifurcation map. Vertical lines are drawn on it corresponding with the values of v_o used and crossing with the final value reached. Since this work takes into account the three macroscopic variables for traffic, the v-v_o bifurcation map is also presented (Figure <ref>).Figure <ref> shows another set of iterations for the three fundamental diagrams when optimum velocity is adjusted to v_o=1.25 and the initial condition is q=k=0.1. Again, a stable point is reached, but unlike the first set of iterations, where free-flow values were reach, in this case a set of congested-flow values are obtained, even though the initial density was for a free-flow regime. This simulates a congestion process in the network, due to characteristics on the road represented by v_o. This time the achieved values are q=k=0.4493 and the velocity is again v=1. Bifurcation maps <ref> and <ref> exhibit this values on their own. Like in the past simulation, if initial condition is changed only the transient is different, but the steady state reached at the end becomes the same again.We have until now two distinct values of v_0 and it must be watched that the first one is a value v_0<1 while the second one v_0>1. Relating with Theorem <ref> it is now possible to see that values v_0<1 represent a road that tends to get empty, while values v_0>1 represent a road that tends to congestion.Keeping the initial condition but changing for v_o=1.75 the situation runs to another stable point, but the iterations become damped cyclic (Figure <ref>). As will be seen in brief, this behavior marks an important limit depicted by the bifurcation maps.If v_o= 2.25 the attractor for traffic variables behaves purely cyclic (Figure <ref>). In fact, values for density and flow fluctuates among q=k=0.3533 and q=k=0.8271 and the values for velocity between v=2.3409 and v=0.4272, once the transients have vanished. This is corresponded in the bifurcation maps, where this value of optimum velocity marks the region where two branches signal a set of pairs of final values. The physical meaning is an oscillatory response of the vehicles, which pass from near optimum values to congested-flow values from one time (iteration) to the next.If optimum velocity is varied again to v_o=2.405 (Figure <ref>), with an initial condition in this case of k=0.275, traffic becomes a 4-period cycle, with values q=k=[0.8496, 0.3330, 0.8806, 0.2692] and v=[3.1560, 0.3919, 2.6446, 0.3057].If attention is focused on the bifurcation maps the branches that correspond with sets of 4 values for each traffic variable runs in an interval that goes from approximately v_o=2.395 to v_o=2.440.This process can continue. If now v_o=2.48, and initial density is k=0.23 the iterations are those of Figure <ref>, and related with the respective optimum velocity in the bifurcation maps. Cyclic behavior has become an 8-period oscillation with q = k = [0.8094,0.4244,0.9021,0.2305,0.8389,0.3654,0.9123,0.2076]andv=[3.8987, 0.5243,2.1256,0.2555,3.6392,0.4356,2.4965,0.2276].It is important to notice that the distance between two adjacent values of v_o decreases for each set of bifurcations, and an n-period bifurcation reach more rapidly the next 2n-period bifurcation, until they are very difficult to distinguish. This situation causes that a very little variation in the v_o parameter could exhibit instabilities among trajectories. Even if the v_o parameter keeps fixed, a very large number of final values can result, which are very sensitive to selection of the initial conditions.An example of this situation is shown in Figure <ref> and Figure <ref>, where v_o=2.585 and k=0.1 for the first of them, which shows what it is called as chaotic trajectories. The cyclic behavior is only apparent, and strictly speaking it does not exist because the different values do not exhibit a period where series of them can repeat themselves. This attractor is known as strange.The final values reached after 300 iterations are q=k=0.8451 and v=4.0519. If optimum velocity v_o is left with no changes but the initial condition is slightly change to q=k=0.101, Figure <ref> depicts an apparent equal image than the last one, but a second sight will reveal that the trajectory is quite different. In fact, the final values after the same number of iterations now are q=k=0.2020 and v=0.2199.This feature that consists in a crescent divergence between two trajectories that has very close initial states is referred as a deterministic but unpredictable behavior, proper of most nonlinear systems under certain set of parameter values. In a more linear set of trajectories, it is expected that for a small difference in initial conditions, the trajectories generated were nearby enough to not take into account that small difference or at least they are proportional. That is to say, for very close initial states, trajectories remain closely enough to achieved predictions about the system.However, as has been seen by the last iterations here included, this difference becomes important for a so-called chaotic system, due to in these cases a set of very close states will diverge. Physically, this means that it is impossible to perform a simulation of such a system with the intention of predicting future states, due to the real states in the beginning of such a simulations cannot be measure and transcribed to our calculations with infinite precision. Those infinitesimal differences among the measured and the real values will grow up rapidly, in an exponentially way as time (iterations) passes. Predictions will only be valid, under a tolerance, for a few future states.§ ANALYSIS OF LYAPUNOV EXPONENTSLyapunov exponents are considered a helpful tool to qualify stability of a nonlinear system (see <cit.>). They are also useful to identify the chaotic feature of those kind of systems.Equation (<ref>) is applied for such a calculation, which in this case n= 10 000 terms.When applying such a concept to an equation like (<ref>) it is possible to confirm the behaviors analyzed and observed in Sections <ref> and <ref>.In that way, the stable trajectories shown in Figures <ref>–<ref> that coincide with the curve that grows up from v_0=0 to v_0=2.0 in the mapping plots <ref> and <ref>, are confirmed by the negative values calculated for the Lyapunov exponents in Figure <ref>.There is a transition from that stable region starting on v_0=2.0 and ending in v_0=2.5 represented by the examples represented in the limit cycles depicted by Figures <ref>–<ref>. Lyapunov exponents are still negative in this interval.However, from v_0=2.5 these exponents modify their sign. From this point, trajectories became chaotic, as depicted by Figures <ref> and <ref>, bifurcations in Figures <ref> and <ref> have become indistinguisable and Lyapunov exponents turn positive. Figure <ref> is a magnification of the region for v_o ≥ 2.5. These values and trajectories have another important feature. They are different from those analogous values obtained for a logistic mapping and are exclusive for the solution of Greenberg's model (<ref>).§ CONCLUDING REMARKS There are different models that deal with the relationship among the macroscopic variables density k, flow q and velocity v, like Greenberg's model, which is well fitted for real data, and even though it is not well suited for free-flow velocities, it has been proved in a wide range of practical values, and its function solution is easy to manipulate to perform proper analyses of different behaviors exhibited as a parameter included in it is varied.A normalization is carried on the variables density k and flow q in order to generalize results. The parameter that is varied in Greenberg's model is the critical velocity v_0, which modifies the scale of the fundamental diagrams that can be produced for that expression. This is equivalent to modify the conditions on a road network represented by those fundamental diagrams, giving as a consequence different behaviors in the modelled traffic.One of the main contributions of these article is to include the fundamental diagrams of the flow–velocity and the velocity–density plots and to perform the respective iterations and analysis on them, instead of using solely the flow–density fundamental diagram as if it was a logistic map.Iterations on values of density k substituted in Greenberg's model to obtain flow q are performed, and by the normalization made those values are directly taken as new values of density, which are again substituted to get new ones. This set of values are plotted with the help of a straight line q=k, emulating the evolution in time of those traffic values. In a similar and parallel manner, values of velicty v are also calculated and plotted. When v_o is changed from low to higher values, iterations draw trajectories that converge to stable, cyclic or chaotic values. It is possible to obtain a (bifurcation) map of final values for each v_o modification. It is easy to recognized in this plot which ranges of the parameter v_o give stable, cyclic or chaotic behavior. Another main contribution of this article is to show the velocity–optimum velocity bifurcation map obtained from the iteration schemes here performed.A main feature of the bifurcations is that they split in 2^n-period cycles, i. e. for the first time a bifurcation map splits in two branches, then it divides in four, latter in eight, sixteen and so on. But this is also done in shorter intervals and cannot continue for ever. Soon, the period 2^n is juxtaposed to the next 2^n+1 and any little variation in the parameter v_o or in the initial conditions will give an unstable situation between branches, which implies big differences between two final states for very close initial values of those states. This sensitivity to initial conditions is a well known feature of chaotic systems, like the cases here presented.Chaos is mainly characterized by the sensitivity of the system to initial conditions, i.e. two trajectories starting from very nearby initial values diverge from each other. A measure of this divergence is the Lyapunov exponents, which are negative for stable and cyclic trajectories, but are positive for chaotic ones.Chaotic behavior is mainly characterized by the sensitivity of the system to initial conditions, that is, two trajectories of variables that start from very nearby initial values diverge from each other. A measure of this divergence is the Lyapunov exponents, which are negative for stable and cyclic trajectories, but is positive for chaotic ones. Further analyses of the limit cycles and chaotic regions can be carried on since a point of view of Lyapunov stability, extending the study of this new approach in further work. 99 [Arasan & Dhivya(2010)]Thamizh2010 T. Arasan & G. Dhivya (2010) Measuring heterogeneous traffic density,International Journal of Engineering and Applied Sciences, 6 (3), 144–148.[Beckmann(2013)]Beckmann2013 M. J. Beckmann (2013) Traffic congestion and what to do about it, Transportmetrica B: Transport Dynamics,1, 103–109.[Daganzo(1997)]Daganzo C. F. Daganzo (1997) Fundamentals of Transportation and Traffic Operations, Pergamon.[Devaney(1987)]Devaney1987 R. L. Devaney (1987) An introduction to chaotic dynamical systems, Westview Press.[Greenberg(1959)]Greenberg59 H. Greenberg (1959) An analysis of traffic flow, Operations Research, 7 (1), 79–85.[Hillborn(2000)]HillbornChaos R. C. Hillborn (2000) Chaos and Nonlinear Dynamics, 2nd. ed. Oxford University Press.[Holmgren(1994)]Holmgren1994 R. A. Holmgren (1994) A First Course in discrete Dynamical Systems, New York: Springer. [ITE(2009)]ITE2009 ITE2009 (2009) Traffic Engineering Handbook, 6th. ed.,Washington DC: Institute of Transportation Engineers.[Kerner(2012)]kerner2012physics B. S. Kerner (2012) The physics of traffic: empirical freeway pattern features, engineering applications and theory, Springer.[Kim & Zhang(2004)]KimZhang T. Kim, & H. M. Zhang (2004)An empirical study on gap time and its relation to the fundamental diagram of traffic flow, In: 7th International IEEE Conference on Intelligent Transportation Systems, Washington, D. C., 94–99.[Korsch & Hartmann(2008)]Korsch1 H. J. Korsch & T. Hartmann (2008) Chaos. A Program Collection for the PC, Springler–Verlag.[Lighthill & Whitham(1955a)]Lighthill1 M. J. Lighthill & G. B. Whitham (1955a) On kinematic waves I. Flood movementin long rivers, Proc. Royal Soc. A, 229 (1178), 281–316.[Lighthill & Whitham(1955b)]Lighthill2 M. J. Lighthill & G. B. Whitham (1955b) On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Royal Soc. A, 229 (1178), 317–345.[Lo & Cho(2005)]ShihChingLo2005 S. C. Lo & H. J. Cho (2005) Chaos and control of discrete dynamic model, Journal of the Franklin Institute, (342), 839–851.[Low & Addison(1998)]Low1998 D. J. Low & P. S. Addison (1998) A nonlinear temporal headway model of traffic dynamics, Nonlinear Dynamics 16 (2), 127–151.[Luo(2012)]Luo2012 A. Luo (2012) Regularity and Complexity in Dynamical Systems. Springer-Verlag.[Lučić & Teodorović(2002)]Lucic2002 P. Lučić & D. Teodorović (2002) Transportation modeling: An artificial life approach. In: 14th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2002 4–6 November, 216–223.[Marchesini & Weijermars(2010)]MW2010 P. Marchesini & W. Weijermars (2010) The relationship between road safety and congestion on motorways, Institute for Road Safety Research, The Netherlands.[May(1990)]May90 A. D. May (1990) Traffic Flow Fundamentals, Prentice Hall.[Ngoc & Hieu(2012)]Ngoc2012 P. H. A. Ngoc, & L. T. Hieu (2012) On stability of discrete-time systems under nonlinear time-varying perturbations, Advances in Difference Equations, (120) 1–10.[Panwai & Dia(2005)]PanwaiDia2005 S. Panwai & H. Dia (2005) Comparative Evaluation of Microscopic Car-Following Behavior, In: 14th IEEE Transactions on Intelligent Transportation Systems, 6(3), 314–325. [Richards(1956)]Richards56 P. I. Richards (1956). Shock waves on the highway, Operation Research, (4) 42–51.[Rosas-Jaimes & Alvarez(2007)]RosasJDS07 O. Rosas-Jaimes, & L. Alvarez (2007). Traffic density and velocity estimation, Nonlinear Dynamics 49(4), 555–566.[Rosas-Jaimes et al.(2016)Rosas-Jaimes, Quezada-Téllez, Fernández-Anaya]RosasPROMET2016O. Rosas-Jaimes, L. A. Quezada-Téllez, and G. Fernández-Anaya (2016). Polynomial Approach and Non-linear Analysis for a Traffic Fundamental Diagram. PROMET, Traffic and Transportation, (4) 321–329.[Treiber & Helbing(2001)]TreiberHelbing01 M. Treiber & D. Helbing (2001) Microsimulations of freeway traffic including control measures, Automatisierungstechnik, 49 478–484.[Villalobos et al.(2007)]Villalobos2010 J. Villalobos, B. A. Toledo, D. Pastén, V. Muñoz, J. Rogan, R. Zarama, N. Lammoglia, & J. A. Valdivia (2010) Characterization of the nontrivial and chaotic behavior that occurs in a simple city traffic model, Chaos, 20 (1) 013109.[Wang et al.(2013)]wang2013stochastic Ni. D. Wang,Q. Y. Chen, & J. Li (2013) Stochastic modeling of the equilibrium speed–density relationship, Journal of Advanced Transportation, 47 (1), 126–150.[Wohl & Martin(1967)]Wohl M. Wohl & B. Martin (1967) Traffic System Analysis, McGraw-Hill Series in Transportation. | http://arxiv.org/abs/1705.09682v1 | {
"authors": [
"Oscar A. Rosas-Jaimes",
"Luis A. Quezada-Téllez",
"Guillermo Fernández-Anaya"
],
"categories": [
"math.DS"
],
"primary_category": "math.DS",
"published": "20170526191007",
"title": "Dynamic analysis in Greenberg's traffic model"
} |
unsrtULB-TH/17-10 Service de Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, BelgiumCLOCKWORK DARK MATTERD. TERESIReceived: date / Accepted: date =================================== I give a pedagogical discussion of thermal dark matter (DM) within the clockwork mechanism. The clockwork mechanism, which is a natural way to generate small numbers starting from order-one couplings, allows to have a long-lived, but unstable, DM particle that nevertheless has O(1) couplings with electroweak- or TeV-scale states. Remarkably, DM decays on time scales much longer than the age of the Universe and has at the same time sizeable couplings with light states, which therefore allow to produce it thermally within the WIMP paradigm. These new particles with large couplings can be searched for at current or future colliders. I also briefly comment on how this setup can minimally emerge from the deconstruction of an extra dimension in flat spacetime.§ INTRODUCTIONThe clockwork mechanism <cit.> is an elegant and economical way to generate tiny couplings much smaller than unity, or equivalently large hierarchies between different dimensionful or dimensionless quantities, in a theory with only O(1) couplings and hierarchies. The important point here is that the mechanism is economical, in the sense that to generate a hierarchy X one only needs a number of fields O(log X). This should be contrasted to what happens in different setups, e.g. the one of Ref. <cit.>, wherea number of fields O(X) are needed.The mechanism was originally introduced in the context of relaxion models <cit.>, to overcome a theoretical difficulty present generically there, namely the fact that the relaxion field needs to have a huge super-Planckian excursion during the relaxation phenomenon, together with the fact that tiny (with respect to the other scales involved) dimensionful couplings are needed in the original formulation of the relaxion mechanism <cit.>. However, it has been recently shown that the clockwork mechanism is much more general than that <cit.>, and that it could be indeed useful for a number of physical situations where large hierarchies/small couplings are indeed present <cit.>: axion models that are “invisible” although the relevant physics is at the weak scale <cit.>, cosmic inflation <cit.>, the flavour puzzle and, most remarkably, the long-standing (and these days more disturbing than ever) gauge hierarchy problem <cit.>.Last but not least, dark matter <cit.>, which is the main focus of this work and that I will now review, following the original study of Ref. <cit.>.Dark matter (DM) must be stable over cosmological time scales. This means that it could be absolutely stable or very long lived.Absolute stability is typically realized by invoking a symmetry, either continuous or discrete, local or global, motivated or ad-hoc. This is the most popular option and is certainly fine. On the other hand, the possibility that DM may be unstable, but long lived, is in turn very interesting as its decay could be probed through indirect-detection searches. In this case the lifetime of a DM candidate in the WIMP mass range, which I will consider in the sequel, must be typically larger than τ∼ 10^26 sec. To obtain such a large lifetime, one may typically assume that the decay is induced by the exchange of very heavy particles. From the effective field theory perspective at low energy, assuming couplings of order one, instability associated to a dimension-5 operator would require that the heavy degrees of freedom have masses above the Planck scale. The situation is better for dimension-6 operators, although still these particles need to be around the GUT scale or heavier. In alternative, one could assume that the particles that trigger the decay of DM are much lighter, but then with tiny coupling. In all these cases, the physics involved in the decay of DM would be essentially impossible to test.The clockwork mechanism can come to our aid. In Ref. <cit.> we have shown that,by making use of the clockwork mechanism, a DM particle could be made very long lived, at these same time having a decay into SM particles induced by O(1) interactions with particles that could be produced at colliders, and therefore tested directly and indirectly. SuchO(1) interactions make DM annihilate into SM or hidden-sector particles fast, so that its relic density is determined by the standard freeze-out mechanism. We arrive at the (at first sight paradoxical) fact that DM is unstable and at the same a WIMP. One may still think that O(log X) fields are too many, and that the clockwork mechanism is not that economical, after all. For the appreciators of minimalism the good news is that the clockwork setup may originate from the deconstruction of an extra dimension, in which only one or few fields are introduced. Although in the original Ref. <cit.> such a construction involved a curved spacetime (with an exponential metric in the extra dimension), we have subsequently shown <cit.> that the clockwork setup can originate from a flat extra dimension and boundary terms, as I will briefly review below. It has been subsequently claimed that this is the only way to obtain consistently the clockwork mechanism from an extra dimension <cit.>, although very recently it has been argued <cit.> that the curved- and flat-spacetime constructions are instead equivalent, at least from the low-energy perspective.§ CLOCKWORK DARK MATTERThe clockwork mechanism is based on the very simple observation that a product such as:1/q × 1/q ×1/q × 1/q ×…× 1/q,with q>1, quickly becomes tiny when the number of factors increases. To implement this idea in quantum field theory, one wants to introduce N fields ϕ_i, which interact schematically as ϕ_01/q2em0.5ptϕ_1 1/q2em0.5ptϕ_2 1/q2em0.5ptϕ_3 1/q2em0.5pt…1/q2em0.5ptϕ_N 2em0.5ptSM ,with couplings 1/q ≲ 1. The key feature of the clockwork mechanism is that, if thanks to an appropriate symmetry one of the mass eigenstates ϕ_ light (typically the lightest one) is essentially given by ϕ_0, its interaction with the Standard Model (SM) will be suppressed exponentially asϕ_ light 2pt1em0.5ptSM ∼ 1/q^N .Therefore, to generate a tiny coupling 1/X by making use of O(1) couplings 1/q, one only need a logarithmically large number of fields N ∼log_q X. In the case of fermions, the convenient symmetry just mentioned can be taken to be the chiral symmetry. Thus, we introduce a set of chiral fermions R_i, L_i interacting schematically asR_0 m1.5em0.5ptL_1 R_1_q mm1.5em0.5ptL_2 R_2_q mm1.5em0.5ptL_3 R_3_q mm1.5em0.5pt … m1.5em0.5ptL_N R_N_q m1.5em0.5pt L_SM ,where L_SM is the SM lepton doublet. Since there are 2 N +1 chiral symmetriesU(1)_R_0× U(1)_L_1× U(1)_R_1×…× U(1)_L_N× U(1)_R_N , with U(1)_R_N≡ U(1)_L_SM ,and 2 N breaking parameters m, q m, there is a massless mode N. Intuitively, for q>1, since R_0 does not have a chiral partner, the massless mode N approximately coincides with R_0. The conclusion is not changed if we add a Majorana mass m_N < q m, so that the light mode N acquires a mass ≈ m_N and will be our dark matter particle. In order to realize the structure (<ref>) we may consider a set of scalars S_i, C_i with chargesS_i ∼ (-1,1) underU(1)_R_i× U(1)_L_i+1, C_i ∼ (1,-1) underU(1)_L_i× U(1)_R_i .Thus, the fermion and scalar fields interact as a chain:R_0 S_12em0.5pt L_1 C_12em0.5ptR_1 S_22em0.5ptL_2C_22em0.5pt…C_N2em0.5ptR_N 2em0.5ptL_SM .When the scalars acquire vacuum expectation values we obtain the clockwork setup (<ref>), with m= y_S ⟨ S_i ⟩, q m= y_C ⟨ C_i ⟩, where y_S,C denote the appropriate Yukawa couplings with the clockwork states.From now on, let us take q ≫ 1 for simplicity, so that an analytic understanding is facilitated. In this limit, one can show that the clockwork mechanism works as long as m_N ⪅ q m. The fermionic spectrum of the theory consists of a the dark-matter Majorana fermionN≈R_0+ 1/q^1 R_1+ 1/q^2 R_2+… + 1/q^N R_N,and a band of N Majorana pairs (the clockwork “gears”), that form pseudo-Dirac states ψ_i with mass ≈ q m:ψ_i ≈1/√(N)∑_k ( O(1) L_k+O(1) R_k).An example of the spectrum is shown in Fig. <ref>. In addition, two sets of N scalars S_i and C_i areexpected in the same mass range. However, these are not necessarily dynamic fields, for more details see Ref. <cit.>.The clockwork gears have sizeable interactions with the SM and the clockwork scalars. The dark-matter particle has sizeable interactions with the scalars too. For q≫ 1, the relevant O(1) interactions are:[c]3cm < g r a p h i c s > [c]2.8cm < g r a p h i c s > [c]2cm < g r a p h i c s >,where h is the SM Brout-Englert-Higgs scalar boson, which in the spirit of the clockwork mechanism (i.e. that one does not have small interactions to start with in the Lagrangian) is expected to mix with the clockwork scalars, in particular S_1. Instead, the key aspect of the mechanism for clockwork dark matter is in the interactions of N with the SM fermions. From (<ref>) we see that the overlap of N with the state R_i is approximately 1/q^i. This is precisely because at each interaction with the clockwork pairs L_i R_i along the chain, one gets a O(1) suppression m/(qm) = 1/q (see (<ref>)). Thus, if the SM leptons interact with the state R_N only (with a O(1) Yukawa coupling y), in the original clockwork basis, the interaction of the mass eigenstate N with them is exponentially clockwork suppressedℒ⊃ - y/q^N L̅_SMH N_R,so that, even though N can decay, e.g. N →ν h , ν Z , l W, one could make the decay lifetime of N longer than the age of the Universe by making use of only 𝒪(1) couplings and ≲ TeV-scale states. Indirect detection requires the lifetime to be typically larger than ≈ 10^26 sec, i.e. q^2N > 1.5 × 10^50 y^2 m_N/GeV. For instance, this can be realized with m_N ∼ 100 GeV, y ∼ 1, q ∼ 10, N ∼ 26. An important point to stress here is that one can show that potentially dangerous loop corrections involving the clockwork gears ψ_i, which have O(1) couplings and could potentially destabilize dark matter by generating large corrections to the decay processes, are also clockwork suppressed. This can be understood by noticing that any decay diagram for N must involve a fermion line starting from the initial state, i.e. N ∼ R_0 itself, and ending into a (lighter) SM fermion in the final state: this can happen only if the fermion line proceeds along the whole clockwork chain in (<ref>), thus inheriting the exponential clockwork suppression. § PHENOMENOLOGYLet us now see how this setup can be used to generate dark matter in the early Universe and the phenomenological consequences of this framework. Let us first consider the case m_S_1 < m_N. In this case, the dominant process for dark matter annihilation in the early Universe is:< g r a p h i c s >Since the eigenstate N has a large overlap with R_0 and the clockwork gears ψ_i have O(1) overlap with L_1, the couplings involved in this diagram are O(1), generated from the Yukawa interaction [In the limit q ≫ 1 the analogous interactions involving L_>1 are instead subleading.]y_S S_1 L_1 R_0. Thus, this process is in thermal equilibrium in the early Universe, freezing out when N is non-relativistic. Although decaying dark matter, N is a WIMP. Since in the clockwork spirit all Lagrangian couplings are expected to be O(1), the WIMP miracle implies that N has to be at the weak scale, up to the TeV range. This is shown in Fig. <ref>, where the Yukawa coupling y_S that yields the correct relic density is plot. For instance, for m_S_1 = 150GeV, the perturbativity condition y_S < √(4 π)≃ 3.5 requires m_N, m_ψ⪅[2]TeV.In alternative, we may consider the case in which m_S is so large that the N N → S_1 S_1, S_1 h are not kinematically allowed for non-relativistic N, so that the relevant process for dark-matter relic density is:< g r a p h i c s >The rate for this process is proportional to (y_S θ_S)^4, where θ_S is the mixing of S_1 with the SM scalar h. The results for the effective coupling y_S θ_S yielding the correct relic density are shown in Fig. <ref>. Since, as discussed below, the mixing of a single singlet scalar has to be less than ≈ 0.4 from collider experiments, in the case of N singlets with universal mixing one would have θ_S ⪅ 0.4/√(N).In this case the required values of y_S from Fig. <ref> would be non-perturbative. Therefore, this scenario works in the presence of non-universal mixing angles for the scalars, or near the h or S resonances, in this case also for universal mixing. The remaining intermediate scenario in which the dominant process is N N → S_1 h is discussed in Ref. <cit.>. In addition to direct-detection experiments, the clockwork dark matter framework discussed here has a very rich phenomenology. First of all, peculiar signals at indirect-detection experiments are possible, because of the unstable nature of dark matter. For instance, the decay N → h ν would give rise to a monochromatic neutrino line. An important feature of the clockwork construction that we are discussing is the presence of a band of fermionic states, the clockwork gears ψ_i, that need to be in the TeV range or less. From the phenomenological point of view, these are essentially a collection of (pseudo-Dirac) heavy neutrinos in the observable mass range, with Yukawa couplings that are large, in the clockwork spirit. Therefore, the relevant indirect constraints are from electroweak precision tests (EWPT), |B_lψ|^2 ≡ y^2 v^2/(2 m_ψ^2) ⪅ 10^-3 and lepton flavour violation (LFV), BR(μ→ e γ)≈ 8 × 10^-4|B_eΨ|^2 |B_μΨ|^2 < 4.2 × 10^-13, although the latter depend on the flavour structure of the couplings. LFV signatures are particularly promising, since planned experiments will improve the existing limits by several orders of magnitude in the near future. As for direct searches at colliders, lepton-number conserving processes will be tested, for couplings allowed by EWPT, up to m_ψ≈ 200 GeV with 300 fb^-1 of data at LHC, whereas lepton-number violating processes will be tested up to m_ψ≈ 300 GeV.§ CLOCKWORK FROM A FLAT EXTRA DIMENSIONThe structure of fields and couplings (<ref>) calls for some justification. Clearly, a possibility is that this is the discretized “deconstructed” version of an extra dimension, as suggested in <cit.>. There, in the fermionic case, the clockwork Lagrangian is obtained by discretizing a single massless fermion in 5D, with a metric d s^2 = e^4 k Z /3 (d x^2 + d Z^2), where Z is the coordinate along the 5th dimension.In Ref. <cit.> we have shown that this is not the only possibility. We have considered a massive fermion field in 5D, with bulk mass M, and a flat metric, with 0 ≤ Z ≤ R. In this case, the light (chiral) mode is introduced separately on one of the branes, say Z=0, and the SM leptons on the other one, sat Z=R. Equivalently, as in the subsequent construction of Ref. <cit.>, the chiral mode can be seen as a boundary term for the bulk action. After discretizing the extra dimension in N points with lattice spacing a = R/N, taking into account the appropriate Wilson term that avoids double counting, we find the 4D Lagrangianℒ ⊃ ∑_i=0^N-11/a L_i+1 R_i - ∑_i=1^N( 1/a + M )L_i R_i.We see that the clockwork setup (<ref>) is realized, by identifyingm ≡1/a ,q m ≡1/a + M.The continuum limit is obtained for N →∞, i.e. a → 0 with R fixed. In this case q → 1 and the total clockwork suppression remains finiteq^N = ( 1 + π R M/N)^N→ e^π R Mi.e. a sensible continuum limit do indeed exist. In this framework, also the scalars S_i, C_i in (<ref>) can come from the discretization of a single field in the bulk. In particular, the fields C_i could originate form a single 5D scalar with a Yukawa coupling with the fermion in the bulk, whereas the fields S_i could come from the link variables of a 5D abelian gauge field, under which the bulk fermion is charged, and these from the 4D perspective are pseudo-scalars.§ CONCLUSIONSAn unstable dark-matter candidate requires a huge suppression of its decay processes to comply with stability on cosmological time scales and bounds from indirect-detection experiments. I have discussed how the clockwork mechanism can provide that, without decoupling the relevant physics. In particular, this framework requires [not “can accommodate”.] the presence of new clockwork states at the TeV range or less and large couplings with the Standard Model, in order to obtain the correct relic density. The very same states mediate both DM annihilation in the early Universe, which fixes the relic density, and the clockwork-suppressed decay processes, thus providing a rather unique connection between them. From the phenomenological point of view, this construction yields rich signatures in a number of experiments. The key aspect is the presence of a band of pseudo-Dirac right-handed neutrinos in the TeV range or less, that can be searched for at colliders and lepton-flavour-violation experiment.I have also discussed how this setup could emerge from the deconstruction of a flat extra dimension. This (de)construction differs form the original one of Ref. <cit.>, which instead makes use of a curved metric. After we originally presented this in Ref. <cit.>, it has been shown very recently <cit.> that the two constructions are indeed equivalent, at least in the bosonic case.Finally, I conclude mentioning that Majorana neutrino masses can be incorporated in the same framework discussed here, and that these are clockwork-suppressed too. For more details on this and the rest of the material discussed here, I refer the reader to the original work <cit.>, where these matters were first discussed.§ ACKNOWLEDGEMENTSI thank the organizers of the Moriond Electroweak 2017 conference, and in particular Jean-Marie Frère, for providing a beautiful environment to discuss fundamental physics. I also thank Thomas Hambye and Michel Tytgat for collaboration on the work discussed in this note. My work has been supported by a postdoctoral fellowship of ULB and by the Belgian Federal Science Policy IAP P7/37. § REFERENCES 99Choi:2015fiu K. Choi and S. H. Im,JHEP1601 (2016) 149 [arXiv:1511.00132 [hep-ph]]. Kaplan:2015fuy D. E. Kaplan and R. Rattazzi,Phys. Rev. D93 (2016) no.8,085007[arXiv:1511.01827 [hep-ph]]. Giudice:2016yja G. F. Giudice and M. McCullough,JHEP1702 (2017) 036 [arXiv:1610.07962 [hep-ph]].Arkani-Hamed:2016rle N. Arkani-Hamed, T. Cohen, R. T. D'Agnolo, A. Hook, H. D. Kim and D. Pinner,Phys. Rev. Lett.117 (2016) no.25,251801 [arXiv:1607.06821 [hep-ph]].Graham:2015cka P. W. Graham, D. E. Kaplan and S. Rajendran,Phys. Rev. Lett.115 (2015) no.22,221801 [arXiv:1504.07551 [hep-ph]].Kehagias:2016kzt A. Kehagias and A. Riotto,Phys. Lett. B767 (2017) 73 [arXiv:1611.03316 [hep-ph]].Farina:2016tgd M. Farina, D. Pappadopulo, F. Rompineve and A. Tesi,JHEP1701 (2017) 095 [arXiv:1611.09855 [hep-ph]].Ahmed:2016viu A. Ahmed and B. M. Dillon,arXiv:1612.04011 [hep-ph].Hambye:2016qkf T. Hambye, D. Teresi and M. H. G. Tytgat,arXiv:1612.06411 [hep-ph].Craig:2017cda N. Craig, I. Garcia Garcia and D. Sutherland,arXiv:1704.07831 [hep-ph].Giudice:2017suc G. F. Giudice and M. McCullough,arXiv:1705.10162 [hep-ph]. | http://arxiv.org/abs/1705.09698v2 | {
"authors": [
"Daniele Teresi"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526195145",
"title": "Clockwork Dark Matter"
} |
M. Mezcua Observational evidence for intermediate-mass black holes Département de Physique, Université de Montréal,C.P. 6128, Succ. Centre-Ville, Montreal, Quebec H3C 3J7, CanadaEmail: [email protected] EVIDENCE FORINTERMEDIATE-MASS BLACK HOLES MAR MEZCUA December 30, 2023 =========================================================Intermediate-mass black holes (IMBHs), with masses in the range 100-10^6 M_⊙, are the link between stellar-mass BHs and supermassive BHs (SMBHs). They are thought to be the seeds from which SMBHs grow, which would explain the existence of quasars with BH masses of up to 10^10 M_⊙ when the Universe was only 0.8 Gyr old. The detection and study of IMBHs has thus strong implications for understanding how SMBHs form and grow, which is ultimately linked to galaxy formation and growth, as well as for studies of the universality of BH accretion or the epoch of reionisation. Proving the existence of seed BHs in the early Universe is not yet feasible with the current instrumentation; however, those seeds that did not grow into SMBHs can be found as IMBHs in the nearby Universe. In this review I summarize the different scenarios proposed for the formation of IMBHs and gather all the observational evidence for the few hundreds of nearby IMBH candidates found in dwarf galaxies, globular clusters, and ultraluminous X-ray sources, as well as the possible discovery of a few seed BHs at high redshift. I discuss some of their properties, such as X-ray weakness and location in the BH mass scaling relations, and the possibility to discover IMBHs through high velocity clouds, tidal disruption events, gravitational waves, or accretion disks in active galactic nuclei. I finalize with the prospects for the detection of IMBHs with up-coming observatories. § INTRODUCTION Black holes (BHs) were found to be the solution to the Einstein field equations of general relativity in the early 1960s[The metric describing a non-rotating BH with no charge (Schwarzschild BH) was found by Karl Schwarzschild in 1916, but it wasn't until 1963 and 1965 that Roy Kerr and Ezra Newman found the solutions for a rotating and a rotating plus electrically charged BH, respectively.] and, according to the no-hair theorem, they can be described by three parameters: mass, spin and charge. Independently of spin and charge, BHs are commonly classified into three types according to their mass: Stellar-mass BHs (3 M_⊙ < M_BH≤ 100 M_⊙). They are the end-product of a massive star (>15 M_⊙) that collapses into a BH when the star's fuel supply is burned out and the internal pressure is insufficient to support the gravitational force. The first solid evidence for the existence of BHs came from X-ray and optical observations in the 1970s and 1980s of X-ray binaries (XRBs; e.g., Cygnus X-1, LMC-X3; see reviews by ; ) whose compact object had a mass above 3 M_⊙ (i.e. too massive for a neutron star or a white dwarf; ). Today, there are more than 20 confirmed stellar-mass BHs in XRBs.Supermassive BHs (SMBHs; M_BH≥ 10^6 M_⊙). SMBHs are the most massive types of BHs and they reside at the center of most massive galaxies in the local Universe (see reviews by ; ; ). The best observational evidence for a SMBH comes from studies of the proper motion of stars around the center of our Galaxy, which reveal the presence of a central BH with a mass of 4 × 10^6 M_⊙ (; ). The masses of SMBHs are observed to correlate with some of their host galaxy properties, such as bulge stellar mass, luminosity, or stellar velocity dispersion, suggesting a co-evolution or synchronized growth between galaxies and their central BHs (e.g. ; ; ; ). SMBHs grow through the accretion of matter, during which they are observable as active galactic nuclei (AGN) or quasars. Quasars with BH masses of up to 10^10 M_⊙ have been detected when the Universe was only 0.8 Gyr old (z ∼ 7, e.g., ; ; ; ; ). To reach this mass in such a short time, SMBHs should have started as lower-mass seed BHs of more than 100 M_⊙ at z > 10 and grow very fast via accretion and mergers (e.g., ; see Sect. <ref>).Intermediate-mass BHs (IMBHs; 100 M_⊙ < M_BH < 10^6 M_⊙). They are the link, thought to be missing for many decades, between stellar-mass and SMBHs and the possible seeds from which SMBHs in the early Universe grew. Finding proof of their existence is thus pivotal for understanding SMBH and galaxy growth. The study of their accretion physics and radiative properties is important for understanding the effects of BH feedback in the formation of the first galaxies and the quenching of star formation (e.g., ; ; ), for studies of the epoch of reionisation (), and for confirming whether accretion is a scale-invariant physical mechanism governing BHs of all masses. The latter has been already inferred from the fundamental plane of accreting BHs (e.g., ; ; ), which is a correlation between BH mass, X-ray luminosity (proxy of accretion flow) and radio luminosity (proxy of jet ejection) that extends all the way from stellar-mass to SMBHs, proving that a disk-jet coupling mechanism takes places in BHs of all masses (). Finally, IMBHs offer the best testbed for investigating tidal disruptions of stars (e.g., ;; see Sect. <ref>) and the coalescence of IMBHs pairs provide the right signal for detection in gravitational wave experiments (e.g., ; ; see Sect. <ref>). Proving the existence of primordial IMBHs at z > 7 is a timely endeavour with the current facilities (e.g., ; ; ; ; ; ). Most of the studies of high-redshift BHs are limited to luminous quasars hosting SMBHs (see review by ). However, observational evidence of those seed BHs that did not grow into SMBHs (the 'leftovers' of the early Universe) should be found in the local Universe (e.g., see review by ; ) and up to z ∼2.4 (; Mezcua et al. in preparation): in dwarf galaxies, as because of their low mass and metallicity they resemble those galaxies formed in the early Universe (e.g., , , ; ; ); in nearby globular clusters, as stellar clusters are one of the sites of possible IMBH formation (e.g., ; ; ; ); or in the form of off-nuclear ultraluminous X-ray sources (ULXs) in the halos and spiral arms of large galaxies, as ULXs could be the stripped nucleus of dwarf galaxies (e.g., ; ; ; , , ; ). The aim of this review is to assemble all the observational evidence found so far for IMBHs. I will focus on globular clusters (Sect. 2.1), ULXs (Sect. 2.2), dwarf galaxies (Sect. <ref>), and seed BH candidates at z > 6 (Sect. <ref>), and mention other pathways for IMBH detection such as tidal disruption events, gravitational waves, accretion disks in AGN, and high velocity clouds (Sect. <ref>). I will first start by providing a brief summary on seed BH formation (see Fig. <ref>; for more details see reviews by e.g., ; ; ; ; ). Throughout the review I will use the terms 'seed BH' and 'IMBH' to refer to those BHs in the same mass regime (100 M_⊙ < M_BH < 10^6 M_⊙). The difference between the two relies on the redshift of the sources: 'seed BH' refers to the early Universe, while the term 'IMBH' will be used for lower redshift objects. Those BHs with M_BH∼10^5-10^6 M_⊙ are sometimes referred to as 'low-mass BHs' or 'low-mass AGN' in the literature (e.g., ; ), and other times as 'IMBHs' (e.g., ; ). Because of this disparity, in this review I define IMBHs as having 100 M_⊙ < M_BH < 10^6 M_⊙ and I use the term 'low-mass AGN' or 'low-mass BH' to refer to those BHs with ∼10^6 M_⊙.§.§ IMBHs: formation scenarios BHs are fed by the accretion of gas in a process in which a small fraction of the energy of the accreted gas is released in the form of radiation. The luminosity at which the outward radiation balances the inward gravitational force is referred to as the Eddington luminosity, or Eddington limit, and can be written as:L_Edd = 4π cGm_pM_BH/σ_T≃ 1.3 × 10^38(M_BH/M_⊙) erg/swhere c is the speed of light, G the gravitational constant, m_p the proton mass and σ_T the Thompson scattering cross-section. The Eddington rate is the rate at which a BH radiating at the Eddington luminosity is accreting mass from its surrounding. A seed BH of 100 M_⊙ < M_BH < 10^6 M_⊙ accreting at the Eddington rate would need more than 0.5 Gyr to reach 10^9 M_⊙ (assuming a typical radiative efficiency of 10%; ). Therefore, the existence of SMBHs of more than 10^9 M_⊙ when the Universe was ∼1 Gyr old implies that the seed IMBHs formed at z ≥ 10, in a primordial cold dark matter Universe in which dark matter halos grow out of the gravitational collapse of small density fluctuations. The first stars formed from the collapse of pristine (metal-free) gas in these dark matter halos. In the absence of metals (elements heavier than He and Li), gas cooling is only possible by means of atomic and molecular hydrogen (H_2; e.g., ; ). In such a early Universe, seed BHs could form from:(i) Population III stars. If H_2 dominates the cooling rate, the primordial gas can cool down to ∼100 K and collapse into protostars (known as Pop III stars) of typically a few hundreds of solar masses (e.g., ) and up to ∼10^3 M_⊙ (e.g., ). Only those Pop III stars over 260 M_⊙ collapse into a BH containing at least half of the initial stellar mass (i.e. M_BH≥100 M_⊙; ; ; ) and thus into IMBHs. However, the existence of numerous isolated stars more massive than 260 M_⊙ has been put into doubt by simulations showing that Pop III stars may instead form in binaries or multiple systems of 10–100 M_⊙ (e.g., ; ; ; see review by ). To reach a BH mass of 10^9 M_⊙ in ∼0.5 Gyr (the time elapsed between z = 10 and z = 6), Pop III stellar BH seeds would have to grow via supra-exponential accretion[Supra-exponential growth can describe Hoyle-Lyttleton wind accretion or spherical Bondi accretion (see e.g.,and references therein for further details).] (e.g., when bound in a nuclear stellar cluster fed by flows of dense cold gas; ) or undergo phases of accretion at super-Eddington rates (e.g., ; ; ; ).(ii) Direct collapse. IMBHs could also form inside the first metal-free (or very metal-poor) protogalaxies by direct collapse of rapidly inflowing dense gas (e.g., ; ; ; ). For the gas to reach the halo center and collapse to occur, fragmentation leading to star formation must be inhibited and the gas must have low angular momentum so that it undergoes gravitational instabilities instead of forming a rotationally-supported disk. The gravitational instabilities and inward gas transport can be achieved by the formation of bars within bars (), while fragmentation can be prevented if H_2 is destroyed by an intense Lyman-Werner (ultraviolet –UV) radiation and atomic hydrogen dominates the cooling rate. In such halos, gas cools down gradually only to ∼10^4 K and can form a supermassive star of ∼10^5 M_⊙. The collapse of such a supermassive star forms a BH of ∼10^4-10^5 M_⊙ (e.g., ), orders of magnitude more massive than Pop III stellar BH seeds, that can grow into a SMBH by z ∼7 without having to invoke super-critical accretion (e.g., ). Given that the cosmic UV background may not be intense enough to prevent H_2 formation and gas fragmentation (), direct collapse can only occur in halos close (within 15 kpc) to luminous star-forming galaxies producing sufficient Lyman-Werner radiation (e.g., ; ; ) and is thus thought to be a much less common mechanism of IMBH formation. Previously-formed direct collapse seed BHs could also provide the radiation necessary to prevent star formation and form additional BHs, in which case they would be more abundant than expected if including only star-forming galaxies as the source of Lyman-Werner radiation (). (iii) Mergers in dense stellar clusters. Another way to form IMBHs is via runaway collisions of stars in dense stellar clusters (e.g., ; ; ). Compact nuclear stellar cluster can form out of the second generation of low-mass stars that formed from the gas metal-polluted (but with still highly sub-solar metallicity) by the first generation of Pop III stars (). Frequent stellar mergers within the cluster can lead to the formation of a supermassive star that will collapse into a BH of ∼10^2-10^4 M_⊙ (). Dense stellar clusters might also form at the center of the protogalaxies previously described when the central density is increased by the inflow of the metal-poor gas. The mass of these clusters is typically of ∼10^5 M_⊙ and runaway stellar collisions can yield the formation of a supermassive star that will collapse into a BH of ∼10^3 M_⊙ (). (iv) Other models: direct SMBH formation. Instead of seed BHs at z > 10 having to grow through accretion and mergers to 10^9 M_⊙ by z ∼7, the existence of SMBHs in such a young Universe can be also explained if these formed directly by mergers of massive protogalaxies at z ∼5–6 (, ; ; but see also ). When the two protogalaxies merge, merger-driven inflows of metal-enriched gas produce a massive (≥ 10^9 M_⊙) compact nuclear gas disk with a high angular momentum. In the inner parsecs an ultra-dense massive disky core is formed, which can turn into a supermassive star and collapse directly into a SMBH of 10^8–10^9 M_⊙ (). Although this avenue of SMBH formation requires initial conditions more complex than those of the direct collapse in less massive halos scenario, it offers an explanation for the existence of high-z SMBHs without having to prevent gas cooling and star formation nor requiring primordial gas composition. Since the detection of z >7 seed BHs is yet challenging with current instrumentation, determining which (if any) of the above seed BH formation scenarios is correct requires to construct models of high-z seed BH formation, predict the leftover populations of IMBHs at low redshift for each scenario, and compare these to the observed number of IMBHs so far available. In the local Universe, leftover IMBHs are expected to reside in dwarf star-forming galaxies, as these have undergone a quieter merging/accretion history than massive galaxies and are thus more likely to resemble the primordial low-metallicity galaxies of the infant Universe. Given that in the early Universe Pop III seed BHs were presumably much more abundant than direct collapse seed BHs, simulations predict a higher BH occupation fraction in today's dwarf galaxies if the Pop III scenario was the dominant seeding mechanism at z >10 (; ; ; see Fig. <ref>, top). The different seed BH formation scenarios should also leave an imprint on the tight correlations found between SMBH mass and host galaxy properties: light Pop III seeds are predicted to be undermassive with respect to the M_BH-σ relation, while the more massive direct collapse seed BHs are expected to lie above it (, ; see Fig. <ref>, bottom). At the high-mass end, no differences are predicted by the different seeding scenarios – in these systems the initial conditions (seed BH mass) have been erased as a consequence of several mergers and accretion phases. Observationally, the M_BH-σ and M_BH-M_bulge correlations are indeed found to be very strong for SMBHs of 10^6-10^9 M_⊙, but they seem to bend or to have a large scatter both at the highest (e.g., ; ; ) and lowest (e.g., ; ; ; ; see Sect. <ref> and Fig. <ref>) mass regimes (although see ; ). Finding observational evidence of IMBHs in the local Universe, deriving their occupation fraction, and measuring the BH mass in the low-mass regime (i.e. in dwarf galaxies) is thus pivotal for understanding how seed BHs formed in the early Universe and evolved until the SMBHs observed today. § OBSERVATIONAL EVIDENCE FOR IMBHS To probe the existence of IMBHs we need to measure their BH mass. The use of stellar or gas dynamics is the most secure way to weight BHs; however, the sphere of influence of a BH of 10^5 M_⊙ is of only 0.5 pc and cannot be resolved, with the current instrumentation, beyond ∼1 Mpc. Dynamical BH masses in the intermediate-mass regime have thus only been obtained for nearby dwarf galaxies (most within the Local Group) and in globular clusters. In the absence of kinematic signatures, radiative signatures of BH accretion (e.g., X-ray and radio emission) must be used to infer the presence of IMBHs and estimate their BH mass. These methods have provided the detection of IMBHs in globular clusters, ultraluminous X-ray sources, and dwarf galaxies in the local Universe and up to z ∼2. At higher redshifts, the near-infrared (NIR) detection of a strong Lyα emission line and the combination of NIR photometry with deep X-ray observations has yielded the identification of a few direct collapse BH candidates. §.§ Globular clusters One of the possible formation scenarios for IMBHs is the runaway core collapse and coalescence of stars in stellar clusters. Globular clusters have thus been common targets in the search for IMBHs. The presence of IMBHs in globular clusters was first suggested by <cit.> when studying their X-ray emission. They argued that the flux of the globular cluster X-ray sources could be explained by accretion onto a 100-1000 M_⊙ BH, which was supported by the finding of high central escape velocities. Since then many studies have aimed at detecting IMBHs in globular clusters through their radiative accretion signatures, but no conclusive results have been obtained (e.g., ; ; ; ; ; ; ). The only globular cluster with detected X-ray and radio emission is G1, in M31 (; ; ), suggesting the presence of an IMBH with a BH mass of (1.8 ± 0.5) × 10^4 M_⊙ estimated from photometric and kinematic observations (). However, later results obtained by <cit.> did not detect any radio emission. Globular clusters have little gas and dust and thus any signatures of accretion from a putative IMBH in the X-ray and radio regimes are expected to be very low. This could explain the lack of X-ray and radio detections, leaving the kinematic signatures as currently the most viable method of probing the presence of IMBHs in these stellar systems. The first IMBH candidate in a globular cluster based on kinematic measurements and dynamical modeling was M15 (; ), which presented a pronounced rise in its velocity dispersion profile and for which a BH mass of (3.9 ± 2.2) × 10^3 M_⊙ was estimated (, ; ). However, the results could also be explained by a central concentration of compact objects (e.g., ; ), which weakened the IMBH scenario for this globular cluster. The use of integral field spectroscopy, to obtain the central velocity-dispersion profile, and of photometric data (e.g., with the Hubble Space Telescope, HST), to obtain the cluster photometric center and surface brightness profile, has allowed estimating the BH mass in a dozen more globular clusters by comparing the data to spherical dynamical models. This is the case of another strong IMBH candidate in a globular cluster, ω Centauri, for which <cit.> claimed the presence of an IMBH of best-fitted mass (4.7 ± 1.0) × 10^4 M_⊙ while <cit.> reported an upper limit of 1.2 × 10^4 M_⊙. <cit.> also found that the velocity dispersion profile of ω Centauri is best fitted by an IMBH of 10^4 M_⊙. Using integral-field spectroscopy and HST photometry,(2012, , ) reported upper limits on the mass of a putative BH in the globular clusters NGC 1851, NGC 2808, NGC 5694, NGC 5824, and NGC 6093 (see Table <ref>) and predicted the presence of an IMBH of (3 ± 1) × 10^3 M_⊙ in NGC 1904, of (2 ± 1) × 10^3 M_⊙ in NGC 6266, and of (2.8 ± 0.4) × 10^4 M_⊙ in NGC 6388. An IMBH of (1.5 ± 1.0) × 10^3 M_⊙ is also suspected in the globular cluster NGC 5286 () and <cit.> reported the possible presence of an IMBH of ∼9400 M_⊙ in NGC 6715 (M54), a globular cluster located at the center of the Sagittarius dwarf galaxy. Nonetheless, no observational evidence for accretion in the form of X-ray or radio emission has been detected for any of these IMBH candidates, indicating that the globular clusters are devoid of gas within the BH radius of influence.<cit.> recently proposed to use measurements of pulsar accelerations, which show an additional component beyond that caused by the gravitational potential of the cluster, together with N-body simulations to obtain stringent constraints on the central BH mass of globular clusters.They apply this method to the globular cluster 47 Tuc, which hosts 25 known millisecond pulsars (; ; ), and find that those models with an IMBH produce pulsar accelerations more consistent with the observed accelerations than models without an IMBH. They infer a BH mass for the IMBH of 2.2 × 10^3 M_⊙ and provide an independent measure of the cluster mass (0.75 × 10^6 M_⊙) that is in agreement with kinematic results (). This method is a promising way to infer the presence of an IMBH in those globular clusters whose lack of gas in their cores does not permit an electromagnetic detection of the BH. Alternative methods include gravitational waves (e.g., ) and gravitational microlensing (e.g., ); however, these have not yet provided any IMBH candidates. §.§ Ultraluminous X-ray sources IMBHs were also proposed to explain the nature of ultraluminous X-ray sources (ULXs). ULXs are extragalactic and off-nuclear X-ray sources with luminosities L_X≥ 10^39 erg s^-1, which corresponds to the Eddington limit for a 10 M_⊙ stellar-mass BH. ULXs could thus host BHs of intermediate masses if accreting isotropically below the Eddington rate (e.g., ). Alternatively, they could be powered by stellar-mass BHs or magnetized neutron stars with near to super-Eddington accretion (e.g., see reviews by ; ; ; ; ).The first evidence for the presence of IMBHs in ULXs came from the fit of the X-ray spectrum by a cool disk (soft excess of 0.1-0.3 keV) plus a power-law tail: if coming from a standard BH, the inverse proportionality between disk temperature and BH mass () implied masses ∼10^4 M_⊙ (e.g., ). However, the later finding of a curved spectrum with a cutoff above a few keV (e.g., ; ; ; ; ) weakened the interpretation of a standard BH with an intermediate mass and supported the scenario in which most ULXs (those with X-ray luminosities < 5 × 10^40 erg s^-1) are stellar-mass BHs in super-Eddington accretion regimes (known as the ultraluminous state; ). Dynamical evidence for the presence of a stellar-mass BH has been found in some ULXs (M101-X1, ; NGC 7793 P13, , but later found to host a neutron star, , ), though the most surprising case is that of M82 X-2. M82 X-2 was thought to host an IMBH of more than 10^5 M_⊙ () based on its high X-ray luminosity (peak at 3 × 10^40 erg s^-1), strong variability on scales of weeks, and low-frequency quasi-periodic oscillations (QPOs; ; ; ). However, the finding of X-ray pulsations indicates that M82 X-2 is a neutron star (). This was the first discovery of a neutron star hosted by a ULX, an unexpected scenario for which there are two more known cases (; ) and which could explain the nature of many more ULXs ().Extreme ULXs, with L_X≥ 5 × 10^40 erg s^-1, and hyperluminous X-ray sources (HLXs), with L_X≥ 10^41 erg s^-1, remain as the best candidates to IMBHs as their high X-ray luminosities can be difficult to explain even by super-Eddington accretion (see Table <ref>). This is the case for HLX-1, the most well-known off-nuclear IMBH candidate. HLX-1 has an isotropic X-ray luminosity of 10^42 erg s^-1 () and, similarly to XRBs, it undergoes periodic outbursts over a timescale of months (; ; ; ) during which spectral state transitions over a timescale of days occur (; ; ; ). The source also exhibits transient jet radio emission following the transition from a powerlaw-like spectrum (X-ray low/hard state) to a thermal-shape spectrum (X-ray high/soft state; ; ; see Fig. <ref>). From the radio emission and applying the fundamental plane of accreting BHs, a BH mass of 9 × (10^3 - 10^4) was estimated by <cit.> and an upper limit of 2.8 × 10^6 M_⊙ by <cit.>. <cit.> suggested that the radio emission is Doppler-boosted and that HLX-1 could be an outlier on the fundamental plane. Based on Eddington arguments, <cit.> set a lower limit on the BH mass of 9 ×10^3 M_⊙, while using accretion disk models the BH mass was estimated to be in the range 6 × 10^3 M_⊙ < M_BH < 3 × 10^5 M_⊙ (; ; ). The redshift of the optical counterpart confirms the association of HLX-1 with the host lenticular galaxy ESO 243-49 and suggests that it could be the nucleus of a dwarf galaxy that underwent a minor merger with ESO 243-49 (; ; ).M82 X-1 is the second strongest IMBH among HLXs because of its variability (Ptak & Griffiths 1999; Kaaret & Feng 2007), peak X-ray luminosity above 10^41 erg s^-1 (; ), spectral transitions similar to those of standard BHs (), and low-frequencyQPOs (; ). Based on the finding of twin-peak QPOs at ∼3 Hz and 5 Hz and extrapolating the inverse scaling between BH mass and frequency that holds for stellar-mass BHs, the BH mass of M82 X-1 was estimated to be 428 ± 105 M_⊙ (). M81 X-1 was also suggested to be the nucleus of a stripped galaxy (), as is the case of the extreme ULX NGC 2276-3c, located in a peculiar arm of the spiral galaxy NGC 2276 (;, 2015).NGC 2276-3c was detected by the XMM-Newton satellite as an extreme ULX of L_X∼6 × 10^40 erg s^-1 blended with two other ULXs (). Later observations with the Chandra X-ray satellite were able to resolve the source, which is strongly variable and whose spectral modeling was consistent with the sub-Eddington hard X-ray state (). Very long baseline interferometry (VLBI) radio observations were performed quasi-simultaneously (one day difference) to the Chandra observations, revealing a radio jet, characteristic of the hard state, with a size several orders of magnitude larger than the typical jet size of stellar-mass BHs but smaller than those of SMBH (Fig. <ref>; ; though Yang et al. 2017 failed to confirm the detection). A BH mass of 5 × 10^4 M_⊙ was estimated using the fundamental plane for accreting BHs, consistent with NGC 2276-3c containing an IMBH in the hard state (). The detection of compact radio jets has suggested the presence of IMBHs in some other ULXs for which the fundamental plane provides a BH mass estimate in the IMBH regime (e.g., N4861-X2, N4088-X1, , ; IC342 X-1, ; N5457-X9, ). However, the luminosity of these ULXs being < 5 × 10^40 erg s^-1 makes them also consistent with the super-Eddington accretion scenario. The presence of jets in the super-Eddington regime is expected from simulations () and was detected for the ULX Holmberg II X-1, which is thought to be powered by a BH of mass in the range 25–100 M_⊙ ().The presence of IMBHs was also suggested in the HLXs 2XMM J011942.7+032421 (; ) and CXO J122518.6+144545 (; ), and in the ULXs NGC 1313 X-1 (e.g., ), NGC 5408 X-1 (e.g., ), and M51 ULX-7 (), among others. The HLX 2XMM J011942.7+032421, with a peak X-ray luminosity of 1.53 × 10^41 erg s^-1, presents short-term variability and its X-ray spectrum can be fitted by multicolor disk emission with a mass ≥ 1900 M_⊙ (). Its optical spectrum confirms the location of the HLX in the spiral arm of the galaxy NGC 470 and shows a high-ionization HeII emission line with a large velocity dispersion, which suggests the presence of a compact (<5 AU) highly ionized accretion disk (). The HLX CXO J122518.6+144545 () reached a peak X-ray luminosity of 2.2 × 10^41 erg s^-1 and its X-ray count rate varies by a factor >60, making it the only second outbursting HLX after HLX-1 (). It also exhibits optical variability likely related to the X-ray variability. Its high X-ray luminosity makes it another strong IMBH candidate. NGC 1313 X-1 shows 3:2 ratio QPOs () and has an X-ray luminosity above 10^40 erg s^-1. The first X-ray spectral studies suggested that NGC 1313 X-1 hosts an IMBH of ∼1000 M_⊙ (). A later analysis and applying the same scaling as for M82 X-1 yielded a BH mass estimate for NGC1313 X-1 of 5000 ± 1300 M_⊙ (). However, the recent detection of a spectral cutoff above 10 keV and of an ultra-fast (0.2c) disk wind indicative of supercritical accretion rule out the IMBH scenario for this source (; ; ; ). Another controversial case was that of NGC 5408 X-1. It also shows QPOs, clear variability (on scales of minutes days, months and years), and a peak X-ray luminosity of 10^40 erg s^-1 (e.g., ; , ; ; ; ); however, the constancy of its X-ray spectral parameters () and variability of the QPO frequency () hampered a robust conclusion on its nature. While <cit.> proposed the presence of an IMBH of at least 800 M_⊙, <cit.> and <cit.> argued in favor of a super-Eddington accreting BH whose winds could explain the spherically-symmetric nebula observed around NGC 5408 X-1 in the radio and optical bands (; ; ; ). The recent detection of an ultra-fast outflow in the X-ray spectrum of NGC 5408 X-1 finally confirmed the super-Eddington accreting nature of this source (). Last but not least, the ULX-7 in the spiral galaxy M51 is suggested to be powered by an IMBH based on its hard spectrum, high rms variability, and the lower limit on the BH mass of 1.6 × 10^3 M_⊙ derived using the relationship between BH mass and high frequency break in the power spectrum (). However, a pulsar nature for this ULX cannot be ruled out.The search for IMBHs in ULXs, and specially in HLXs, is an active field in which many strong IMBH candidates are being found either by cross-correlating X-ray and optical catalogs (e.g., ; ) or serendipitously (e.g., ). Detailed analysis is yet required in order to distinguish between other possible scenarios proposed to explain their X-ray luminosity, such as super-Eddington accretion. §.§ Dwarf galaxies Unlike massive galaxies, dwarf galaxies (M_*≤ 3 × 10^9 M_⊙) have not significantly grown through mergers/accretion and thus resemble those galaxies formed in the early Universe. They constitute thus one of the best places where to look for seed BHs (e.g., ; ; ). The first observational evidence for IMBHs in dwarf galaxies were identified in the late 80s in the spiral galaxy NGC 4395 (M_*∼1.3 × 10^9 M_⊙; Fig. <ref>) and the elliptical galaxy Pox 52 (M_*∼1.2 × 10^9 M_⊙; Fig. <ref>) by the finding of high-ionization narrow emission lines and broad Balmer emission lines in their optical spectrum (; ). These, together with the detection of hard X-ray emission, indicate the presence of an AGN with a BH mass of M_BH∼3 × 10^5 M_⊙ estimated from the width of the broad emission lines under the assumption that the gas is virialized (; ; ; ; ; ). NGC 4395 has, in addition, a compact radio jet () and is one of the few dwarf galaxies for which a dynamical BH mass measurement has been possible (M_BH = 4^+8_-3× 10^5 M_⊙; ). An upper limit on the BH mass of M_BH = 1.5 × 10^5 M_⊙ was also estimated from dynamical modeling for the dwarf S0 galaxy NGC 404 (; ) and is in agreement with that derived using the fundamental plane of BH accretion (; but see ). Further evidence for the presence of an accreting BH in NGC 404 comes from its optical classification as a LINER[Low-ionization nuclear emission line region (LINERs) are associated with low-luminosity AGN (e.g., ; )], and from the finding of a hard X-ray core (), UV variability (), unresolved and variable hot dust emission (), and mid-infrared (MIR) AGN-like emission lines (). The detection of hard X-ray emission spatially coincident with core radio emission is, in the absence of dynamical mass measurements, a very strong tracer of BH accretion. This yielded the discovery of the first AGN in a blue compact dwarf galaxy (Henize 2-10; ). Although Henize 2-10 is a starburst galaxy whose optical emission is dominated by star formation, it hosts a low-mass BH (M_BH∼10^6 M_⊙) at its center as revealed by Chandra X-ray observations and VLA and VLBI radio observations ( 2011, ; ). A low-mass BH was also found based on spatially coincident Chandra X-ray emission and VLA radio emission in Mrk 709, a low-metallicity blue compact dwarf formed by a pair of interacting galaxies (). Mrk 59, the compact core of the blue compact dwarf galaxy NGC 4861, has been also found to host an IMBH of ∼5 × 10^4 M_⊙ based on high-resolution X-ray and radio observations with Chandra and the European VLBI Network, respectively, and the detection of high-excitation emission lines (e.g., HeII) typically associated with gas photoionized by AGN (; Yang et al. in preparation)The first searches for low-mass BHs had already begun with the arrival of the Sloan Digital Sky Survey (SDSS), which provided optical spectra for more than 100,000 galaxies. This allowed a systematic search for low-mass BHs through the kinematics and the ionization properties of the excited gas: the broad-line widths of the gas provide an estimate of the BH mass under the assumption that the gas is virialized, while narrow emission line diagnostics (e.g.,[OIII]/Hβ versus [NII]/Hα; ; ) are used to distinguish between AGN and starburst emission (; ; ; ). This yielded the identification of 229 () and 309 () low-mass AGN with BH masses < 2 × 10^6 M_⊙. Most of the host galaxies of these low-mass BHs are of late-type and more massive than typical dwarf galaxies. It was the discovery of Henize 2-10 that invigorated the quest for low-mass AGN in dwarf galaxies: <cit.> found 136 optically selected dwarf galaxies at z < 0.055 that qualified as either AGN or composite objects in the narrow-line emission diagnostic diagram (see Fig. <ref>) and provided an AGN fraction (not corrected for incompleteness) of 0.5%. Ten of these sources present broad optical emission lines, from which a range of BH masses of ∼7 × 10^4-1 × 10^6 M_⊙ was estimated, and have X-ray luminosities significantly higher than what would be expected from star formation, thus confirming the presence of accreting BHs in these galaxies (). Follow-up studies of the source RGG 118 in <cit.>, a dwarf galaxy classified as a composite object and for which there was a hint of broad emission in the SDSS spectrum, revealed the presence of hard X-ray emission and broad Hα line emission, from which a BH mass of ∼5 × 10^4 M_⊙ was estimated (). This makes RGG 118 the lightest nuclear BH known. Using also optical emission line diagnostics and SDSS data, <cit.> identified 18 additional IMBH candidates with a minimum BH mass in the range 10^3-10^4 M_⊙. Most of their host galaxies have stellar masses above the typical threshold of 3 × 10^9 M_⊙. <cit.> identified 3 additional AGN candidates using a combined criterion that includes MIR color cuts in addition to the classical narrow-line diagnostic diagram. MIR color searches rely on the different colors of dust when heated by AGN or by stars or non-active galaxies and have become a very common tool for identifying AGN, specially since the arrival of the Wide-Field Infrared Survey Explorer (WISE; e.g., for the WISE bands W1 and W2 at 3.4μm and 4.6μm, respectively, AGN can be identified as having W1-W2 ≥ 0.8, ). Although several studies have made use of MIR colors cuts for selecting AGN in low-mass galaxies (e.g., ; ), caution should be taken when using this selection technique as star-forming dwarf galaxies can show similar MIR colors to those of luminous AGN (). Other MIR searches are based on the detection of the high-ionization emission line [NeV] 14 μm or the 24 μm line using Spitzer spectral observations. This yielded the detection of 9 AGN in bulgeless or late-type (with a minimal bulge) galaxies, for which a lower limit on the BH mass ranging from ∼3 × 10^3-1.5 × 10^5 M_⊙ was estimated assuming sub-Eddington accretion ( 2007, , ). Follow-up observations with the XMM-Newton X-ray satellite revealed the presence of hard, unresolved X-ray emission in one of these bulgeless dwarf galaxies (J1329+3234), with an X-ray luminosity (L_X = 2.4 × 10^40 erg s^-1) two orders of magnitude larger than that expected from star formation and consistent with an accreting BH (). NGC 4178 is another late-type bulgeless disk galaxy that has a prominent [NeV] emission line suggesting the presence of an AGN (). Chandra observations of NGC 4178 reveal the presence of unresolved nuclear X-ray emission spatially coincident with the dynamical center of the galaxy and suggest that the AGN is heavily absorbed and accreting at high rates (). Using the fundamental plane of accreting BHs, the correlation between nuclear stellar cluster mass and BH mass, and the bolometric luminosity, the authors estimate a range of BH masses for this source of ∼10^4-10^5 M_⊙. Using reverberation mapping, <cit.> found another low-mass BH in a nearby late-type galaxy, UGC 06728, which hosts a low-luminosity Seyfert 1 AGN with a BH mass of M_BH = (7.1 ± 4.0) × 10^5 M_⊙ The detection of broad Hα and Hβ emission lines so commonly used to estimate BH masses should not be used as the only tool for identifying AGN, as the emission of broad optical lines might come from transient stellar processes rather than the AGN ionized gas. Evidence for this is provided by the finding that the broad Hα emission of those objects from <cit.> classified as star-forming in the narrow-line diagnostic diagram has faded over a time range of 5-14 years, while those falling in the AGN region of the diagram present persistent broad Hα emission (). An intriguing case is that found by <cit.>, who studied a sample of low-metallicity dwarf galaxies with broad Hα emission: although their sources present long-lived (> 10 yr) broad Hα emission lines incompatible with a supernova origin (see Fig. <ref>), they lack the strong X-ray emission and non-thermal hard UV emission characteristic of AGN. This implies that these sources are either a particular case of AGN with very weak X-ray and UV emission, with fully obscured accretion disks, or that they are not AGN and the persistent broad emission lines are produced by very extreme stellar processes.Unlike optical studies, which are often skewed toward high Eddington ratios, X-ray searches offer the advantage of probing low rates of accretion and detecting AGN so faint as XRBs (L_X∼10^38 erg s^-1; e.g., ). X-ray surveys also cover larger volumes, and make possible the detection of IMBHs in dwarf galaxies at intermediate redshifts (Mezcua et al. in preparation), at an epoch when cosmic star formation and AGN activity reached their peak (; ). One of the first searches for accreting BHs in low-mass galaxies that made use of deep Chandra and XMM-Newton X-ray surveys is that of <cit.>, who found 32 objects out to z ∼1. The authors identified low-mass galaxies as having M_* < 2 × 10^10 M_⊙ and considered only sources with L_X > 10^42 erg s^-1, hence their sample is formed by SMBHs with an average BH mass of 3 × 10^6 M_⊙. Using the AGN Multiwavelength Survey of Early-Type Galaxies (AMUSE; in the Virgo cluster, 454 ks, ; in the field within 30 Mpc, 479 ks, ), <cit.> found nuclear X-ray emission possibly coming from accreting BHs in 7 early-type galaxies with M_* < 10^10 M_⊙ and obtain a lower limit on the BH occupation fraction of > 20%. Using a stacking analysis of early-type galaxies in the Chandra COSMOS survey (0.9 deg^2, 1.8 Ms; ; ) and after removing the contribution from XRBs and hot gas to the X-ray emission, <cit.> also found that highly absorbed AGN are present in low-mass early-type galaxies, with BH masses ranging from 10^6 - 10^8 M_⊙. <cit.> applied the same stacking technique to a sample of ∼50,000 starburst and late-type dwarf galaxies up to redshift ∼1.5 in the Chandra COSMOS Legacy survey (2.2 deg^2, 4.6 Ms; ), finding also an X-ray excess that can be explained by accreting BHs with M_BH∼10^5 M_⊙ and X-ray luminosities as low as 10^39 erg s^-1 (see Fig. <ref>). The authors concluded that a population of IMBHs exists in dwarf starburst galaxies but that their detection beyond the local Universe is challenging due to their low luminosity and mild obscuration. Yet, the use of wide-area X-ray surveys such as COSMOS Legacy provide one of the best tools for detecting IMBHs at intermediate redshifts: Mezcua et al. (in preparation) find 47 dwarf galaxies in the COSMOS Legacy with AGN X-ray luminosities L_0.5-10 keV ranging ∼4 × 10^39 erg s^-1 to 10^44 erg s^-1 and redshifts as high as z ∼2.4. The BH masses range from ∼10^4 M_⊙ to ∼8 × 10^5 M_⊙, indicating that all the sources likely host an IMBH. This constitutes the largest sample of IMBHs beyond the local Universe so far discovered. Making use of the 4 Ms Chandra Deep Field South (CDF-S; ) and also applying the stacking technique, <cit.> found that the unresolved 6-8 keV cosmic X-ray background is mostly produced by low-mass galaxies with obscured AGN at z ∼1-3. The discovery of three individual IMBHs with M_BH∼2 × 10^5 M_⊙ in dwarf galaxies at z < 0.3 in the Extended CDF-S (0.3 deg^2) was reported by <cit.>. <cit.> found ten more individual detections up to z < 0.6 in the area of the All-Wavelength Extended Groth Strip International Survey (AEGIS) field (∼200–800 ks; ) covered by Chandra (0.1 deg^2), and derived an AGN fraction for dwarf galaxies with 10^9 < M_* < 3 × 10^9 M_⊙ at 0.1 < z < 0.6 of ∼3 %. <cit.> found that 19 out of ∼44,000 dwarf galaxies in the NASA-Sloan Atlas[<http://www.nsatlas.org>] have Chandra hard X-ray detections. Eight of these sources present enhanced X-ray emission with respect to that of star formation and are potential IMBH candidates with L_X∼10^38-10^40 erg s^-1, while the X-ray luminosity of the rest of the sources can be explained by XRBs. Using also the NASA-Sloan Atlas, <cit.> identified 19 low-mass galaxies (with stellar masses up to 10^10 M_⊙) with XMM-Newton X-ray emission and radio emission spatially coincident with the galaxy center. Using the fundamental plane of BH accretion, the authors derive a range of BH masses of 10^4-2 × 10^8 M_⊙. A few more tens of IMBH candidates have been found using pointed observations. Chandra observations of 66 of the 229 low-mass AGN identified by <cit.> revealed the detection of X-ray emission in 52 sources and confirmed their AGN nature (; ; ; ; ). <cit.> carried out XMM-Newton observations of six Lyman Break Analogs, which are the local analogs to the high-redshift star-forming Lyman-Break galaxies[Lyman-Break galaxies are high redshift (z > 6) massive galaxies that are expected to host a BH by that time (e.g., ).]. The intermediate starburst-type 2 AGN classification of their optical emission line spectra, the detection of hard X-ray emission with L_X∼10^42 erg s^-1, and MIR to [OIII] luminosity ratios higher than those of type 2 AGN indicate the possible presence of low-mass AGN in these targets, with a BH mass ranging between 10^5 - 10^6 M_⊙ assuming the sources radiate at the Eddington limit. Based on rapid X-ray variability, <cit.> presented an additional sample of 15 low-mass AGN candidates, 7 of which have M_BH < 2 ×10^6 M_⊙ and are thus candidates to IMBHs. Optical spectroscopy of 12 of these low-mass BHs revealed the presence of broad Hα emission lines, from which BH masses in the range 10^5 - 10^6 M_⊙ were estimated (). §.§.§ X-ray weak IMBHs: hiding behind the dust? Both for the <cit.> sample and the <cit.> sample studied in X-rays by <cit.>, the X-ray observations show that most of the low-mass AGN have X-ray to UV ratios (α_OX) below the correlation between α_OX and luminosity density at 2500 Å (l_2500) defined for more luminous sources hosting SMBHs (; see Fig. <ref>, left). This behavior was also found for broad absorption-line quasars whose X-ray weakness is caused by absorption (e.g., ; ). The puzzling X-ray weak tail of low-mass AGN does not seem to be caused by differences between low- and high-Eddington rates nor slim disk accretion (; ), while variability would scatter the sources around the correlation instead of them lying systematically below it. Are then these low-mass AGN intrinsically X-ray weak? Or are their accretion disks fully obscured along our line-of-sight? The latter scenario is supported by the [OIII] deficit of some of the low-mass AGN compared to their X-ray luminosity (i.e., they lie below the relation between 2-10 keV X-ray luminosity and [OIII]λ5007 optical line luminosity defined by higher luminosity unobscured AGNs; see Fig. <ref>, right), a behavior typical of heavily obscured AGN (e.g., ). Similar results were found by Simmonds et al. (2016), whose low-mass AGN lie at least ∼1-2 dex below the relation between [OIII]λ5007 luminosity and 2-10 keV luminosity. Hard X-ray (≥10 keV) observations with NuSTAR could test whether these weak low-mass AGN are indeed obscured. <cit.> searched for low-mass AGN using the 40-month NuSTAR serendipitous survey, finding 10 sources with median stellar mass <M_*> = 4.6 × 10^9 M_⊙, X-ray luminosity <L_2-10 keV> = 3.1 × 10^42 erg s^-1 and BH masses in the range (1.1 - 10.4) × 10^6 M_⊙. Although 30% of their sources do not show AGN-like optical narrow emission lines, only one source is found to be heavily obscured (with a column density N_H > 10^23 cm^-2). Similar results were found by <cit.>, who studied XMM-Newton data of 14 low-mass AGN drawn from the <cit.> sample and found that only two of them show evidence for significant absorption.§.§.§ Do IMBHs follow the trends?Whether or not IMBHs follow the BH-host galaxy correlations found for spheroidal galaxies hosting SMBHs is directly connected to the formation of seed BHs: the low-mass end of the M_BH-σ_* correlation (or also of the M_BH-M_* relation; ) is expected to flatten towards what looks like a plume of ungrown BHs if seed BHs are massive (i.e. formed from direct collapse), while this asymptotic flattening or plume would lie at BH masses that cannot be currently measured observationally (e.g., ; ; see Fig. <ref>) if seed BHs are light (Pop III seeds).In Table <ref> I compile all the IMBHs and low-mass AGN with M_BH≲ 10^6 M_⊙ for which a stellar velocity dispersion is available, and plot them on the M_BH versus σ_* diagram in Fig. <ref>: a tentative plume similar to that expected from massive BH seeds (Fig. <ref>, left) seems to be observed at BH masses ∼10^5-10^6 M_⊙, which suggests that the direct collapse scenario is the main formation mechanism of seed BHs. However, this could just be a bias caused by the easier detection of massive (i.e. of ∼10^5 M_⊙) than light (i.e. of ∼10^3 M_⊙) BHs. The virial BH masses of the 10 low-mass AGN from <cit.> are found to follow the M_BH-σ_* relation (if the considerably large scatter in the pseudobulge regime is considered, e.g., ; grey squares in Fig. <ref>), as found for other samples of low-mass AGN (also included in Table <ref> and Fig. <ref>; e.g., ; ; ; ). IMBHs tend also to sit on the extrapolation of the M_BH-σ_* relation for early-type galaxies or AGN (), indicating that the M_BH-σ_* relation extends over five orders of magnitude in BH mass (see Fig. <ref>). However, this is not the case for the M_BH-M_bulge relation: the BH mass of IMBHs/low-mass AGN tends to be lower at a given bulge mass than expected from an extrapolation of the relation found for classical bulges (e.g., ; ;; ; ). SMBHs in inactive galaxies with pseudobulges (; ) and spiral galaxies with megamaser BH mass measurements (; ) are also found to fall below the M_BH-M_bulge relation of early-type galaxies (), which suggests that most IMBHs/low-mass AGN are hosted by late-type galaxies (; ). Finally, low-mass late-type galaxies tend to lie as well below the M_BH-M_* relation compared to bulge-dominated and elliptical (early-type) galaxies (), which could be explained by IMBH quenching of the star formation during an early gas-rich phase in the life of the dwarf galaxy (). The lower normalization found by <cit.>, if it holds at high redshift, could explain the dearth of BH detections at z > 6 (e.g., ; ; see next section).§.§ Seed BHs at high redshifts The largest samples of IMBHs have been found in the local Universe. Although a few hundred sources have been identified, it is not yet clear which seeding mechanism of SMBHs dominated in the early Universe. To better understand the formation of SMBHs at high redshifts, several campaigns have aimed at discovering faint AGN at z ≥ 5. Optical and IR surveys (e.g., GOODS[The Great Observatories Origins Deep Survey, <http://www.stsci.edu/science/goods/>], CANDELS[Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey, <http://candels.ucolick.org/index.html>]), which provide an estimate of the photometric redshift, have been combined with deep X-ray surveys (e.g., the CDF-S; ); however, no AGN candidates have convincingly been detected at z ≥ 5 individually nor via stacking (e.g., ; ; ). This is in agreement with the extrapolation of the 3 ≤ z ≤ 5 X-ray luminosity function to z ≥ 5, which predicts < 1 AGN in the CDF-S (). The use of NIR photometry has allowed us to reach fainter X-ray sources than direct X-ray surveys: using an H-band luminosity selection, <cit.> identified three faint AGN candidates at z > 6 with X-ray luminosities in the 2-10 keV band above 10^43 erg s^-1. Two of these high-z AGN candidates are also selected from the CANDELS/GOODS-South survey and identified as potential direct collapse BHs by <cit.>, who used a novel photometric method combined with radiation-hydrodynamic simulations that predict a steep IR spectrum. However, none of these candidates was identified by <cit.> using different thresholds, which indicates that the identification of high-z AGN candidates is very sensitive to the selection procedure (). The dearth of detections at z > 6 could be explained by a low BH occupation fraction, by smaller BHs predicted by the M_BH-M_* relation (; ) than by the M_BH-M_bulge relations, or by heavy obscuration (e.g., ; ; ; ). The short phases of super-Eddington growth, followed by longer periods of quiescence, expected to occur in IMBHs (e.g., ; ; ) could also decrease the probability of detecting accreting BHs (). Modeling the X-ray emission of accreting BHs at z ∼6 and taking into account super-Eddington accretion, <cit.> found that faint AGN progenitors at z ∼6 should be luminous enough to be detected in current X-ray surveys even when accounting for maximum obscuration. The authors concluded that the limited number of high-z detections is caused by a low active BH occupation fraction that results from the short episodes of super-Eddington growth and suggested that wide-area surveys with shallow sensitivities such as COSMOS Legacy (instead of deeper, smaller-area surveys as the CDF-S) are better for detecting the progenitors of SMBHs at high-z (as also concluded by ; Mezcua et al. in preparation).The detection of a strong Ly α emission line, virtually the only line available to confirm high redshifts, has been commonly used to detect galaxies at z > 6 (e.g., ; ; ) and has recently emerged as another powerful tool to discover seed BHs at high-z. Those galaxies in which the formation of Pop III stars has recently taken place should emit strong Ly α and He II lines but no metal lines (e.g., ). Similarly, pristine haloes, where direct collapse BHs can form in the presence of intense Lyman-Werner radiation, are expected to cool predominantly via Ly α line emission (e.g., ; ; ).The discovery of strong He II line emission from the most luminous Ly α emitter at z > 6 could constitute the first detection of a (high-z) seed BH using this method (). The source, CR7, is found in the COSMOS field with a redshift z = 6.6 () and is spatially extended (∼16 kpc; ). Its Ly α and He II lines are narrow (FWHM ≤ 200 km s^-1), which disfavors their origin from an AGN or Wolf-Rayet stars, and no metal lines are detected. This initially suggested that CR7 could host a population of Pop III stars. However, the finding that CR7 is formed by a blue galaxy (component A) lying close to two redder galaxies (components B and C) that could provide the Lyman-Werner radiation necessary to suppress star formation in component A seems to favor the direct collapse scenario (see Fig. <ref>; ; ; ; ; ; ). Using deeper IR observations, <cit.> found evidence for metal enrichment in component A and claimed that this rules out the presence of Pop III stars or a pristine direct collapse BH. The authors suggested alternative scenarios such as the presence of a low-mass AGN or a young, low-metallicity starburst galaxy; however, using analytic models <cit.> showed that signatures of metals do not rule out the existence of a direct collapse BH: metal pollution of the direct collapse BH is inevitable, but the BH could form (in component A) before it is metal polluted by the same galaxies that provide the intense UV radiation required to prevent the formation of young stars (components B and C). Further observations are required to find more sources like CR7 and better constrain their nature and the conditions under which they form.Once the upcoming James Webb Space Telescope (JWST) comes online, color-color cuts in the mid-IR regime could also provide unambiguous detections of direct collapse BHs that have acquired a stellar component, termed as 'obese black hole galaxies' (OBGs). Up to 10 such OBG candidates are predicted to be detected in the CANDELS field at z = 6-10 (; ). This will allow us to discriminate between the two main formation mechanisms of seed BHs in the early Universe, and possibly understand the onset of the M_BH-σ relation. §.§ Other pathways to detection * Tidal disruption events Tidal disruption events (TDEs) occur when a star passing too close (within the tidal disruption radius) of a BH is ripped apart by a tidal force that exceeds the star's self-gravity (). When the bound debris is accreted by the BH, it generates a powerful flare observable from radio to γ-rays () and that peaks in UV or soft X-rays. The flare emission declines on the timescale of months to years, producing a long-term lightcurve with a typical time decay of t^-5/3 (). For SMBHs above 10^8 M_⊙, the tidal disruption radius is smaller than the Schwarschild radius and solar-mass stars are not disrupted but swallowed whole without emitting any flares (; ; ; ). SMBHs with M_BH < 10^8 M_⊙ can disrupt solar-type stars; however, the disruption of compact stars such as a white dwarf can be only produced by BHs< 10^5 M_⊙ (). IMBHs, with BH masses below 10^6 M_⊙, should have higher rates of stellar disruption than SMBHs (; ); nonetheless, most TDEs (∼50; ) have been associated with SMBHs (e.g., ; ; ; ; ; ; ; ; ; ; seefor a review). The few TDEs suggested to occur in IMBHs come from events observed in dwarf galaxies: the gamma-ray burst GRB 110328A, or Swift J164449.3+573451, discovered by the Swift Burst Alert Telescope (BAT) took place in a dwarf galaxy at z = 0.354 and with log M_* = 9.14 (; ). A BH of mass 10^5-10^7 M_⊙ is estimated to be the responsible for this TDE (; ; ; ; ; ). An X-ray flare detected in the dwarf galaxy WINGS J1348 (M_*∼3 × 10^8 M_⊙; ), in the Abell cluster 1795, indicates the tidal disruption of a star by an IMBH of log M_BH∼ 5.3-5.7 M_⊙ (; ; see Fig. <ref>). <cit.> suggested that the tidal disruption of a white dwarf by an IMBH of ∼10^4 M_⊙ is the responsible for the gamma-ray burst GRB060218 and associated supernova SN2006aj observed in a dwarf galaxy at a redshift z=0.0335 (e.g., ; ; ). The variability of RBS 1032, a ROSAT X-ray source associated with the inactive dwarf galaxy SDSS J114726.69+494257.8, is suggested to be an accretion flare from a possible IMBH (). TDEs in dwarf galaxies may thus be a potential tool for detecting IMBHs, even during quiescence, and for constraining their occupation fraction out to high redshifts (). * Gravitational waves The double detection by the advanced LIGO[Laser Interferometer Gravitational-wave Observatory.] of gravitational waves (GWs) produced by the coalescence of two heavy (compared to those in our Galaxy; i.e., ≥ 25 M_⊙) stellar-mass BHs (signal GW150914; ) and of two ∼10 M_⊙ BHs (signal GW151226; ) opened a new era in astronomy, revolutionizing the way in which we can detect BHs. Since the more massive the BHs the lower the frequency of the gravitational waves emitted by the binary system, the inspiral of an IMBH into a SMBH or of two IMBHs with BH masses ≥10^3 M_⊙ will produce GWs with frequencies too low for the current ground-based GW interferometers but in the observational range of future space-based interferometers (10^-5–10 Hz; e.g., ). The ring-down GW of IMBH binaries with total masses in between ∼200-2 × 10^3 M_⊙ would produce a significant signal-to-noise ratio in the advanced LIGO and VIRGO interferometers and the future Einstein Telescope (e.g., ; ; ; ) with a detection rate in the advanced detectors of ∼10 mergers a year. The binary formed by an IMBH and a stellar-mass BH would also be detectable by these ground-based interferometers, with an estimated detection rate of up to tens of events per year (). In this case, it would be possible to claim the detection of GWs from an IMBH at 95% confidence if the merging IMBH has a mass of at least 130 M_⊙ (). The detection of IMBHs through GW emission seems to be thus on the horizon.* Accretion disks in AGNIMBHs could also be found in the accretion disks of AGN if they grow in a manner similar to planets in protoplanetary disks (, , ; ). The accretion disks around SMBHs contain nuclear cluster objects (i.e., stars, compact objects, XRBs) that can collide with each other, merge, and exchange angular momentum with the gas in the disk. This can make them migrate within the disk and accrete from it, which may form an IMBH seed (; ). The GWs emitted by the rate of stellar-mass BH mergers required to produce the IMBH could naturally account for the heavy stellar-mass BHs observed by LIGO (e.g., ; ) and is also consistent with the inferred 9-240 Gpc^-3 yr^-1 rate from LIGO ().IMBH seeds in AGN disks can then grow very rapidly and reach super-Eddington accretion rates via slow collisions with nuclear cluster objects in the disk (). If the IMBH exhausts the nuclear cluster objects around it, it can migrate in the disk and expand its feeding zone so that super-Eddington growth can continue. This growth mechanism of IMBHs in AGN disks can open a gap in the disk, which would result in several observational signatures such as oscillations on the broad Fe Kα line profile of the AGN caused by the presence of the secondary BH (i.e., the IMBH; ) or a reduction of the ionizing continuum luminosity of the AGN if the disk is removed to large radii. The later can yield luminosities consistent with those of LINERs and change the optical lines ratio from AGN-like to LINER-like (), suggesting that many LINERs could consist of binary BHs: a SMBH and a secondary BH (the IMBH) that opens a large cavity in the SMBH disk. Such binary systems would emit GWs during the IMBH inspiral, which could be detected with the up-coming eLISA (). The non-detection of IMBH-SMBH binaries by eLISA would imply that this is not an efficient channel for producing IMBHs.The blackbody spectrum of the accretion disk of an IMBH peaks in the soft X-ray band (∼0.1-1 keV for BH masses ranging 10^4-10^2 M_⊙). IMBHs in AGN disks could thus also be detected as an excess of soft X-rays relative to the soft X-ray emission expected from an extrapolation of a power-law fit from hard X-rays (). Soft X-rays could also be generated by the bow shock associated with the headwind of a non-gap-opening IMBH on a retrograde orbit (). Additionally, IMBHs in AGN disks could also produce QPOs, which have been occasionally observed in AGN (), asymmetric X-ray intensity distributions that may be detected as AGN transits (e.g., ; ; ), and UV/X-ray flares caused by a TDE. * High velocity cloudsThe presence of an IMBH has been also proposed in the dense interstellar gas surrounding the nucleus of the Milky Way within a few hundred parsecs (). Using the Nobeyama Radio Observatory 45 m radio telescope, <cit.> found a compact (<5 pc) molecular cloud named CO-040-022 whose kinematical structure consists of an intense region with a shallow velocity gradient plus a less intense high-velocity wing. The kinematical appearance and very high velocity width (∼100 km s^-1) of CO-040-022 can be explained by a gravitational kick imparted by a compact object not visible at other wavelengths and with a mass of 10^5 M_⊙ (). This suggests the presence of an IMBH and opens a new window in the search for these objects: whether they are active or not, they leave a kinematic imprint when encountering a molecular cloud. The finding of compact high-velocity features when studying molecular line kinematics could thus yield the detection of many more IMBHs.§ CONCLUSION AND OUTLOOK According to cosmological models of BH growth, the determination of the number of present-day IMBHs can elucidate how seed BHs formed in the early Universe: either from the death of Pop III stars, the direct collapse of pristine gas, or by stellar mergers in dense stellar clusters (e.g., ). A few hundreds of IMBH candidates have been now identified in a variety of objects, from dwarf galaxies and ULXs to TDEs and high velocity clouds; nonetheless the BH occupation fraction has not yet been constrained to a level that allows us to draw conclusive results about the dominant seeding mechanism at high redshift. <cit.> provide an AGN fraction for optically selected dwarf galaxies at z < 0.055 of 0.5%, a value which is not corrected for incompleteness and which is much lower than the lower limit of > 20% found by <cit.> for dwarf early-type galaxies with Eddington ratios down to 10^-4. <cit.> apply an incompleteness correction to their sample of dwarf galaxies with 10^9 M_⊙ < M_* < 3 × 10^9 M_⊙ at 0.1 < z < 0.6 and find an active fraction of 0.6%-3%, which is in agreement with the 3.1% obtained from semi-analytic models of galaxy formation that seed halos with 10^4 M_⊙ BHs. However, to obtain the true BH occupation fraction we need a larger cosmological volume, a wider range of masses, and to know the distribution of Eddington ratios across the mass scale. The distribution of AGN X-ray detections can be used to infer the BH occupation fraction, e.g., using a mock catalog with 15,000 galaxies and ∼300 X-ray AGN, <cit.> find an X-ray AGN fraction of 11.9% in dwarf galaxies with full occupation but of 6.1% if they have half occupation. Wide-area X-ray surveys, such as the one that will be carried out by the Wide Field Imager instrument onboard of the up-coming ATHENA X-ray satellite, can provide the larger volumes required to constrain the BH occupation fraction.The BH masses of the IMBH candidates so far found are always in the range 10^4-10^6 M_⊙, which is the typical mass of the seed BHs formed from direct collapse and adopted in large-scale cosmological simulations that show how their growth could explain the highest-redshift SMBHs (e.g., ; ; ). This, together with the discovery of CR7 (e.g., ; ) and the plume-like distribution of IMBHs/low-mass AGN around M_BH∼10^5-10^6 M_⊙ on the M_BH-σ relation (see Sect. <ref>, Fig. <ref>), seems to favor the direct collapse scenario (e.g., ). However, the number density of BHs expected from direct collapse is relatively low (e.g., ); so there might just be a bias towards the direct collapse scenario caused by the easier detection of heavy seeds with large BH masses. Light seeds formed from Pop III stars have not grown much since their formation, hence, although they should have a higher occupation fraction, their detection is harder (e.g., ; ).Additional constraints on the mass and growth of seed BHs can be provided by the X-ray non-detections of faint AGN at z ≥ 5 (e.g., ; ; ) nor of Lyman Break Galaxies (e.g., ; ; ). The current non-detections can be explained by a low BH occupation fraction, BH masses ≤ 10^5 M_⊙ (), heavy obscuration, or intrinsic X-ray weakness (). This later possibility has been also proposed to explain the unusual behavior of local low-mass AGN whose ratio of UV to X-ray is lower than 'normal' AGN and are relatively X-ray weak compared to the AGN power expected from ionized gas emission, and it would imply that these sources are governed by a different physical regime than that producing the characteristic strong X-ray-emitting corona of AGN (e.g., ; ). The advent of the next generation of ground and space observatories (ATHENA, SKA, JWST, E-ELT) and dedicated surveys will provide a giant leap on the detection of faint AGN at high redshifts and of direct collapse BHs (e.g., ; ); yet, BHs fainter than 3 × 10^43 erg s^-1 will be extremely difficult to detect at z=7 even by this new generation of telescopes (see Fig. <ref>). It is the multi-wavelength study of IMBHs in dwarf galaxies (e.g., such as the multiple-method approach performed by ), in ULXs or in globular clusters, and of AGN in high-redshift quasars the one that coupled with simulations of BH formation models will allow us to understand how seed BHs formed and grew to become the SMBHs we know today. § ACKNOWLEDGMENTSI would like to thank the anonymous referee for carefully reading the manuscript as well as Pau Diaz Gallifa and Marios Karouzos for constructive comments and illuminating discussions. I am also thankful to Bhaskar Agarwal, Vivienne Baldassare, Barry McKernan, Amy Reines, and Marta Volonteri for useful suggestions that have helped improve the manuscript and, together with James Aird, Andrea Comastri, Jenny Greene, Richard Plotkin, and David Sobral, for allowing me to reproduce their figures in this review. apj | http://arxiv.org/abs/1705.09667v1 | {
"authors": [
"Mar Mezcua"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170526180016",
"title": "Observational evidence for intermediate-mass black holes"
} |
A Sampling Theory Perspective of Graph-based Semi-supervised Learning Aamir Anis, Student Member, IEEE, Aly El Gamal, Member, IEEE, Salman Avestimehr, Senior Member, IEEE, and Antonio Ortega, Fellow, IEEEThis work is supported in part by NSF under grants CCF-1410009, CCF-1527874, CCF-1408639, NETS-1419632 and by AFRL and DARPA under grant 108818. S. Avestimehr and A. Ortega are with the Ming Hsieh Department of Electrical Engineering, University of Southern California. A. Anis is currently with Google Inc., he was affiliated with the University of Southern California at the time this work was completed. A. El Gamal is with the Department of Electrical and Computer Engineering, Purdue University. E-mail: [email protected], [email protected], [email protected], [email protected]. Copyright (c) 2017 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Biomedical events describe complex interactions between various biomedical entities. Event trigger is a word or a phrase which typically signifies the occurrence of an event. Event trigger identification is an important first step in all event extraction methods. However many of the current approaches either rely on complex hand-crafted features or consider features only within a window. In this paper we propose a method that takes the advantage of recurrent neural network (RNN) to extract higher level features present across the sentence. Thus hidden state representation of RNN along with word and entity type embedding as features avoid relying on the complex hand-crafted features generated using various NLP toolkits. Our experiments have shown to achieve state-of-art F1-score on Multi Level Event Extraction (MLEE) corpus. We have also performed category-wise analysis of the result and discussed the importance of various features in trigger identification task.§ INTRODUCTIONBiomedical events play an important role in improving biomedical research in many ways. Some of its applications include pathway curation <cit.> and development of domain specific semantic search engine <cit.>. So as to gain attraction among researchers many challenges such as BioNLP'09 <cit.>, BioNLP'11 <cit.>, BioNLP'13 <cit.> have been organized and many novel methods have also been proposed addressing these tasks. An event can be defined as a combination of a trigger word and arbitrary number of arguments. Figure 1 shows two events with trigger words as “Inhibition” and “Angiogenesis” of trigger types “Negative Regulation” and “Blood Vessel Development” respectively.Pipelined based approaches for biomedical event extraction include event trigger identification followed by event argument identification. Analysis in multiple studies <cit.> reveal that more than 60% of event extraction errors are caused due to incorrect trigger identification.Existing event trigger identification models can be broadly categorized in two ways: rule based approaches and machine learning based approaches. Rule based approaches use various strategies including pattern matching and regular expression to define rules <cit.>. However, defining these rules are very difficult, time consuming and requires domain knowledge. The overall performance of the task depends on the quality of rules defined. These approaches often fail to generalize for new datasets when compared with machine learning based approaches. Machine learning based approaches treat the trigger identification problem as a word level classification problem, where many features from the data are extracted using various NLP toolkits <cit.>or learned automatically <cit.>. In this paper, we propose an approach using RNN to learn higher level features without the requirement of complex feature engineering. We thoroughly evaluate our proposed approach on MLEE corpus. We also have performed category-wise analysis and investigate the importance of different features in trigger identification task.§ RELATED WORKMany approaches have been proposed to address the problem of event trigger identification. <cit.> proposed a model where various hand-crafted features are extracted from the processed data and fed into a Support Vector Machine (SVM) to perform final classification. <cit.> proposed a novel framework for trigger identification where embedding features of the word combined with hand-crafted features are fed to SVM for final classification using multiple kernel learning. <cit.> proposed a pipeline method on BioNLP'13 corpus based on Conditional Random Field (CRF) and Support vector machine (SVM) where the CRF is used to tag valid triggers and SVM is finally used to identify the trigger type. The above methods rely on various NLP toolkits to extract the hand-crafted features which leads to error propagation thus affecting the classifier's performance. These methods often need to tailor different features for different tasks, thus not making them generalizable. Most of the hand-crafted features are also traditionally sparse one-hot features vector which fail to capture the semantic information.<cit.> proposed a neural network model where dependency based word embeddings <cit.> within a window around the word are fed into a feed forward neural network (FFNN) <cit.> to perform final classification. <cit.> proposed another model based on convolutional neural network (CNN) where word and entity mention features of words within a window around the word are fed to a CNN to perform final classification. Although both of the methods have achieved good performance they fail to capture features outside the window. § MODEL ARCHITECTUREWe present our model based on bidirectional RNN as shown in Figure <ref> for the trigger identification task. The proposed model detects trigger word as well as their type. Our model uses embedding features of words in the input layer and learns higher level representations in the subsequent layers and makes use of both the input layer and higher level features to perform the final classification. We now briefly explain about each component of our model. §.§ Input Feature LayerFor every word in the sentence we extract two features, exact word w ∈ W and entity type e ∈ E. Here W refers the dictionary of words and E refers to dictionary of entities. Apart from all the entities, E also contains a None entity type which indicates absence of an entity. In some cases the entity might span through multiple words, in that case we assign every word spanned by that entity the same entity type.§.§ Embedding or Lookup LayerIn this layer every input feature is mapped to a dense feature vector. Let us say that E_w and E_e be the embedding matrices of W and E respectively. The features obtained from these embedding matrices are concatenated and treated as the final word-level feature (l) of the model. The E_w ∈ℝ^n_w × d_w embedding matrix is initialized with pre-trained word embeddings and E_e ∈ℝ^n_e × d_e embedding matrix is initialized with random values. Here n_w, n_e refer to length of the word dictionary and entity type dictionary respectively and d_w, d_e refer to dimension of word and entity type embedding respectively. §.§ Bidirectional RNN LayerRNN is a powerful model for learning features from sequential data. We use both LSTM <cit.> and GRU <cit.> variants of RNN in our experiments as they handle the vanishing and exploding gradient problem <cit.> in a better way. We use bidirectional version of RNN <cit.> where for every word forward RNN captures features from the past and the backward RNN captures features from future, inherently each word has information about whole sentence. §.§ Feed Forward Neural NetworkThe hidden state of the bidirectional RNN layer acts as sentence-level feature (g), the word and entity type embeddings (l) act as a word-level features, are both concatenated (<ref>) and passed through a series of hidden layers (<ref>), (<ref>)with dropout <cit.> and an output layer. In the output layer, the number of neurons are equal to the number of trigger labels. Finally we use Softmax function (<ref>) to obtain probability score for each class.f =g^k ⊕ l^k h_0 =tanh(W_0f+b_0) h_i =tanh(W_ih_i-1+b_i) p(y|x) = Softmax(W_oh_i+b_o)Here k refers to the k^th word of the sentence, i refers to the i^th hidden layer in the network and ⊕ refers to concatenation operation. W_i,W_o and b_i,b_o are parameters of the hidden and output layers of the network respectively. §.§ Training and HyperparametersWe use cross entropy loss function and the model is trained using stochastic gradient descent. The implementation[Implementation is available at <https://github.com/rahulpatchigolla/EventTriggerDetection>] of the model is done in python language using Theano <cit.> library. We use pre-trained word embeddings obtained by <cit.> using word2vec tool <cit.>.We use training and development set for hyperparameter selection. We use word embeddings of 200 dimension, entity type embeddings of 50 dimension, RNN hidden state dimension of 250 and 2 hidden layers with dimension 150 and 100. In both the hidden layers we use dropout of 0.2.§ EXPERIMENTS AND DISCUSSION§.§ Dataset DescriptionWe use MLEE <cit.> corpus for performing our trigger identification experiments. Unlike other corpora on event extraction it covers events across various levels from molecular to organism level. The events in this corpus are broadly divided into 4 categories namely “Anatomical”, “Molecular”, “General”, “Planned” which are further divided into 19 sub-categories as shown in Table <ref>. Here our task is to identify correct sub-category of an event. The entity types associated with the dataset are summarized in Table <ref>. §.§ Experimental DesignThe data is provided in three parts as training, development and test sets. Hyperparameters are tuned using development set and then final model is trained on the combined set of training and development sets using the selected hyperparameters. The final results reported here are the best results over 5 runs.We have used micro averaged F1-score as the evaluation metric and evaluated the performance of the model by ignoring the trigger classes with counts ≤ 10 in test set while training and considered them directly as false-negative while testing. §.§ Performance comparison with Baseline ModelsWe compare our results with baseline models shown in Table <ref>. <cit.> defined a SVM based classifier with hand-crafted features. <cit.> also defined a SVM based classifier with word embeddings and hand-crafted features. <cit.> defined window based CNN classifier. Apart from the proposed models we also compare our results with two more baseline methods FFNN and CNN^ψ which are our implementations. Here FFNN is a window based feed forward neural network where embedding features of words within the window are used to predict the trigger label <cit.>. We chose window size as 3 (one word from left and one word from right) after tuning it in validation set. CNN^ψ is our implementation of window based CNN classifier proposed by <cit.> due to unavailability of their code in public domain. Our proposed model have shown slight improvement in F1-score when compared with baseline models. The proposed model's ability to capture the context of the whole sentence is likely to be one of the reasons of improvement in performance.We perform one-side t-test over 5 runs of F1-Scores to verify our proposed model's performance when compared with FFNN and CNN^Ψ. The p value of the proposed model (GRU) when compared with FFNN and CNN^ψ are 8.57× 10^-07 and 1.178× 10^-10 respectively. This indicates statistically superior performance of the proposed model. §.§ Category Wise Performance AnalysisThe category wise performance of the proposed model is shown in Table <ref>. It can be observed that model's performance in anatomical and molecular categories are better than general and planned categories. We can also infer from the confusion matrix shown in Figure <ref> that positive regulation, negative regulation and regulation among general category and planned category triggers are causing many false positives and false negatives thus degrading the model's performance.§.§ Further Analysis In this section we investigate the importance of various features and model variants as shown in Table <ref>. Here E_w and E_e refer to using word and entity type embedding as a feature in the model, l and g refer to using word-level and sentence-level feature respectively for the final prediction. For example, E_w+E_e and g means using both word and entity type embedding as the input feature for the model and g means only using the global feature (hidden state of RNN) for final prediction.Examples in Table <ref> illustrate importance of features used in best performing models. In phrase 1 the word “knockdown”, is a part of an entity namely “round about knockdown endothelial cells” of type “Cell” and in phrase 2 it is trigger word of type “Planned Process”, methods 1 and 2 failed to differentiate both of them because of no knowledge about the entity type. In phrase 3 “impaired” is a trigger word of type “Negative Regulation” methods 1 and 3 failed to correctly identify but when reinforced with word-level feature the model succeeded in identification. So, we can say that E_e feature and l+g model variant help in improving the model's performance.§ CONCLUSION AND FUTURE WORKIn this paper we have proposed a novel approach for trigger identification by learning higher level features using RNN. Our experiments have shown to achieve state-of-art results on MLEE corpus. In future we would like to perform complete event extraction using deep learning techniques.acl_natbib | http://arxiv.org/abs/1705.09516v1 | {
"authors": [
"Patchigolla V S S Rahul",
"Sunil Kumar Sahu",
"Ashish Anand"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170526103612",
"title": "Biomedical Event Trigger Identification Using Bidirectional Recurrent Neural Network Based Models"
} |
[email protected]@cftp.tecnico.ulisboa.ptCentro de Física Teórica de Partículas, CFTP, and Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais nr. 1, 1049-001 Lisboa, Portugal [email protected] d'Estructura i Constituents de la Matèria and Institut de Ciencies del Cosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain We extend the standard model with two extra Higgs doublets. Making use of a symmetry principle, we present flavour symmetries based on cycle groups Z_N that oblige each Higgs doublet to contribute to the mass of only one generation. The Higgs doublets couple to the fermions with different strengths and in this way accommodate the quark mass hierarchy. We systematically search for all charge configurations that naturally lead to the alignment in flavour space of the quark sectors, resulting in a CKM matrix near to the identity, determined by the quark mass hierarchy, and with the correct overall phenomenological features. The minimal realization is by the group Z_7. We show that only a limited number of solutions exist, and that any accidental global symmetry that may occur together with the discrete symmetry is necessarily anomalous. A phenomenological study of each class of solutions concerning predictions to the flavour changing neutral current (FCNC) phenomena is also performed: for some solutions, it is possible to obtain realistic quark masses and mixing, while the flavour violating neutral Higgs are light enough to be accessible at LHC. Exploring the quark flavor puzzle within the three-Higgs double model Nuno Rosa Agostinho=====================================================================§ INTRODUCTION The discovery of the Higgs boson in 2012 at the LHC has attested the success of the standard model (SM) in describing the observed fermions and their interactions. However, there exist many theoretical issues or open questions that have no satisfactory answer. In particular, the observed flavour pattern lacks of a definitive explanation, i.e., the quark Yukawa coupling matrices Y_u and Y_d, which in the SM reproduce the six quark masses, three mixings angles and a complex phase to account for CP violation phenomena, are general complex matrices, not constrained by any gauge symmetry.Experimentally the flavour puzzle is very intricate. First, there is the quark mass hierarchy in both sectors. Secondly, the mixings in the SM, encoded in the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix, turns out to be close to the identity matrix. If one takes also the lepton sector into account, the hierarchy there is even more puzzling <cit.>. On the other hand, in the SM there is in general no connection between the quark masses hierarchy and the CKM mixing pattern. In fact, if one considers the Extreme Chiral Limit, where the quark masses of the first two generations are set to zero, the mixing does not necessarily vanish <cit.>, and one concludes that the CKM matrix V being close to the identity matrix has nothing to do with the fact that the quark masses are hierarchical. Indeed, in order to have V≈1, one must have a definite alignment of the quark mass matrices in the flavour space, and to explain this alignment, a flavour symmetry or some other mechanism is required <cit.>.Among many attempts made in the literature to address the flavour puzzle, extensions of the SM with new Higgs doublet are particularly motivating. This is due to fact that the number of Higgs doublets is not constrained by the SM symmetry. Moreover, the addition of scalar doublets gives rise to new Yukawa interactions and as a result it provides a richer framework in approaching the theory of flavour. On the other hand, any new extension of the Higgs sector must be very much constrained, since it naturally leads to flavour changing neutral currents. At tree level, in the SM, all the flavour changing transitions are mediated through charged weak currents and the flavour mixing is controlled by the CKM matrix <cit.>. If new Higgs doublets are added, one expects large FCNC effects already present at tree level. Such effects have not been experimentally observed and they constrain severely any model with extra Higgs doublets, unless a flavour symmetry suppresses or avoids large FCNC <cit.>.Minimal flavour violating models <cit.> are examples of a multiHiggs extension where FCNC are present at tree-level but their contributions to FCNC phenomena involve only off-diagonal elements of the CKM matrix or their products. The first consistent models of this kind were proposed by Branco, Grimus and Lavoura (BGL) <cit.>, and consisted of the SM with two Higgs doublets together with the requirement of an additional discrete symmetry. BGL models are compatible with lower neutral Higgs masses and FCNC's occur at tree level, with the new interactions entirely determined in terms of the CKM matrix elements.The goal of this paper is to generalize the previous BGL models and to, systematically, search for patterns where a discrete flavour symmetry naturally leads to the alignment of the flavour space of both the quark sectors. Although the quark mass hierarchy does not arise from the symmetry, the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features, determined by the quark mass hierarchy, <cit.>. To do this we extend the SM with two extra Higgs doublets to a total of three Higgs ϕ _a. The choice for discrete symmetries is to avoid the presence of Goldstone bosons that appear in the context of any global continuous symmetry, when the spontaneous electroweak symmetry breaking occurs. For the sake of simplicity, we restrict our search to the family group Z_N, and demand that the resulting up-quark mass matrix M_u is diagonal. This is to say that, due to the expected strong up-quark mass hierarchy, we only consider those cases where the contribution of the up-quark mass matrix to quark mixing is negligible.If one assumes that all Higgs doublets acquire vacuum expectation values with the same order of magnitude, then each Higgs doublet must couple to the fermions with different strengths. Possibly one could obtain similar results assuming that the vacuum expectation values (VEVs) of the Higgs have a definite hierarchy instead of the couplings, but this is not considered here. Combining this assumption with the symmetry, we obtain the correct ordered hierarchical pattern if the coupling with ϕ _3 gives the strength of the third generation, the coupling with ϕ _2 gives the strength of the second generation and the coupling with ϕ _1 gives the strength of the first generation. Therefore, from our point of view, the three Higgs doublets are necessary to ensure that there exists three different coupling strengths, one for each generation, to guarantee simultaneously an hierarchical mass spectrum and a CKM matrix that has the correct overall phenomenological features e.g. | V_cb| ^2+| V_ub| ^2=O(m_s/m_b)^2,and denoted here by V≈1.Indeed, our approach is within the BGL models, and such that the FCNC flavour structure is entirely determined by CKM. Through the symmetry, the suppression of the most dangerous FCNC's, by combinations of the CKM matrix elements and light quark masses, is entirely natural.The paper is organised as follows. In the next section, we present our model and classify the patterns allowed by the discrete symmetry in combination with our assumptions.In Sec. <ref>, we give a brief numerical analysis of the phenomenological output of our solutions. In Sec. <ref>, we examine the suppression of scalar mediated FCNC in our framework for each pattern. Finally, in Sec. <ref>, we present our conclusions.§ THE MODEL We extend the Higgs sector of the SM with two extra new scalar doublets, yielding a total of three scalar doublets, as ϕ _1, ϕ _2, ϕ _3. As it was mentioned in the introduction, the main idea for having three Higgs doublets is to implement a discrete flavour symmetry, that leads to the alignment of the flavour space of the quark sectors. The quark mass hierarchy does not arise from the symmetry, but together with the symmetry the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features, determined by the quark mass hierarchy.Let us start by considering the most general quark Yukawa coupling Lagrangian invariant in our setup -ℒ_Y=(Ω _a)_ij Q_Liϕ_a u_R_j+(Γ _a)_ij Q_Li ϕ _ad_R_j+h.c.,with the Higgs labeling a=1,2,3 and i,j are just the usual flavour indexes identifying the generations of fermions. In the above Lagrangian, one has three Yukawa coupling matrices Ω _1, Ω _2, Ω _3 for the up-quark sector and three Yukawa coupling matrices Γ _1, Γ _2, Γ _3 for the down sector, corresponding to each of the Higgs doublets ϕ _1, ϕ _2, ϕ _3. Assuming that only the neutral components of the three Higgs doublets acquire vacuum expectation value (VEV), the quark mass M_u and M_d are then easily generated as M_u=Ω _1⟨ϕ _1⟩ ^∗+ Ω _2⟨ϕ _2⟩ ^∗+ Ω _3 ⟨ϕ _3⟩ ^∗,M_d=Γ _1⟨ϕ _1⟩ + Γ _2⟨ϕ _2⟩ + Γ _3⟨ϕ _3⟩ ,where VEVs ⟨ϕ _i⟩ are parametrised as⟨ϕ _1⟩ =v_1/√(2),⟨ϕ _2⟩ =v_2e^iα _2/√(2) ,⟨ϕ _3⟩ =v_3e^iα _3/ √(2),with v_1, v_2 and v_3 being the VEV moduli and α _2, α _3 just complex phases. We have chosen the VEV of ϕ _1 to be real and positive, since this is always possible through a proper gauge transformation. As stated, we assume that the moduli of VEVs v_i are of the same order of magnitude, i.e., v_1∼ v_2∼ v_3. Each of the ϕ _a couples to the quarks with a coupling (Ω _a)_ij,(Γ _a)_ij which we take be of the same order of magnitude, unless some element vanishes by imposition of the flavour symmmetry. In this sence, each ϕ _a and (Ω _a,Γ _a) will generate it's own respective generation: i.e., our model is such that by imposition of the flavour symmmetry, ϕ _3, Ω _3, Γ _3 will generate m_t respectivelly m_b, that ϕ _2, Ω _2, Γ _2 will generate m_c respectivelly m_s, and that ϕ _1, Ω _1, Γ _1 will generate m_u respectivellym_d. Generically, we have v_1| (Ω _1)_ij| ∼ m_u, v_2| (Ω _2)_ij|∼ m_c, v_3| (Ω _3)_ij|∼ m_t, v_1| (Γ _1)_ij| ∼ m_d, v_2| (Γ _2)_ij|∼ m_s, v_3| (Γ _3)_ij|∼ m_b,which together with Eq. <ref> implies a definite hierarchy amongst the non-vanishing Yukawa coupling matrix elements:| (Ω _1)_ij| ≪| (Ω _2)_ij|≪| (Ω _3)_ij| ,| (Γ _1)_ij|<| (Γ _2)_ij|≪| (Γ _3)_ij| . Next, we focus on the required textures for the Yukawa coupling matrices Ω _a and Γ _a that naturally lead to an hierarchical mass quark spectrum and at the same time to a realistic CKM mixing matrix. These textures must be reproduced by our choice of the flavour symmetry. As referred in the introduction, we search for quark mass patterns where the mass matrix M_u is diagonal. Therefore, one derives from Eqs. (<ref>), (<ref>) the following textures for Ω _a Ω _1= [ 𝗑 0 0; 0 0 0; 0 0 0 ] , Ω _2= [ 0 0 0; 0 𝗑 0; 0 0 0 ] , Ω _3= [ 0 0 0; 0 0 0; 0 0 𝗑 ] .The entry 𝗑 means a non zero element. In this case, the up-quark masses are given by m_u=v_1| (Ω _1)_11|, m_c=v_2| (Ω _2)_22| and m_t=v_3| (Ω _3)_33|.Generically, the down-quark Yukawa coupling matrices must have the following indicative textures Γ _1= [ 𝗑 𝗑 𝗑; 𝗑 𝗑 𝗑; 𝗑 𝗑 𝗑 ] , Γ _2= [ 0 0 0; 𝗑 𝗑 𝗑; 𝗑 𝗑 𝗑 ] , Γ _3= [ 0 0 0; 0 0 0; 𝗑 𝗑 𝗑 ] .We distinguish rows with bold 𝗑 in order to indicate that it is mandatory that at least one of matrix elements within that row must be nonvanishing. Rows denoted with 𝗑 may be set to zero, without modifying the mass matrix hierarchy. These textures ensure that not only isthe mass spectrum hierarchy respected but it also leads to the alignment of theflavor space of both the quark sectors <cit.> and to a CKM matrix V≈1. For instance, if one would not have a vanishing, or comparatively very small, (1,3) entry in the Γ _2 , this would not necessarily spoil the scale of m_s, but it would dramatically change the predictions for the CKM mixing matrix.In order to force the Yukawa coupling matrices Ω _a and Γ _a to have the indicative forms outlined in Eqs. (<ref>) and (<ref>), we introduce a global flavour symmetry. Since any global continuous symmetry leads to the presence of massless Goldstone bosons after the spontaneous electroweak breaking, one should instead consider a discrete symmetry. Among many possible discrete symmetry constructions, we restrict our searches to the case of cycle groups Z_N. Thus, we demand that any quark or boson multiplet χ transforms according to Z_N as χ→χ ^'=e^i 𝒬(χ ) 2π/N χ ,where 𝒬(χ )∈{0,1,… ,N} is the Z_N-charge attributed for the multiplet χ.We have chosen the up-quark mass matrix M_u to be diagonal. This restricts the flavour symmetry Z_N. We have found that, in order to ensure that all Higgs doublet charges are different, and to have appropriate charges for fields Q_L_i and u_R_i, we must have N≥ 7. We simplify our analysis by fixing N=7 and choose: 𝒬(Q_L_i)=(0,1,-2),𝒬(u_R_i)=(0,2,-4),In addition, we may also fix𝒬(Q_L_i)=𝒬(ϕ _i)It turns out that these choices do not restrict the results, i.e. the possible textures that one can have for the Γ _i matrices. Other choices would only imply that we reshuffle the charges of the multiplets.With the purpose of enumerating the different possible textures for the Γ _i matrices implementable in Z_7, we write down the charges of the trilinears 𝒬(Q_L_iϕ _ad_R_j) corresponding to each ϕ _a as𝒬(Q_L_iϕ _1d_R_j)= [ d_1 d_2 d_3; d_1-1 d_2-1 d_3-1; d_1+2 d_2+2 d_3+2 ] , 𝒬(Q_L_iϕ _2d_R_j)= [ d_1+1 d_2+1 d_3+1; d_1 d_2 d_3; d_1+3 d_2+3 d_3+3 ] , 𝒬(Q_L_iϕ _3d_R_j)= [ d_1-2 d_2-2 d_3-2; d_1-3 d_2-3 d_3-3; d_1 d_2 d_3 ] ,where d_i≡𝒬(d_R_i). One can check that, in order to have viable solutions, one must vary the values of d_i∈{0,1,-2,-3}.We summarise in Table <ref> all the allowed textures for the Γ _a matrices and the resulting M_d mass matrix texture, excluding all cases which are irrelevant, e.g. matrices that have too much texture zeros and are singular, or matrices that do not accommodate CP violation. It must be stressed, that these are the textures obtained by the different charge configurations that one can possibly choose. However, if one assumes a definite charge configuration, then the entire texture, M_d and M_u and the respective phenomenology are fixed. As stated, the list of textures in Table <ref> remains unchanged even if one chooses any other set than in Eqs. (<ref>), (<ref>). As stated, that all patterns presented here are of the Minimal Flavour Violation (MFV) type <cit.>.Pattern I in the table was already considered in Ref. <cit.> in the context of Z_8. We discard Patterns IV, VII and X, because contrary to our starting point, at least one of three non-zero couplings with ϕ _1 will turn out be of the same order as the larger coupling with ϕ _2 in order to meet the phenomenological requirements of the CKM matrix.Notice also, that the structure of other M_d's cannot be trivially obtained, e.g. from Pattern I, by a transformation of the right-handed down quark fields.Our symmetry model may be extended to the charged leptons and neutrinos, e.g. in the context of type one see-saw. Choosing for the lepton doublets L _i the charges 𝒬(L_i)=(0,-1,2), opposite to the Higgs doublets in Eq. (<ref>), and e.g. for the charges 𝒬(e_R _i)=(0,-2,4) of the right-handed fields e_R_i, we force the charged lepton mass matrix to be diagonal. Then for the right-handed neutrinos ν _R_i, choosing 𝒬(ν _R_i)=(0,0,0), we obtain for the neutrino Dirac mass matrix a pattern similar to pattern I. Of course, for this case, the heavy right-handed neutrino Majorana mass matrix is totally arbitrary. In other cases, i.e. for other patterns and charges, in particular for the right-handed neutrinos, we could introduce scalar singlets with suitable charges, which would then lead to certain heavy right-handed neutrino Majorana mass matrices.Next, we address an important issue of the model, namely, whether accidentalU(1) symmetries may appear in the Yukawa sector or in the potential. One may wonder whether a continuous accidental U(1) symmetry could arise, once the Z_7 is imposed at the Lagrangian level in Eq. (<ref>). This is indeed the case, i.e., for all realizations of Z_7, one has the appearance of a global U(1)_X. However, any consistent global U(1)_X must obey to the anomaly-free conditions of global symmetries <cit.>, which read for the anomalies SU(3)^2× U(1)_X, SU(2)^2× U(1)_X and U(1)_Y^2× U(1)_X as A_3≡1/2∑_i=1^3(2X(Q_L_i)-X(u_R_i)-X( d_R_i))=0, A_2≡1/2∑_i=1^3(3X(Q_L_i)+X(ℓ _L _i))=0, A_1≡1/6∑_i=1^3(X(Q_L_i)+3X(ℓ _L _i)-8X(u_R_i)-2X(d_R_i)-6X(e_R_i))=0,where X(χ ) is the U(1)_X charge of the fermion multiplet χ. We have properly shifted the Z_7-charges in Eq.(<ref>) and in Table <ref> so that X(χ )=𝒬(χ ), apart of an overall U(1)_X convention. In general to test those conditions, one needs to specify the transformation laws for all fermionic fields. Looking at the Table 1, we derive that all the cases, except the first case corresponding to d_i=(0,0,0), violate the condition given in Eq. (<ref>) that depends only on coloured fermion multiplets. In the case d_i=(0,0,0), if one assigns the charged lepton charges as X(ℓ _L _i)=X(Q_L_i) one concludes that the condition given in Eq. (<ref>) is violated. One then concludes that the global U(1)_X symmetry is anomalous and therefore only the discrete symmetry Z_7 persists.We also comment on the scalar potential of our model. The most general scalar potential with three scalars invariant under Z_7 reads asV(ϕ )=∑_i[ -μ _i^2ϕ _i^†ϕ _i+λ _i(ϕ _i^†ϕ _i)^2] +∑_i<j[ +C_i(ϕ _i^†ϕ _i)(ϕ _j^†ϕ _j)+ C̅ _i|ϕ _i^†ϕ _j| ^2] ,where the constants μ _i^2, λ _i, C_i and C̅_i are taken real for i,j=1,2,3. Analysing the potential above, one sees that it gives rise to the accidental global continuous symmetry ϕ _i→ e^iα _iϕ _i, for arbitrary α _i, which upon spontaneous symmetry breaking leads to a massless neutral scalar, at tree level. Introducing soft-breaking terms like m_ij^2ϕ _i^† ϕ _j +H.c. can erase the problem. Another possibility without spoiling the Z_7 symmetry is to add new scalar singlets, so that the coefficients m_ij^2 are effectively obtained once the scalar singlets acquire VEVs. § NUMERICAL ANALYSIS In this section, we give the phenomenological predictions obtained by the patterns listed in Table <ref>. Note that, although these patterns arrize directly from the chosen discrete charge configuration of the quark fields, one may further preform a residual flavour transformation of the right-handed down quark fields, resulting in an extra zero entry in M_d. Taking this into account, all the parameters in each pattern may be uniquely expressed in terms of down quark masses and the CKM matrix elementsV_ij. This follows directly from the diagonalization equation of M_d:V ^†M_d W=diag(m_d,m_s,m_b)⟹ M_d=V diag(m_d,m_s,m_b) W^†with V being the CKM mixing matrix, since M_u is diagonal. Because of the zero entries in M_d, it is easy to extract the right-handed diagonalization matrix W, completely in terms of the down quark masses and the V_ij. Thus, all parameters, modulo the residual transformation of the right-handed down quark fields, are fixed, i.e., all parameters in each pattern may be uniquely expressed in terms of down quark masses and the CKM matrix elements V_ij, including the right-handed diagonalization matrix W of M_d. More precisely, all matrix elements of V are written in terms of Wolfenstein real parameters λ, A, ρ andη, defined in terms of rephasing invariant quantities asλ≡|V_us|/√(|V_us|^2+|V_ud|^2), A≡1/λ|V_cb/V_us| , ρ+i η≡ -V_ud^∗ V_ub^∗/V_cd^∗V_cb^∗and diag(m_d,m_s,m_b) in Eq. (<ref>)[ √(m_d/m_s)=√(k_d/k_s) λ;;m_s/m_b=k_s λ ^2 ] ⟹[ m_d=k_d λ ^4m_b;; m_s=k_s λ ^2m_b ]with phenomenologically, k_d and k_s being factores of order one. Writing W^† in Eq. (<ref>) as W^†=(v_1,v_2,v_3), with the v_i vectors formed by the i-th column of W^†, we find e.g. for pattern II, v_3=1/n_3([ m_d/m_bV_11; m_s/m_bV_12;V_13 ] ) ×([ m_d/m_bV_31; m_s/m_bV_32;V_33 ] )where n_3 is the norm of the vector obtained from the external product of the two vectors. Taking into account the extra freedom of transformation of the right-handed fields, we may choose M_31^d=0, corresponding to c_1=0 in Table <ref>, and we conclude thatv_1=1/n_1([ m_d/m_bV_31; m_s/m_bV_32;V_33 ] ) × v_3^∗Obviously, then v_2=1/n_2v_1^∗× v_3^∗. This process is replicated for all patterns. Thus, V and W, are entirely expressed in terms of Wolfenstein parameters and k_d and k_s of Eq. (<ref>). These two matrices will be later used to compute the patterns of the FCNC's in Table <ref>. Indeed, in this way, we find e.g. for pattern II,in leading order order, M_d=m_b ([ -k_d λ ^3 ( ρ-i η) A λ ^3 0; -k_d λ ^2A λ ^2 -k_s λ ^3; 0 1 0 ] )which corresponds to the expected power series where the couplings in Γ _1 to the first Higgs ϕ _1 are comparatively smaller than then couplings in Γ _2, and these smaller to the couplings in Γ _3. Similar results are obtained for all patterns in Table <ref>, except for patterns IV, VII and X, where e.g. for pattern IV, we find that the coupling in (Γ _1)_33 is proportional toλ, which is too large and contradicts our initial assumption that all couplings in Γ _1 to the first Higgs ϕ _1 must be smaller than the couplings in Γ _2 to the second Higgs ϕ _2. Therefore, we exclude Patterns IV, VII and X.We give in Table <ref> a numerical example of a Yukawa coupling configuration for each pattern. We use the following quark running masses at the electroweak scale M_Z:m_u=1.3_-0.2^+0.4 MeV, m_d=2.7± 0.3 MeV , m_s=55_-3^+5 MeV, m_c=0.63± 0.03 GeV, m_b=2.86_-0.04^+0.05 GeV , m_t=172.6± 1.5 GeV.which were obtained from a renormalisation group equation evolution at four-loop level <cit.>, which, taking into account all experimental constrains <cit.>, implies: λ=0.2255± 0.0006, A=0.818± 0.015,ρ=0.124± 0.024,η=0.354± 0.015. § PREDICTIONS OF FLAVOUR CHANGING NEUTRAL CURRENTS In the SM, flavour changing neutral currents (FCNC) are forbidden at tree level, both in the gauge and the Higgs sectors. However, by extending the SM field content, one obtains Higgs Flavour Violating Neutral Couplings <cit.>. In terms of the quark mass eigenstates, the Yukawa couplings to the Higgs neutral fields are:-ℒ_Neutral Yukawa=H_0/v( d_LD_d d_R + u_LD_u u_R ) + 1/v'd_LN^d_1 ( R_1 + iI_1 ) d_R+ 1/v'u_LN^u_1 ( R_1 - iI_1 ) u_R + 1/v”d_LN^d_2 ( R_2 + iI_2 ) d_R + 1/v”u_LN^u_2 ( R_2 - iI_2 ) u_R + h.c.where the N_i^u,d are the matrices which give the strength and the flavour structure of the FCNC,N_1^d=1/√(2)V^† ( v_2Γ _1-v_1e^i α _2Γ _2)W,N_2^d=1/√(2)V^†( v_1Γ _1+v_2e^i α _2Γ _2-v_1^2+v_2^2/v_3 e^i α _3Γ _3)W,N_1^u=1/√(2)( v_2Ω _1-v_1e^-i α _2Ω _2) ,N_2^u=1/√(2)( v_1Ω _1+v_2e^-i α _2Ω _2-v_1^2+v_2^2/v_3e^-i α _3Ω _3) .Since in our case the N_i^u are diagonal, there are no flavour violating terms in the up-sector. Therefore, the analysis of the FCNC resumes only to the down-quark sector. One can use the equations of the mass matrices presented in Eq. (<ref>) to simplify the Higgs mediated FCNC matrices for the down-sector:N_1^d=v_2/v_1D_d-v_2/√(2)( v_2 /v_1+v_1/v_2) e^iα _2 V^† Γ _2 W-v_2 v_3/v_1√(2)e^iα _3V^†Γ _3 W N_2^d=D_d-v^2/v_3√(2)e^iα _3 V^† Γ _3 W In order to satisfy experimental constraints arising from K^0- K^0, B^0-B^0 and D^0-D^0, the off-diagonal elements of the Yukawa interactions N_1^d and N_2^d must be highly suppressed <cit.> <cit.> . For each of our 10 solutions in Table <ref>, we summarize in Table <ref> all FCNC patterns, for each solution, and for v_1=v_2=v_3 and α _2=α _3=0. These patterns are of the BGL type, since in Eq. (<ref>) all matrices can be expressed in terms of the CKM mixing matrix elements and the down quark masses. As explained, to obtain these patterns, we express the CKM matrix V and the matrix W in terms of Wolfenstein parameters.The tree level Higgs mediated Δ S=2 amplitude must be suppressed. This may allways be achieved if one chooses the masses of the flavour violating neutral Higgs scalars sufficiently heavy. However, from the experimental point of view, it would be interesting to have these masses as low as possible. Therefore, we also estimate the lower bound of these masses, by considering the contribution to B^0-B^0 mixing. We choose this mixing, since for our patterns, the (3,1) entry of the matrix N_1^d is the less suppressed in certain cases and would require very heavy flavour violating neutral Higgses. The relevant quantity is the off-diagonal matrix element M_12, which connects the B meson with the corresponding antimeson. This matrix element, M_12^NP, receives contributions <cit.> both from a SM box diagram and a tree-level diagram involving the FCNC:M_12=M_12^SM+M_12^NP,where the New Physics (NP) short distance tree level contribution to the meson-antimeson contribution is: M_12^NP= ∑_i^2 f_B^2 m_B/ 96v^2 m^2_R_i{[ (1+ ( m_B/m_d+m_b)^2 )(a^R_i)_12] - [ (1+ 11 ( m_B/m_d+m_b)^2 ] )(b^R_i)_12}+∑_i^2 f_B^2 m_B/ 96v^2 m^2_I_i{[ ( 1+ ( m_B/m_d+m_b)^2)(a^I_i)_12] - [ (1+ 11 ( m_B/m_d+m_b)^2]) (b^I_i)_12}with v^2=v_1^2+v_2^2+v_2^2 and [( a_i^R) _12=[ ( N_i^d) _31^∗+( N_i^d) _13] ^2; ( a_i^I) _12=-[ ( N_i^d) _31^∗-( N_i^d) _13] ^2 ] ,[( b_i^R) _12=[ ( N_i^d) _31^∗-( N_i^d) _13] ^2; ( b_i^I) _12=-[ ( N_i^d) _31^∗+( N_i^d) _13] ^2 ] , i=1,2In order to obtain a conservative measure, we have tentatively expanded the original expression in <cit.> and, for the three Higgs case, included all neutral Higgs mass eigenstates.Adopting as input values the PDG experimental determinations of f_B, m_B and Δ m_B and considering a common VEV for all Higgs doublets, we impose the inequality M_12^NP<Δ m_B. The following plots show an estimate of the lower bound for the flavour-violating Higgs masses for two different patterns. We plot two masses chosen from the set ( m_1^R,m_2^R,m_1^I,m_2^I), while the other two are varied over a wide range. In Fig. 1, we illustrate these lower bounds for Pattern III, which are restricted by the (3,1) entry of N_1^d matrix and suppressed by a factor of λ. For Pattern VIII, in Fig. 2 we find the flavour violating neutral Higgs to be much lighter and possibly accessible at LHC.§ CONCLUSIONS We have presented a model based on the SM with 3 Higgs and an additional flavour discrete symmetry. We have shown that there exist flavour discrete symmetry configurations which lead to the alignment of the quark sectors.By allowing each scalar field to couple to each quark generation with a distinctive scale, one obtains the quark mass hierarchy, and although this hierarchy does not arise from the symmetry, the effect of both is such that the CKM matrix is near to the identity and has the correct overall phenomenological features. In this context, we have obtained 7 solutions fulfilling these requirements, with the additional constraint of the up quark mass matrix being diagonal and real.We have also verified if accidental U(1) symmetries may appear in the Yukawa sector or in the potential, particularly the case where a continuous accidental U(1) symmetry could arise, once the Z_7 is imposed at the Lagrangian level. This was indeed the case, however we shown that the anomaly-free conditions of global symmetries are violated. Thus, the global U(1)_X symmetry is anomalous and therefore only the discrete symmetry Z_7 persists.As in this model new Higgs doublets are added, one expects large FCNC effects, already present at tree level. However, such effects have not been experimentally observed. We show that, for certain of our specific implementations of the flavour symmetry, it is possible to suppress the FCNC effects and to ensure that the flavour violating neutral Higgs are light enough to be detectable at LHC. Indeed, in this respect, our model is a generalization of the BGL models for 3HDM, since the FCNC flavour structure is entirely determined by CKM.This work is partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CERN/FP/123580/2011, PTDC/FIS-NUC/0548/2012, EXPL/FIS-NUC/0460/2013, and CFTP-FCT Unit 777 (PEst-OE/FIS/UI0777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. The work of D.E.C. is also supported by Associação do Instituto Superior Técnico para a Investigação e Desenvolvimento (IST-ID). N.R.A is supported by European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896. N.R.A is grateful to CFTP for the hospitality during his stay in Lisbon.ieeetr | http://arxiv.org/abs/1705.09743v2 | {
"authors": [
"Nuno Rosa Agostinho",
"David Emmanuel-Costa",
"J. I. Silva-Marcos"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170527000342",
"title": "Exploring the Quark Flavour Puzzle within the 3 Higgs Model"
} |
CASENet: Deep Category-Aware Semantic Edge DetectionZhiding YuThe authors contributed equally.Carnegie Mellon [email protected] Chen Feng[1]Ming-Yu LiuThis work was done during the affiliation with MERL.Srikumar Ramalingam[2]Mitsubishi Electric Research Laboratories (MERL)[email protected], [email protected], [email protected] ===========================================================================================================================================================================================================================================================================================================================empty Boundary and edge cues are highly beneficial in improving a wide variety of vision tasks such as semantic segmentation, object recognition, stereo, and object proposal generation. Recently, the problem of edge detection has been revisited and significant progress has been made with deep learning. While classical edge detection is a challenging binary problem in itself, the category-aware semantic edge detection by nature is an even more challenging multi-label problem. We model the problem such that each edge pixel can be associated with more than one class as they appear in contours or junctions belonging to two or more semantic classes. To this end, we propose a novel end-to-end deep semantic edge learning architecture based on ResNet and a new skip-layer architecture where category-wise edge activations at the top convolution layer share and are fused with the same set of bottom layer features. We then propose a multi-label loss function to supervise the fused activations. We show that our proposed architecture benefits this problem with better performance, and we outperform the current state-of-the-art semantic edge detection methods by a large margin on standard data sets such as SBD and Cityscapes. § INTRODUCTION Figure <ref> shows an image of a road scene from Cityscapes dataset <cit.> with several object categories such as building, ground, sky, and car. In particular, we study the problem of simultaneously detecting edge pixels and classifying them based on association to one or more of the object categories <cit.>. For example, an edge pixel lying on the contour separating building and pole can be associated with both of these object categories. In Figure <ref>, we visualize the boundaries and list the colors of typical category combinations such as “building+pole” and “road+sidewalk”. In our problem, every edge pixel is denoted by a vector whose individual elements denote the strength of pixel's association with different semantic classes. While most edge pixels will be associated with only two object categories, in the case of junctions <cit.> one may expect the edge pixel to be associated with three or even more. We therefore do not restrict the number of object categories a pixel can be associated with, and formulate our task as a multi-label learning problem. In this paper, we propose CASENet, a deep network able to detect category-aware semantic edges.Given K defined semantic categories, the network essentially produces K separate edge maps where each map indicates the edge probability of a certain category. An example of separately visualized edge maps on a test image is given in Figure <ref>.The problem of edge detection has been shown to be useful for a number of computer vision tasks such as segmentation <cit.>, object proposal <cit.>, 3d shape recovery <cit.>, and 3d reconstruction <cit.>. By getting a better understanding of the edge classes and using them as prior or constraints, it is reasonable to expect some improvement in these tasks. With a little extrapolation, it is not difficult to see that a near-perfect semantic edge, without any additional information, can solve semantic segmentation, depth estimation <cit.>, image-based localization <cit.>, and object detection <cit.>. We believe that it is important to improve the accuracy of semantic edge detection to a certain level for moving towards a holistic scene interpretation.Early work tends to treat edge information as low-level cues to enhance other applications. However, the availability of large training data and the progress in deep learning methods have allowed one to make significant progress for the edge detection problem in the last few years. In particular, there have been newer data sets <cit.>. The availability of large-scale semantic segmentation data sets <cit.> can also be easily processed to obtain semantic edge data set as these two problems can be seen as dual problems. §.§ Related works The definition of boundary or edge detection has evolved over time from low-level to high-level features: simple edge filters <cit.>, depth edges <cit.>, object boundaries <cit.>, and semantic contours <cit.>. In some sense, the evolution of edge detection algorithms captures the progress in computer vision from simple convolutional filters such as Sobel <cit.> or Canny <cit.> to fully developed deep neural networks. Low-level edges Early edge detection methods used simple convolutional filters such as Sobel <cit.> or Canny <cit.>. Depth edges Some previous work focuses on labeling contours into convex, concave, and occluding ones from synthetic line drawings <cit.> and real world images under restricted settings <cit.>. Indoor layout estimation can also be seen as the identification of concave boundaries (lines folding walls, ceilings, and ground) <cit.>. By recovering occluding boundaries <cit.>, it was shown that the depth ordering of different layers in the scene can be obtained. Perceptual edges A wide variety of methods are driven towards the extraction of perceptual boundaries <cit.>. Dollar et al. <cit.> used boosted decision trees on different patches to extract edge maps. Lim et al. <cit.> computed sketch tokens which are object boundary patches using random forests. Several other edge detection methods include statistical edges <cit.>, multi-scale boundary detection <cit.>, and point-wise mutual information (PMI) detector <cit.>. More recently, Dollar and Zitnick <cit.> proposed a realtime fast edge detection method using structured random forests. Latest methods <cit.> using deep neural networks have pushed the detection performance to state-of-the-art. Semantic edges The origin of semantic edge detection can be possibly pinpointed to <cit.>. As a high level task, it has also been used implicitly or explicitly in many problems related to segmentation <cit.> and reconstruction <cit.>. In some sense, all semantic segmentation methods <cit.> can be loosely seen as semantic edge detection since one can easily obtain edges, although not necessarily an accurate one, from the segmentation results. There are papers that specifically formulate the problem statement as binary or category-aware semantic edge detection <cit.>. Hariharan et al. <cit.> introduced the Semantic Boundaries Dataset (SBD) and proposed inverse detector which combines both bottom-up edge and top-down detector information to detect category-aware semantic edges. HFL <cit.> first uses VGG <cit.> to locate binary semantic edges and then uses deep semantic segmentation networks such as FCN <cit.> and DeepLab <cit.> to obtain category labels. The framework, however, is not end-to-end trainable due to the separated prediction process. DNNs for edge detection Deep neural networks recently became popular for edge detection. Related work includes SCT based on sparse coding <cit.>, N^4 fields <cit.>, deep contour <cit.>, deep edge <cit.>, and CSCNN <cit.>. One notable method is the holistically-nested edge detection (HED) <cit.> which trains and predicts edges in an image-to-image fashion and performs end-to-end training. §.§ ContributionsOur work is related to HED in adopting a nested architecture but we extend the work to the more difficult category-aware semantic edge detection problem. Our main contributions in this paper are summarized below: * To address edge categorization, we propose a multi-label learning framework which allows improved edge learning than traditional multi-class framework.* We propose a novel nested architecture without deep supervision on ResNet <cit.>, where bottom features are only used to augment top classifications. We show that deep supervision may not be beneficial in our problem.* We outperform previous state-of-the-art methods by significant margins on SBD and Cityscapes datasets. § PROBLEM FORMULATIONGiven an input image, our goal is to compute the semantic edge maps corresponding to pre-defined categories. More formally, for an input image 𝐈 and K defined semantic categories, we are interested in obtaining K edge maps {𝐘_1,⋯,𝐘_K}, each having the same size as 𝐈. With a network having the parameters 𝐖, we denote 𝐘_k(𝐩|𝐈,𝐖) ∈ [0,1] as the network output indicating the computed edge probability on the k-th semantic category at pixel 𝐩. §.§ Multi-label loss function Possibly driven by the multi-class nature of semantic segmentation, several related works on category-aware semantic edge detection have more or less looked into the problem from the multi-class learning perspective. Our intuition is that this problem by nature should allow one pixel belonging to multiple categories simultaneously, and should be addressed by a multi-label learning framework.We therefore propose a multi-label loss. Suppose each image 𝐈 has a set of label images {𝐘̅_1,⋯,𝐘̅_K}, where 𝐘̅_k is a binary image indicating the ground truth of the k-th class semantic edge. The multi-label loss is formulated as:ℒ(𝐖)=∑_kℒ_k(𝐖) =∑_k∑_𝐩{ - β𝐘̅_k(𝐩)log𝐘_k(𝐩|𝐈;𝐖)- (1-β) (1-𝐘̅_k(𝐩))log(1-𝐘_k(𝐩|𝐈;𝐖))},where β is the percentage of non-edge pixels in the image to account for skewness of sample numbers, similar to <cit.>.§ NETWORK ARCHITECTUREWe propose CASENet, an end-to-end trainable convolutional neural network (CNN) architecture (shown in Fig. <ref>) to address category-aware semantic edge detection. Before describing CASENet, we first propose two alternative network architectures which one may come up with straightforwardly given the abundant previous literature on edge detection and semantic segmentation. Although both architectures can also address our task, we will analyzeissues associated with them, and address these issues by proposing the CASENet architecture. §.§ Base networkWe address the edge detection problem under the fully convolutional network framework. We adopt ResNet-101 by removing the original average pooling and fully connected layer, and keep the bottom convolution blocks. We further modify the base network in order to better preserve low-level edge information. We change the stride of the first and fifth convolution blocks (“res1” and “res5” in Fig. <ref>) in ResNet-101 from 2 to 1.We also introduce dilation factors to subsequent convolution layers to maintain the same receptive field sizes as the original ResNet, similar to <cit.>. §.§ Basic architecture A very natural architecture one may come up with is the Basic architecture shown in Fig. <ref>. On top of the base network, we add a classification module (Fig. <ref>) as a 1 × 1 convolution layer, followed by bilinear up-sampling (implemented by a K-grouped deconvolution layer) to produce a set of K activation maps {𝐀_1, ⋯, 𝐀_K}, each having the same size as the image. We then model the probability of a pixel belonging to the k-th class edge using the sigmoid unit given by 𝐘_k(𝐩)= σ(𝐀_k(𝐩)), which is presented in the Eq. (<ref>). Note that 𝐘_k(𝐩) is not mutually exclusive. §.§ Deeply supervised architecture One of the distinguishing features of the holistically-nested edge detection (HED) network <cit.> is the nested architecture with deep supervision <cit.>. The basic idea is to impose losses to bottom convolution sides besides the top network loss. In addition, a fused edge map is obtained by supervising the linear combination of side activations.Note that HED only performs binary edge detection. We extended this architecture to handle K channels for side outputs and K channels for the final output. We refer to this as deeply supervised network (DSN), as depicted in Fig. <ref>. In this network, we connect an above-mentioned classification module to the output of each stack of residual blocks, producing 5 side classification activation maps {𝐀^(1),…,𝐀^(5)}, where each of them has K-channels. We then fuse these 5 activation maps through a sliced concatenation layer (the color denotes the channel index in Fig. <ref>) to produce a 5K-channel activation map:𝐀^f = {𝐀^(1)_1,…,𝐀^(5)_1,𝐀^(1)_2,…,𝐀^(5)_2,…,𝐀^(5)_K}𝐀^f is fed into our fused classification layer which performs K-grouped 1 × 1 convolution (Fig. <ref>) to produce a K-channel activation map 𝐀^(6). Finally, 6 loss functions are computed on {𝐀^(1),…,𝐀^(6)} using the Equation <ref> to provide deep supervision to this network.Note that the reason we perform sliced concatenation in conjunction with grouped convolution instead of the corresponding conventional operations is as follows. Since the 5 side activations are supervised, we implicitly constrain each channel of those side activations to carry information that is most relevant to the corresponding class.With sliced concatenation and grouped convolution, the fused activation for a pixel 𝐩 is given by:𝐀^(6)_k(𝐩) = W_k^T[𝐀^(1)_k(𝐩)^T, ⋯, 𝐀^(5)_k(𝐩)^T]This essentially integrates corresponding class-specific activations from different scales as the finally fused activations. Our experiments empirically support this design choice. §.§ CASENet architecture Upon reviewing the Basic and DSN architectures, we notice several potential associated issues in the category-aware semantic edge detection task:First, the receptive field of the bottom side is limited. As a result it may be unreasonable to require the network to perform semantic classification at an early stage, given that context information plays an important role in semantic classification. We believe that semantic classification should rather happen on top where features are encoded with high-level information.Second, bottom side features are helpful in augmenting top classifications, suppressing non-edge pixels and providing detailed edge localization and structure information. Hence, they should be taken into account in edge detection.Our proposed CASENet architecture (Fig. <ref>) is motivated by addressing the above issues. The network adopts a nested architecture which to some extent shares similarity to DSN but also contains several key improvements. We summarize these improvements below:* Replace the classification modules at bottom sides to the feature extraction modules.* Put the classification module and impose supervision only at the top of the network.* Perform shared concatenation (Fig. <ref>) instead of sliced concatenation. The difference between side feature extraction and side classification is that the former only outputs a single channel feature map 𝐅^(j) rather than K class activations. The shared concatenation replicates the bottom features 𝐅 = {𝐅^(1),𝐅^(2),𝐅^(3)} from Side-1-3 to separately concatenate with each of the K top activations:𝐀^f= {𝐅, 𝐀^(5)_1, 𝐅, 𝐀^(5)_2, 𝐅, 𝐀^(5)_3,…, 𝐅, 𝐀^(5)_K}.The resulting concatenated activation map is again fed into the fused classification layer with K-grouped convolution to produce a K-channel activation map 𝐀^(6).In general, CASENet can be thought of as a joint edge detection and classification network by letting lower level features participating and augmenting higher level semantic classification through a skip-layer architecture.§ EXPERIMENTSIn this paper, we compare CASENet[Source code available at: <http://www.merl.com/research/license#CASENet>.] with previous state-of-the-art methods, including InvDet <cit.>, HFL <cit.>, weakly supervised object boundaries <cit.>, as well as several baseline network architectures. §.§ DatasetsWe evaluate the methods on SBD <cit.>, a standard dataset for benchmarking semantic edge detection. Besides SBD, we also extend our evaluation to Cityscapes <cit.>, a popular semantic segmentation dataset with pixel-level high quality annotations and challenging street view scenarios. To the best of our knowledge, our paper is the first work to formally report semantic edge detection results on this dataset. SBD The dataset consists of 11355 images from the PASCAL VOC2011 <cit.> trainval set, divided into 8498 training and 2857 test images[There has been a clean up of the dataset with a slightly changed image number. We also report the accordingly updated InvDet results.]. This dataset has semantic boundaries labeled with one of 20 Pascal VOC classes. Cityscapes The dataset contains 5000 images divided into 2975 training, 500 validation and 1525 test images. Since the labels of test images are currently not available, we treat the validation images as test set in our experiment. §.§ Evaluation protocolOn both SBD and Cityscapes, the edge detection accuracy for each class is evaluated using the official benchmark code and ground truth from <cit.>. We keep all settings and parameters as default, and report the maximum F-measure (MF) at optimal dataset scale (ODS), and average precision (AP) for each class. Note that for Citiscapes, we follow <cit.> exactly to generate ground truth boundaries with single pixel width for evaluation, and reduce the sizes of both ground truth and predicted edge maps to half along each dimension considering the speed of evaluation. §.§ Implementation detailsWe trained and tested CASENet, HED <cit.>, and the proposed baseline architectures using the Caffe library <cit.>. Training labels Considering the misalignment between human annotations and true edges, and the label ambiguity of pixels near boundaries, we generate slightly thicker ground truth edges for network training. This can be done by looking into neighbors of a pixel and seeking any difference in segmentation labels. The pixel is regarded as an edge pixel if such difference exists. In our paper, we set the maximum range of neighborhood to be 2. Under the multi-label framework, edges from different classes may overlap. Baselines Since several main comparing methods such as HFL and HED use VGG or VGG based architectures for edge detection and categorization, we also adopt the CASENet and other baseline architectures on VGG (denoted as CASENet-VGG). In particular, we remove the max pooling layers after conv4, and keep the resolutions of conv5, fc6 and fc7 the same as conv4 (1/8 of input). Similar to <cit.>, both fc6 and fc7 are treated as convolution layers with 3×3 and 1×1 convolution and dimensions set to 1024. Dilation factors of 2 and 4 are applied to conv5 and fc6.To compare our multi-label framework with multi-class, we generate ground truth with non-overlapping edges of each class, reweight the softmax loss similar to our paper, and replace the top with a 21-class reweighted softmax loss. Initialization In our experiment, we initialize the convolution blocks of ResNet/VGG in CASENet and all comparing baselines with models pre-trained on MS COCO <cit.>. Hyper-parameters We unify the hyper-parameters for all comparing methods with the same base network, and set most of them following HED. In particular, we perform SGD with iteration size of 10, and fix loss weight to be 1, momentum 0.9, and weight decay 0.0005. For methods with ResNet, we set the learning rate, step size, gamma and crop size to 1e-7 / 5e-8, 10000 / 20000, 0.1 / 0.2 and 352× 352 / 472× 472 respectively for SBD and Cityscapes. For VGG, the learning rate is set to 1e-8 while others remain the same as ResNet on SBD. For baselines with softmax loss, the learning rate is set to 0.01 while other parameters remain the same. The iteration numbers on SBD and Cityscapes are empirically set to 22000 and 40000. Data augmentation During training, we enable random mirroring and cropping on both SBD and Cityscapes. We additionally augment the SBD data by resizing each image with scaling factors {0.5, 0.75, 1.0, 1.25, 1.5}, while no such augmentation is performed on Cityscapes. §.§ Results on SBDTable <ref> shows the MF scores of different methods performing category-wise edge detection on SBD, where CASENet outperforms previous methods. Upon using the benchmark code from <cit.>, one thing we notice is that the recall scores of the curves are not monotonically increasing, mainly due to the fact that post-processing is taken after thresholding in measuring the precision and recall rates. This is reasonable since we have not taken any postprocessing operations on the obtained raw edge maps. We only report the MF on SBD since AP is not well defined under such situation. The readers may kindly refer to supplementary materials for class-wise precision recall curves. Multi-label or multi-class? We compare the proposed multi-label loss with the reweighted softmax loss under the Basic architecture. One could see that using softmax leads to significant performance degradation on both VGG and ResNet, supporting our motivation in formulating the task as a multi-label learning problem, in contrast to the well accepted concept which addresses it in a multi-class way. Is deep supervision necessary? We compare CASENet with baselines network architectures including Basic and DSN depicted in Fig. <ref>. The result empirically supports our intuition that deep supervision on bottom sides may not be necessary. In particular, CASENet wins frequently on per-class MF as well as the final mean MF score. Our observation is that the annotation quality to some extent influenced the network learning behavior and evaluation, leading to less performance distinctions across different methods. Such distinction becomes more obvious on Cityscapes. Is top supervision necessary? One might further question the necessity of imposing supervision on Side-5 activation in CASENet. We use CASENet^- to denote the same CASENet architecture without Side-5 supervision during training. The improvement upon adding Side-5 supervision indicates that a supervision on higher level side activation is helpful. Our intuition is that Side-5 supervision helps Side-5 focusing more on the classification of semantic classes with less influence from interacting with bottom layers. Visualizing side activations We visualize the results of CASENet, CASENet^- and DSN on a test image in Fig. <ref>. Overall, CASENet achieves better detection compared to the other two. We further show the side activations of this testing example in Fig. <ref>, from which one can see that the activations of DSN on Side-1, Side-2 and Side-3 are more blurry than CASENet features. This may be caused by imposing classification requirements on those layers, which seems a bit aggressive given limited receptive field and information and may caused performance degradation. Also one may notice the differences in “Side5-Person” and “Side5-Boat” between CASENet^- and CASENet, where CASENet's activations overall contain sharper edges, again showing the benefit of Side-5 supervision. From ResNet to VGG CASENet-VGG in Table <ref> shows comparable performance to HFL-FC8. HFL-CRF performs slightly better with the help of CRF postprocessing. The results to some extent shows the effectiveness our learning framework, given HFL uses two VGG networks separately for edge localization and classification. Our method also significantly outperforms the HED baselines from <cit.>, which gives 44 / 41 on MF / AP, and 49 / 45 with detection. Other variants We also investigated several other architectures. For example, we kept the stride of 2 in “res1”. This downgrades the performance for lower input resolution. Another variant is to use the same CASENet architecture but impose binary edge losses (where a pixel is considered lying on an edge as long as it belongs to the edge of at least one class) on Side-1-3 (denoted as CASENet-edge in Fig. <ref>). However we found that such supervision seems to be a divergence to the semantic classification at Side-5. §.§ Results on Cityscapes We also train and test both DSN and CASENet with ResNet as base network on the Cityscapes. Compared to SBD, Cityscapes has relatively higher annotation quality but contains more challenging scenarios. The dataset contains more overlapping objects, which leads to more cases of multi-label semantic boundary pixels and thus may be better to test the proposed method. In Table <ref>, we provide both MF and AP of the comparing methods. To the best of our knowledge, this is the first paper quantitatively reporting the detection performance of category-wise semantic edges on Cityscapes. One could see CASENet consistently outperforms DSN in all classes with a significant margin. Besides quantitative results, we also visualize some results in Fig. <ref> for qualitative comparisons.§ CONCLUDING REMARKSIn this paper, we proposed an end-to-end deep network for category-aware semantic edge detection. We show that the proposed nested architecture, CASENet, shows improvements over some existing architectures popular in edge detection and segmentation. We also show that the proposed multi-label learning framework leads to better learning behaviors on edge detection. Our proposed method improves over previous state-of-the-art methods with significant margins. In the future, we plan to apply our method to other tasks such as stereo and semantic segmentation. ieee § MULTI-LABEL EDGE VISUALIZATIONIn order to effectively visualize the prediction quality of multi-label semantic edges, the following color coding protocol is used to generate results in Fig. <ref>, Fig. <ref>, and Fig. <ref>. First, we associate each of the K semantic object class a unique value of Hue, denoted as 𝖧≜[𝖧_0,𝖧_1,⋯,𝖧_K-1]. Given a K-channel output 𝐘 from our CASENet's fused classification module, where each element 𝐘_k(𝐩) ∈ [0,1] denotes the pixel 𝐩's predicted confidence of belonging to the k-th class, we return an HSV value for that pixel based on the following equations:𝐇(𝐩) =∑_k𝐘_k(𝐩)𝖧_k/∑_k𝐘_k,𝐒(𝐩) =255 max{𝐘_k(𝐩)|k=0,⋯,K-1},𝐕(𝐩) =255,which is also how the ground truth color codes are computed (by using Ŷ instead). Note that the edge response maps of testing results are thresholded with 0.5, with the two classes having the strongest responses selected to compute hue based on Eq. (<ref>).For Cityscapes, we manually choose the following hue values to encode the 19 semantic classes so that the mixed Hue values highlight different multi-label edge types:𝖧≜ [ 359,320,40,80,90,10,20,30,140,340,280,330,350,120,110,130,150,160,170]The colors and their corresponding class names are illustrated in following Table <ref>. The way Hue is mixed in equation <ref> indicates that any strong false positive response or incorrect response strength can lead to hue values shifted from ground truth. This helps to visualize false prediction.§ ADDITIONAL RESULTS ON SBD§.§ Early stage loss analysisFig. <ref> shows the losses of different tested network configurations between iteration 100-500. Note that for Fig. <ref>, loss curves between iteration 0-8000 is not available due to the large averaging kernel size. One can see CASENet's fused loss is initially larger than its side5 loss. It later drops faster and soon become consistently lower than the side5 loss (see Fig. <ref>).§.§ Class-wise prediction examplesWe illustrate 20 typical examples of the class-wise edge predictions of different comparing methods in Fig. <ref> and <ref>, with each example corresponding to one of the SBD semantic category. One can observe that the proposed CASENet slightly but consistently outperforms ResNets with the basic and DSN architectures, by overall showing sharper edges and often having stronger responses on difficult edges.Meanwhile, Fig. <ref> shows several difficult or failure cases on the SBD Datasets. Interestingly, while the ground truth says there is no “aeroplane” in the first row and “dining table” in the second, the network is doing decently by giving certain level of edge responses, particularly in the “dining table” example. The third row shows an example of the false positive mistakes often made by the networks on small objects. The networks falsely think there is a sheep while it is in fact a rock. When objects become smaller and lose details, such mistakes in general happen more frequently. §.§ Class-wise precision-recall curvesFig. <ref> shows the precision-recall curves of each semantic class on the SBD Dataset. Note that while post-processing edge refinement may further boost the prediction performance <cit.>, we evaluate only on the raw network predictions to better illustrate the network performance without introducing other factors. The evaluation is conducted fully based on the same benchmark code and ground truth files released by <cit.>. Results indicate that CASENet slightly but consistently outperforms the baselines. §.§ Performance at different iterationsWe evaluate the Basic, DSN, CASENet on SBD for every 2000 iterations between 16000-30000, with the MF score shown in Fig. <ref>. We found that the performance do not change significantly, and CASENet consistently outperforms Basic and DSN. < g r a p h i c s > figureTesting Performance vs. different iterations. §.§ Performance with a more standard splitConsidering that many datasets adopts the training + validation + test data split, we also randomly divided the SBD training set into a smaller training set and a new validation set with 1000 images. We used the average loss on validation set to select the optimal iteration number separately for both Basic and CASENet. Their corresponding MFs on the test set are 71.22% and 71.79%, respectively.§ ADDITIONAL RESULTS ON CITYSCAPES §.§ Additional qualitative resultsFor more qualitative results, the readers may kindly refer to our released videos on Cityscapes validation set, as well as additional demo videos. §.§ Class-wise precision-recall curvesFig. <ref> shows the precision-recall curves of each semantic class on the Cityscapes Dataset. Again the evaluation is conducted only on the raw network predictions. Since evaluating the results at original scale (1024 × 2048) is extremely slow and is not necessary, we bilinearly downsample both the edge responses and ground truths to 512 × 1024. Results indicate that CASENet consistently outperforms the ResNet with the DSN architecture. | http://arxiv.org/abs/1705.09759v1 | {
"authors": [
"Zhiding Yu",
"Chen Feng",
"Ming-Yu Liu",
"Srikumar Ramalingam"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170527033536",
"title": "CASENet: Deep Category-Aware Semantic Edge Detection"
} |
Instytut Automatyki, Wydział Automatyki Elektroniki i Informatyki, Politechnika Śląska, Akademicka 16, 44-100 Gliwice, Poland [email protected] Instytut Astronomiczny, Uniwersytet Wrocławski, Kopernika 11, 51-622 Wrocław, PolandInstytut Elektroniki, Wydział Automatyki Elektroniki i Informatyki, Politechnika Śląska, Akademicka 16, 44-100 Gliwice, Poland Institute of Communication Networks and Satellite Communications, Graz University of Technology, Inffeldgasse 12, 8010 Graz, Austria Institut für Astrophysik, Universität Wien, Türkenschanzstrasse 17, 1180 Wien, Austria Département de physique, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Québec, H3C 3J7, Canada Centre de recherche en astrophysique du Québec (CRAQ), Canada Centrum Astronomiczne im. M. Kopernika, Polska Akademia Nauk, Bartycka 18, 00-716 Warszawa, Poland European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching, Germany Department of Physics, Royal Military College of Canada, PO Box 17000, Station Forces, Kingston, Ontario, K7K 7B4, Canada LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada Centrum Badań Kosmicznych, Polska Akademia Nauk, Bartycka 18A, 00-716 Warszawa, Poland Institut für Astro- und Teilchenphysik, Universität Innsbruck, Technikerstrasse 25/8, 6020 Innsbruck, Austria The BRITE mission is a pioneering space project aimed at the long-term photometric monitoring of the brightest stars in the sky by means of a constellation of nano-satellites.Its main advantage is high photometric accuracy and time coverage inaccessible from the ground. Its main drawback is the lack of cooling of the CCD and the absence of good shielding that would protect sensors from energetic particles.The main aim of this paper is the presentation of procedures used to obtain high-precision photometry from a series of images acquired by the BRITE satellites in two modes of observing, stare and chopping.The other aim is comparison of the photometry obtained with two different pipelines and comparison of the real scatter with expectations. We developed two pipelines corresponding to the two modes of observing. They are based on aperture photometry with a constant aperture, circular for stare mode of observing and thresholded for chopping mode. The impulsive noise is a serious problem for observations made in the stare mode of observing and therefore in the pipeline developed for observations made in this mode, hot pixels are replaced using the information from shifted images in a series obtained during a single orbit of a satellite. In the other pipeline, the hot pixel replacement is not required because the photometry is made in difference images.The assessment of the performance of both pipelines is presented. It is based on two comparisons, which use data from six runs of the UniBRITE satellite: (i) comparison of photometry obtained by both pipelines on the same data, which were partly affected by charge transfer inefficiency (CTI), (ii) comparison of real scatter with theoretical expectations. It is shown that for CTI-affected observations, the chopping pipeline provides much better photometry than the other pipeline. For other observations, the results are comparable only for data obtained shortly after switching to chopping mode. Starting from about 2.5 years in orbit, the chopping mode of observing provides significantly better photometry for UniBRITE data than the stare mode.This paper shows that high-precision space photometry with low-cost nano-satellites is achievable. The proposed methods, used to obtain photometry from images affected by high impulsive noise, can be applied to data from other space missions or even to data acquired from ground-based observations.BRITE-Constellation: Data processing and photometryBased on data collected by the BRITE Constellation satellite mission, designed, built, launched, operated and supported by the Austrian Research Promotion Agency (FFG), the University of Vienna, the Technical University of Graz, the Canadian Space Agency (CSA), the University of Toronto Institute for Aerospace Studies (UTIAS), the Foundation for Polish Science & Technology (FNiTP MNiSW), and National Science Centre (NCN). A. Popowicz<ref> A. Pigulski<ref> K. Bernacki<ref> R. Kuschnig<ref>,<ref> H. Pablo<ref>,<ref> T. Ramiaramanantsoa<ref>,<ref> E. Zocłońska<ref> D. Baade<ref> G. Handler<ref> A. F. J. Moffat<ref>,<ref> G. A. Wade<ref> C. Neiner<ref> S. M. Rucinski<ref> W. W. Weiss<ref> O. Koudelka<ref> P. Orleański<ref> A. Schwarzenberg-Czerny<ref> K. Zwintz<ref> Received; accepted ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTIONBRITE-Constellation[BRITE stands for BRIght Target Explorer.] is a group of five nano-satellites, launched in 2013 – 2014, aimed at obtaining long uninterrupted time-series precision photometry in two passbands for the brightest (typically V < 5 mag) stars in the sky <cit.>. This is the first space astronomy mission accomplished with nano-satellites, cubes of an edge length of 20 cm.The scientific goals of the mission focus on variability of intrinsically luminous stars, especially those that are located at the upper main sequence, i.e., hot massive stars. They show a variety of variability types which are caused by pulsations, mass loss, rotation, stellar winds and interactions in binary and multiple systems. There are several advantages of studying bright stars. First, spectroscopic observations of such stars either exist in large number or can be easily obtained even with a small or a midsize telescope. Next, they usually have a long record of observations which in many cases allows for a study of secular variability and long-term processes. Finally, these stars are usually relatively close enough to us, so that additional information, like parallax or interferometric orbit for a binary, is available. Up to the time of writing (April 2017), photometric data for over 400 stars in seventeen fields located along the Galactic plane, were delivered. The first scientific results of the mission have been published <cit.>.The present paper is regarded as the third in a series devoted to the description of the technical aspects of the mission. Paper I describes the history, concept, and the main objectives of the BRITE mission, while Paper II <cit.> explains design and characteristics of BRITE optics and detectors, pre-flight tests, launches, commissioning phase and in-orbit performance. In Paper II, particular attention is paid to the effects of radiation damage and ways to mitigate them. It also introduces the two modes of observing in which the BRITE satellites observe, a stare and a chopping mode. In the present paper, we give details of the two pipelines that are currently used for obtaining photometry from BRITE images and discuss the quality of the final photometry by a comparison with theoretical expectations.§ DATA FROM THE BRITE SATELLITESThe data downloaded from the BRITE satellites consist of pre-defined small parts of the full CCD image, which we will call subrasters throughout the paper. The subrasters are either square (in the stare mode of observing) or rectangular (in the chopping mode) having sides that range from 24 to 54 pixels depending on the star that is in it and depending on the satellite. With 27^'' per pixel resolution, this corresponds to 11^' – 24^' in the sky. Full-frame images are downloaded only occasionally, e.g., to verify the pointing. The full field of view of the CCD covers 30^∘ × 20^∘ in the sky, but is limited by vignetting to the area of about 24^∘ wide and 20^∘ high (Fig. <ref>). Stars are intentionally defocussed in BRITE images. Without defocussing, a star's image would be recorded within a small fraction of a pixel, making the stellar profile undersampled and thus the photometry less precise. The price of defocussing are position-dependent and highly non-symmetric stellar profiles. The complexity of profiles increases with the distance from the image centre. A sample configuration of subrasters in the CCD plane is presented in Fig. <ref> for the UBr satellite[As in Papers I and II, the following abbreviations for BRITE satellites will be used: BRITE-Austria (BAb), UniBRITE (UBr), BRITE-Toronto (BTr), BRITE-Lem (BLb) and BRITE-Heweliusz (BHr).] and Orion 1 field. Stellar profiles in this field are presented in Fig. <ref>. In spite of defocussing one can see that profiles sometimes show large gradients, even within a single pixel (see Fig. <ref>), and therefore the images can be considered as undersampled. The high-resolution profile templates shown in the right-hand part of Fig. <ref> were obtained by processing the raw frames with the filtering scheme implemented in our pipeline and then by performing resolution enhancement with the Drizzle algorithm <cit.>. They reveal the subtle structure of the profiles. Drizzle parameters were chosen as follows: the drop size () was set to 0.2 and the output image had 10 times higher sampling than the original image.The next critical problem of BRITE data is the presence of detector defects, which can be observed as distinctively brighter pixels and columns. While the former are related to the generation of dark current during exposure, the latter result from a similar process in the serial register during the readout phase. These are permanent defects induced by energetic protons hitting the CCD sensor <cit.>. In contrast to imagers employed in ground-based observations, the dark-current generation rate in cameras located in space is unstable because of the presence of so-called random telegraph signals <cit.>, a sudden step-like transitions between two or more metastable levels of noise that occur at random times. Thus, a standard dark frame subtraction will not work. For more details on the degradation of BRITE CCDs, the interested reader is referred to Paper II. Finally, the precision of tracking is far from perfect, resulting in occasional blurring of images or shifting the stars outside their subrasters. An example of such behaviour is presented in Fig. <ref>. The same figure shows also an example of tracking performance as judged by the motion of the image centroids during the first observing run of UBr.§.§ Modes of observingTwo years following the launch of the first BRITE satellites (UBr and BAb) in February 2013, the observations were carried out in so-called `stare' mode, in which star trackers kept satellites in a fixed orientation, so that in consecutive images stars were expected to be located at approximately the same position on the CCD, close to the centres of subrasters. An unavoidable jitter in position was also observed. For this observing mode, the pipeline presented in Sect. <ref> was developed. As mentioned in Paper II, the number of hot pixels for BRITE detectors increased with time, which severely affected photometry in stare mode, especially for observations made at higher temperatures. Moreover, charge transfer inefficiency (CTI) emerged, introducing a blurring of stellar profiles and hot pixels along columns of the detector as shown in the bottom left panel of Fig. <ref>. As a remedy, a new mode of observing, called `chopping' mode, was introduced. In this mode, the satellite pointing is switched between two positions separated by about 20 pixels (9^' in the sky). In consequence, a star is placed either on the left or the right side of the subraster. After each move, a new image is taken. Subtracting consecutive frames provides difference images which are used for the photometry. Difference images contain no or very few defective pixels. Those which are left occur as a consequence of the RTS phenomenon. Sample original and difference images are presented in Fig. <ref>. In chopping mode, the subrasters had to be enlarged approximately twice with respect to the stare mode, to encompass two complete stellar profiles in the difference image. Chopping mode was successfully adopted in each BRITE satellite and is currently the default mode of observing. The pipeline for chopping mode is presented in Sect. <ref>. § THE PIPELINE FOR OBSERVATIONS MADE IN STARE MODEThe unpredictable generation of charge in hot pixels and imperfect tracking have put high demands on the image processing pipeline. In the adopted pipeline, both the conventional image processing routines and novel approaches were implemented. Up to now, the pipeline was used to obtain light curves for over 200 stars in eight BRITE fields. The algorithm used in the pipeline can be divided into four main parts: classification of images (Sect. <ref>), image processing and photometry (Sect. <ref>), compensation for intra-pixel sensitivity (Sect. <ref>), and optimization of parameters (Sect. <ref>). The first part aims at dividing the raw images into three types: useful, dark, and defective. In the second part, the useful images are processed in order to obtain centres of gravity (centroids) of stellar profiles and consequently to estimate stellar flux within a circular aperture. The variations of intra-pixel sensitivity are compensated in the subsequent part of the pipeline. Finally, the pipeline parameters are optimized to achieve the highest photometric precision. §.§ Classification of subrasters and localization of stellar imagesThe first part of the pipeline included the routines developed for finding and rejecting the images disturbed by bad tracking or data transfer problems. At this step, the dark images[There is no shutter in the optical path. Therefore, these are not real dark images but the images ofsmall parts of the sky devoid of bright stars. They include bias, dark, and the (negligible) stellar background.] were also identified. After the first few months of observation, it was decided that dark images would be regularly taken during each orbit by swinging the satellite, so as to move the stars outside their pre-defined subrasters. This helped to reveal the locations of hot pixels.The dark current generation in the vertical charge transfer register results in columns with higher signal (see examples in the top row of Fig. <ref>). To account for this effect, the column offsets were compensated by subtracting the median intensity estimated from all pixels in a given column of the subraster. The median was used because it is highly robust with respect to the occurence of hot pixels. Since pixels containing stellar profile may also affect the median, their effect is diminished by excluding pixels which fall into stellar profile; see below. The results of this step are presented in the second row of Fig. <ref>.In the next step, the image is smoothed by applying a 3 × 3 pixel median filter, in which each pixel intensity is replaced by the median of its eight neighbours. Such a procedure suppresses the hot pixels so that they do not affect the identification of the stellar region in the next step. A slight smearing of stellar profiles induced by this procedure is acceptable for the next step of defining the stellar aperture area. An example of the application of such median filtering is given in the third row of Fig. <ref>. The estimation of the area covered by a stellar profile (hereafter called stellar area, A_ s, expressed in pixels) is based on a simple thresholding of the median-filtered image obtained in the previous step. From the analysis of a few dozen stars, a value of 100 ADU was selected as a reasonable threshold. This warrants the robustness against electronic noise while keeping the ability to detect the faintest targets. In order to consistently identify the stellar profile, only the region containing the largest number of contiguous pixels is identified as a star. Smaller, isolated regions, detected during the thresholding, are rejected. This step constitutes an additional protection against identifying hot pixels or their clusters as real stars. It also discards fainter nearby stars, which are occasionally present within subrasters.Although the median is the robust measure of a column offset, it may give an overestimated value if a stellar profile extends over a large fraction of a subraster. In this case, its subtraction may result in a dimmer column (see the rightmost images in the second and third row of Fig. <ref>). As a consequence, one obtains a slightly underestimated stellar area after thresholding. In order to avoid this problem, the procedure of column subtraction and star detection is iterated. In each iteration, the median in a column is calculated ignoring the pixels included in the stellar area during the previous iteration. If two consecutive stellar areas are identical, the procedure stops and the solution is adopted. The procedure usually converges after two or three iterations. Examples of images after the first and the second iteration are shown in Fig. <ref>. Next, the values of the stellar areas for a series of images are analyzed. An example is given in Fig. <ref>. As one can see, A_ s clusters around 80 – 100 pixels with some outliers on both sides. The A_ s markedly larger than the mean come from blurred images. Those with A_ s significantly smaller than the mean appear due to the stellar profile located at the subraster edge. Finally, the images for which A_ s = 0, i.e. no stellar area was detected, are good candidates for dark frames. The final step in classification is setting two threshold values, lower, A_ low, and upper, A_ up. The images with A_ s = 0 are classified as dark, those with A_ low≤ A_ s≤ A_ up as useful, and those with 0 < A_ s < A_ low and A_ s > A_ up are marked as defective and rejected. Several methods, e.g. based on σ-clipping or local standard deviation, were tested to automatically set the two thresholds. Unfortunately, the values of A_ s may occasionally fluctuate due to the intrinsic variability and temperature-dependent changes of the size of the stellar profile. Therefore, it was decided that for each star the thresholds will be set manually. They were chosen carefully with relatively large margins to prevent excluding possible short-term intrinsic brightenings or dimmings of a star. §.§ Image processing and photometryThe next part of the pipeline is aimed at obtaining aperture photometry of a star from a series of images. Only images classified previously as useful are considered. The dark frames are median-averaged to create a master dark frame, in which the pixels are classified as hot if their intensity is above a given threshold, S_ bad. As a result of this thresholding, a mask of bad pixels is obtained; see the second column in Fig. <ref>. The S_ bad parameter is optimized in the last part of the pipeline (Sect. <ref>). At this stage, the part of a subraster corresponding to the stellar area (let us call it the stellar mask) is derived in the same iterative way as was performed during the classification (columns 3 to 6 in Fig. <ref>). Once this is done, the image corrected for column offsets is multiplied by the stellar mask and each hot pixel within the mask is replaced by the median intensity of the neighbouring pixels. The resulting image will be referred to as I_ col (last column in Fig. <ref>). Then, the initial coordinates of the stellar centroid (x_ centr, y_ centr) are calculated as a centre of gravity from the following equations:x_ centr= ∑_x=1^X∑_y=1^Yx I_ col(x,y)∑_x=1^X∑_y=1^YI_ col(x,y), y_ centr= ∑_x=1^X∑_y=1^Yy I_ col(x,y)∑_x=1^X∑_y=1^YI_ col(x,y),where X and Y are the widths of the I_ col image, in the x and y coordinate, respectively. A precise estimation and replacement of the signal in hot pixels is a crucial part of the pipeline. For the purpose of image classification and initial estimation of the stellar centre, the hot pixels were replaced by a simple median of intensities of eight neighbouring pixels. Although this is sufficient for the classification and initial determination of the stellar centre, the final photometry requires a more sophisticated filtering scheme. Multiple interpolation approaches utilizing the intensity estimation based on a pixel neighbourhood have been tested <cit.>. Unfortunately, due to the significant undersampling of BRITE images, the fitting algorithms produced unsatisfactory results. This is why methods based on the neighbouring pixels were abandoned in favour of the approach based on the analysis of a series of exposures obtained during the same orbit.The method we propose utilizes the fact that during a single orbit typically more than 40 frames are taken. Imperfect tracking causes considerable changes of the position of a star within a subraster (see Fig. <ref>). Surprisingly, this can help in obtaining a good estimation of the signal, which can be used to replace the one in a hot pixel. The adopted procedure of hot pixel interpolation is the following. Let us consider that we want to perform corrections for hot pixels in the k-th subraster, I_k, of a series of N subrasters taken during a single orbit. For this purpose, the I_ col subrasters are used. Let the centroid calculated with Eq. (<ref>) be equal to (x_k,y_k) for that subraster. Let us now denote the centroids of the other images in the series as (x_i,y_i),i = , ..., N. The differential centroids (Δ x_i , Δ y_i ) can now be defined as follows:{Δ x_i= x_i-x_k, Δ y_i= y_i-y_k, i = , ..., N. .Let us also define fractional shifts (δ x_i, δ y_i):{δ x_i= Δ x_i - (Δ x_i), δ y_i= Δ y_i - (Δ y_i), i = , ..., N, .wherestands for rounding a number to the nearest integer. By definition, fractional shifts range between -0.5 and 0.5. When in two subrasters δ x_i and δ y_i are close to each other, we may assume that in these subrasters stellar profiles are sampled in a similar way. Since stellar profiles are in general under-sampled in BRITE images, the fractional shifts play a very important role, especially when the intensities are to be replaced using other subrasters. The idea of the method is to replace a hot pixel using only subrasters shifted with respect to the considered one by an amount close to an integer number of pixels in both directions. This means that the replacement procedure will involve only those stellar profiles which are sampled in a similar way. Adopting this procedure leads us to the following formula for the signal that will be used to replace a hot pixel at the position (x_h,y_h) in I_k (I_k is the I_k subraster with hot pixels corrected):I_k(x_h,y_h)= ∑_i=1, i≠ k ^N w_i I_i(x_h+Δ x_i, y_h+Δ y_i)∑_i=1^N w_i,w_i= 1√(δ x_i^2+δ y_i^2).The pixels flagged as hot in the thresholded master dark frame are excluded from the sum in Eq. (<ref>).By trial and error we came to the conclusion that it is optimal to use only about ten subrasters with the highest weights in the sum in Eq. (<ref>). Using a larger number of subrasters leads to a decrease of the photometric precision because more subrasters with larger fractional shifts (albeit with smaller weights) contribute to the sum. On the other hand, reducing the number of summed pixels would lead to an increase of noise. As a compromise, N= 10 was used.Once the intensity of a hot pixel is replaced with the value calculated from Eq. (<ref>), the centre of gravity of the stellar profile may change. Therefore, the procedure of replacement of hot pixels was iterated. In general, there was no recognizable improvement of the light curve quality after the second iteration, so that it was assumed that two iterations are sufficient.The presented approach is based on the assumption that stellar flux does not change significantly during a single-orbit observation (10 – 50 minutes). This assumption is well justified for most of the observed stars. However, even if this is not the case, the presented procedure introduces only a very small bias, which has a negligible effect on the final light curve. This is because the number of hot pixels in a given subraster is small in comparison with the number of pixels falling into the aperture containing the stellar profile.In the last step, circular aperture photometry with the radius optimized as explained in Sect. <ref> is performed, wherein only final, filtered subrasters I_k are used. The intensities of all the pixels, whose positions are within the aperture, are summed.§.§ Compensation for intra-pixel sensitivity variationsIt is well known that sensitivity of a pixel varies slightly across its surface <cit.>. Usually, the sensitivity is the highest in the pixel centre, decreasing towards its edges. Since stellar profiles in BRITE images have complex shapes, it was expected that the variations of intra-pixel sensitivity would noticeably affect the photometry. In Fig. <ref>, a sample high-resolution stellar profile is shown. It was obtained by averaging1000 interpolated low-resolution images, an example of which is depicted in the right bottom image of Fig. <ref>. As one can clearly see, there are significant intensity gradients within some pixels, and it is highly probable that the photometry will be affected by the sub-pixel shifts of a stellar profile. In order to assess and then compensate for the intra-pixel sensitivity variations, the following procedure was performed for each star. First, the median stellar flux in each orbit was computed. Then, for each measurement the deviation from the median was plotted against the fractional centroid position (δ x_ centr, δ y_ centr), defined as follows:{δ x_ centr = (x_ centr) ≡ x_ centr - ⌊x_ centr⌋, δ y_ centr =(y_ centr) ≡ y_ centr - ⌊y_ centr⌋, . where ⌊ ⌋ is the floor function. The dependencies are approximated by a fourth-order polynomial. An example of such a fit is shown in Fig. <ref>. The larger effect in the y coordinate is not a surprise because there are larger gradients in the cross section along this coordinate in comparison with the x coordinate (Fig. <ref>). The fitted polynomials are used in the next step to compensate for the intra-pixel variability. This is the only correction to the derived fluxes implemented in this pipeline. Intra-pixel corrections account only for the effect averaged over all pixels falling in the aperture. In other words, it is assumed that dependency between the fractional centroid position and the photometric offset is the same for the whole subraster. In reality, this is not always the case because stellar profiles change due to the temperature effect and smearing. In consequence, intra-pixel correction accounts only for a part of the position-dependent instrumental effect. It was decided that the remaining instrumental effects (such as possible flux dependencies on CCD temperature, centroid position, etc.) will be accounted independently of the photometric pipeline by the end-user. Examples of the required corrections are shown e.g. by <cit.> and <cit.>. Decorrelations with position would account for most of the residual dependencies provided that the correlation function is relatively smooth. A combination of both corrections, one implemented in the pipeline, the other accounted by decorrelations, seems to be the best solution when dealing with position-dependent instrumental effects.§.§ OptimizationThe last part of the algorithm is actually the loop which involves procedures described in Sects <ref> and <ref>, run for a set of different hot pixel thresholds S_ bad and various aperture radii R. In our estimation of quality, we used the median absolute deviation (MAD) calculated for all N_ orb orbits. The quality parameter, Q, is the median of normalized MADs, defined as follows:Q = median{MAD_i/√(N_i)}, i = 1, ..., N_ orb, MAD_i = median{|J_ki - median(𝐉_𝐢)|},k = 1, ..., N_i,where J_ki is k-th measured stellar flux in the i-th orbit, N_i is the number of points in the i-th orbit and 𝐉_𝐢 is the set of all measurements in i-th orbit. Using median-based indicators allowed us to reduce the influence of outliers within orbits and the worst orbits, corrupted e.g. by stray light, on the value of Q (the median of MADs for all orbits is used in its definition, see Eq. (<ref>)). The normalization by √(N_i) reflects the noise reduction due to the averaging of N_i measurements in the i-th orbit. The values of N_i range between several and over 160 depending on the satellite and field. The number of orbits N_ orb per setup (see Sect. <ref> for the definition of setups) has a wide range and amounts from about a dozen up to almost 1500.The optimization was performed for each star and setup combination independently. It was carried out for a set of four values of S_ bad, S_ bad∈ {50, 100, 150, 200} ADU and eight values of R, R∈ {3, 4, 5, 6, 7, 8, 9, 10} pixels. The adopted range always encompassed the minimum Q value. An example of optimization is presented in Fig. <ref>.The pipeline described in this section was applied to all data obtained in the stare mode of observing, i.e. the first eight fields observed by the BRITE satellites (see Table <ref>) and to the test data in the Perseus field obtained in the chopping mode. The photometry made with this pipeline is now available as Data Release 2; see Appendix <ref> for the explanation of data releases.§ THE PIPELINE FOR OBSERVATIONS MADE IN THE CHOPPING MODEAs explained in Sect. <ref>, the chopping mode was introduced as a remedy for the appearance of CTI and the increasing number of hot pixels in BRITE detectors. The photometry for images taken in this mode is carried out with difference images (Fig. <ref>). A difference image is defined as an image obtained by a subtraction of two consecutive images: I_k-I_k+1. In order to secure the same number of difference images as in the stare mode, each image is used twice, i.e., the next difference image is I_k+1-I_k+2. For the last image in each orbit the order of subtraction is reversed: I_ last-I_ last-1. The pipeline developed for the chopping mode is much less complex than that proposed for stare mode. A set of adaptive apertures was introduced to better fit the stellar profiles. The algorithm for chopping mode can be divided into two main parts: fully automatic classification and photometry with both intra-pixel sensitivity compensation and optimization. §.§ Automatic classificationThe classification of images in chopping mode is based on the positions of centroids of the positive and negative stellar profiles in a difference image (Fig. <ref>). In particular, we use the difference between the coordinates of the two stellar centroids in the direction in which a satellite is moved in the chopping mode.[Since CTI smears the stellar profiles and hot pixels along the direction of charge transfer (i.e., along CCD columns) during the CCD readout, it was decided that the satellites working in the chopping mode would be moved roughly perpendicularly to this direction, along the x coordinate.] Since the movements are done along the x coordinate, the difference is denoted Δ_x. A difference image is classified as useful if the following three conditions are fulfilled: (i) Δ_ low < Δ_x < Δ_ high. The lower and upper limits for the difference in x coordinate between the centroids of the positive and negative profiles are chosen to be Δ_ low = 0.25 X and Δ_ high = 0.75X, respectively, where X is the length of the subraster along the x coordinate, (ii) the two stellar profiles do not overlap, and (iii) the positive profile does not touch the subraster edge[The negative profile can touch the raster edge because the photometry is performed using the positive profile.]. Otherwise, the image is classified as defective.To obtain the positions of centroids, the difference image is first median-filtered with a 3 × 3 pixel sliding window and then thresholded (above 100 ADU for the positive profile and below -100 ADU for the negative profile, respectively).The purpose of filtering was to reduce occasional flickering of hot pixels. The thresholds are used to obtain the two stellar masks, positive, M_+, and negative, M_-, see Fig. <ref>. For both the positive and negative stellar profile, the centroid is calculated in the median-filtered and thresholded image. The differences, Δ_x, for a sample set of images are presented in Fig. <ref>. In order to check if the positive and negative profiles overlap, the two masks are independently dilated by one pixel in each direction. Then, if there are pixels that belong to both dilated masks, the image is classified as defective. The same is done if any pixel of an extended, positive mask touches the subraster edge. Figure <ref> shows two examples of the classification process for one of the faintest stars observed by BRITEs (V= 5.8 mag) and one of the highest temperatures of the detector (T= 38^∘C) resulting in a strong dark-noise component. Such images would be lost in the stare pipeline because of the large number of hot pixels. Almost all hot pixels cancel out in the resulting difference image. Those which remain occur as a consequence of the RTS phenomenon. On the left-hand side, the process that leads to classifying an image as a useful one is shown. On the right-hand side, we obtain a defective image due to the proximity of the profiles.§.§ Photometry with optimizationHaving classified the images, a high-resolution stellar profile template is created from all useful images. For this purpose, for each difference image only the M_+ mask is retrieved and linearly interpolated onto a four times finer grid, i.e., each real pixel is replaced by a 4 × 4 grid of square sub-pixels. Then, the profiles were co-aligned using the information on centroids and averaged. An example of the final stellar profile template is shown in Fig. <ref>. Photometry for observations in stare mode was made with circular apertures. Given the complex shapes of stellar profiles, such apertures may not be optimal from the point of view of the error budget, since circular apertures frequently include many pixels with very low or even no signal from a star. We therefore decided to obtain photometry in the pipeline for chopping data both with circular and non-circular apertures. The non-circular apertures were defined in the following way. In the first step, the apertures are created from the high-resolution template by simply thresholding it with levels ranging from 25 to 200 ADU with a step equal to 25 ADU plus another threshold at 10 ADU. In this way, nine apertures are created. Furthermore, the largest aperture (obtained with a threshold equal to 10 ADU) is enlarged by a dilation ranging from 2 to 18 sub-pixels with a step of 4 sub-pixels (the operation is performed on a finer grid). This gives five additional apertures. In effect, we define 14 non-circular apertures. An example of their shapes is shown in Fig. <ref>. In addition, a series of 11 circular apertures with radii in the range between 4 and 44 sub-pixels with a step of 4 sub-pixels is used. Having defined the apertures, the final photometry is made with positive profiles in difference images. For each series of difference images, the photometry is done with all 25 apertures, both circular and non-circular. Each aperture is first shifted to the centroid of a positive profile in a difference image. Then, the aperture is binned to the lower, original resolution, and signals from pixels in a difference image confined by the aperture are co-added. Signals from pixels falling partially into aperture are added proportionally to the area of this pixel falling into the aperture. Ultimately, time-series photometry in all 25 apertures is obtained. Similarly to the pipeline for the stare mode of observing, the last step of the process consists of applying corrections for intra-pixel variability in the same way as described in Sect. <ref>. Next, the optimal aperture is chosen by calculating the Q_m parameter; see Eq. (<ref>). Photometry with the smallest Q_m is regarded as optimal. We observed that circular apertures were chosen as optimal only in the rare cases when the stellar profile was round or when observations were strongly affected by tracking problems (stellar profile varied between exposures). The pipeline described in this section was applied to all BRITE chopping data which have been reduced up to now, i.e. ten fields (see Table <ref>). The only exception were test data in the chopping mode obtained in the Perseus field which were reduced with the pipeline for the stare mode of observing (Sect. <ref>). The resulting photometry is now available as Data Releases 3 (DR3), 4 (DR4), and 5 (DR5). A detailed explanation of data releases is available in Appendix <ref>. The pipeline is essentially the same for these three data releases, but outputs include different parameters for decorrelation (Table <ref>). The only change to the pipeline, applied to the data provided in DR5, is a compensation for a magnitude offset between the two positions in chopping during the optimization phase of reduction. This change could have affected the selection of optimal apertures and therefore the final photometry.§ THE QUALITY OF BRITE PHOTOMETRYThe final quality of the BRITE photometry must be assessed from real data. It is, however, good to know what can be expected given the parameters of detectors and observing strategy, and check if the real data meet expectations. It is also important to know if the quality of the photometry meets the mission requirements defined in Paper II. We therefore start with the theoretical expectations which are given in Sect. <ref> followed by the estimation of the uncertainties from real data. §.§ Statistical uncertaintiesA standard formula for the photometric uncertainties associated with CCD photometry of stars usually includes four sources of noise: Poisson (or shot) noise related to the signal from a star (σ_ star), Poisson noise related to sky background (σ_ sky), dark current noise (σ_ dark), and read-out noise (σ_ RON). Assuming that all sources of noise are independent, the formula for the total theoretical noise, σ_ N, can be expressed as:σ_ N^2 = σ_ star^2 + σ_ sky^2 + σ_ dark^2 + σ_ RON^2.The only component in Eq. (<ref>) that can be neglected for BRITE data is σ_ sky since for such bright stars as those observed by BRITE, the background contribution to error budget is negligible. The total signal measured within the apertures for stars observed by BRITEs, S, ranges between 10^4 and 2 × 10^6 electrons (e^-). This corresponds to σ_ star = √(S) = 100 – 1500 e^-, depending on the stellar magnitude. In order to estimate the remaining two sources of noise, it is good to recall the temperature dependence of the detector-related parameters. They were taken from Tables 3, 4, and 5 of Paper II and are given in Table <ref>. The temperature variations are relatively large for BRITE satellites. Taking into account all five satellites and 11 fields observed up to August 2016, the temperatures of the CCD detector, T, ranged between 2^∘C and 41^∘C. Using the data from Table <ref>, we derived the following empirical relations for the dependencies of the dark current generation rate measured on the ground, D_0, and the read-out noise, r_ n, on temperature:D_0(T) = 10^0.0365T + 0.562, r_ n(T) = 12.924 + 0.20213T - 0.010175T^2 + 0.00034524T^3,where T is given in ^∘C. Given the quoted range of temperatures, we get a range of D_0 between 4 and 110 e^- s^-1 per pixel. Most of the BRITE data are taken with exposure time t_ exp = 1 s and the aperture radii, R, range between 2 and 12 pixels. The dark current contribution to Eq. (<ref>) σ_ dark^2 = AD_0t_ exp, where A is the aperture area in pixels. Given the values of A and D_0, σ_ dark ranges between 11 and 186 e^-. Finally, the contribution of the read-out-noise can be estimated from Eq. (<ref>). For the adopted ranges of temperature and aperture radius, σ_ RON = √(Ar_ n^2) ranges between 69 and 485 e^-.Taking t_ exp= 1 s for most BRITE observations, the final formula for theoretical uncertainty expressed in e^- is the following:σ_ N = √(S+A(D_0+r_ n^2))or, if given in magnitudes:σ_ N^ mag≈ 2.5log(1 + √(S+A(D_0+r_ n^2))/S). The quoted numbers show that for the range of temperatures under consideration, σ_ RON > σ_ dark. Theoretically, read-out noise is therefore the dominating source of noise for faint stars (in the BRITE magnitude range) while for bright stars it is the shot noise that principally determines the error budget. Figure <ref> shows the theoretical uncertainty σ_ N^ mag as a function of instrumental magnitude defined as m_ inst = -log(S[^-]) + ≈ -log(S[]) +. The constant 16.0 was chosen to tie very roughly the BRITE red filter instrumental magnitudes to r magnitudes of the SDSS ugriz system using the calibration of <cit.>. The choice was made due to the similarity of the BRITE red filter and SDSS r passbands. The curves for different satellites were calculated for median values of D_0 and r_ n, which are temperature dependent and therefore vary from one satellite to another (see Paper II). They are given in Table <ref> together with mean values of A. All parameters in this table were derived for the observations in stare mode. While σ_ N^ mag represents the theoretical uncertainty of a single measurement obtained with a 1-second BRITE exposure, the right-hand ordinate in Fig. <ref> shows the uncertainty for the average of all measurements within an orbit. The number of measurements per orbit ranges between several and over 160 and depends on the satellite and field[Sometimes a satellite switches between two fields in a single orbit. This usually results in a smaller number of points per orbit in each of the two fields.], but a mean value amounts to about 40 measurements per orbit. Therefore, in Fig. <ref>, σ_ orb = σ_ N^ mag/√(). As one can see, the σ_ orb < 1 mmag theoretical uncertainty can be achieved for stars brighter than 4th magnitude.Equations (<ref>) and (<ref>) are valid for the stare mode of observing. In the chopping mode, the photometry is done with difference images, so that the dark noise and read-out noise contributions in Eq. (<ref>) have to be multiplied by a factor of √(2) and the equation for the chopping mode has the following form:σ_ N^ mag≈ 2.5log(1 + √(S+2A(D_0+r_ n^2))/S). Equation (<ref>) does not include some other possible sources of noice. One of them could be the contribution related to the lack of flat-fielding <cit.>, which cannot be done properly in orbit. Using the pre-flight flat-fields obtained in the UTIAS-SFL[University of Toronto Institute for Aerospace Studies, Space Flight Laboratory.] clean room by a fully integrated instrument aboard UBr, we estimated that the variation of sensitivity over the CCD chip is constant to within ±0.27% rms. There is no known mechanism, which would increase this value in orbit. We assumed that the stellar profile is a truncated Gaussian with a 3-σ radius and covers 100 pixels, a typical value of stellar area in BRITE photometry (Fig. <ref>). The profile was moved randomly on a pixel grid. We obtained that the noise component due to the lack of flat-fielding can be estimated to approximately 0.4 mmag for a single measurement and about 0.07 mmag for orbit-averaged measurements if we adopt that 40 points per orbit are measured. This is well below the Poisson noise, even for the brightest objects. A similar test was performed with several high-resolution profile templates obtained from the real BRITE images. The result was similar; the noise for orbit-averaged measurements did not exceed 0.1 mmag. More importantly, the utilized decorrelations with centroid positions effectively compensate for most of the possible effects due to the lack of flat-fielding, especially in the chopping mode. The final comparison of scatter in Sect. <ref> is performed with decorrelated data. Thus, the 0.1 mmag per orbit flat-field noise should be regarded only as an upper limit. Consequently, this source of noise was recognized as negligible and not taken into account in Eq. (<ref>); see also Fig. <ref>. We also realize that for BRITE observations a significant source of noise is impulsive noise. Nevertheless, we do not include it into the noise equation either because it is of discrete nature (hot pixels), strongly depends on the temperature and time (number of CCD defects increases with time), and the associated RTS phenomenon has a complicated character. Therefore, it is very hard to characterize this source of noise with real numbers. We regard this source of noise as the main factor causing the real scatter to be higher than expected (Sect. <ref>). §.§ Stare vs. chopping mode of observing: the assessment of qualityA simple comparison of Eq. (<ref>) and (<ref>) shows that the stare mode of observing should lead to a smaller noise level and thus better photometry than the chopping mode. As we will show below, usually the opposite is true. This can be explained by the presence of an additional source of noise, not accounted for by Eq. (<ref>). This is an impulsive noise which occurs in the form of hot pixels and depends primarily on temperature: higher temperatures lead to both larger density and higher amplitude of the impulsive noise. In addition, as explained in Sect. 6 of Paper II, the impulsive noise increases with time because of the growing number of CCD defects caused by proton hits. A detailed study of the temporal behaviour of the impulsive noise is out of the scope of this paper. However, in order to justify the decision of switching from the stare to the chopping mode of observing, we show here how photometry obtained for the same data with the two pipelines compares. For this purpose, we ran both pipelines using data obtained in the chopping mode. This is because the pipeline developed to obtain photometry from images made in the stare mode can be applied to the chopping data, while the opposite is not possible. For simplicity, from now on we will use the notions `stare pipeline' and `chopping pipeline' for the two pipelines presented in this paper. In order to make the comparison, we chose six setups[Data from a given satellite pointing are split into parts, called setups, during reductions. See Appendix <ref> for the explanation of setups.] of the UBr satellite runs in the Scorpius I, Cygnus II, Crux/Carina I, and Cygnus/Lyra I fields. The total span of the data is 543 d, between March 20, 2015 and September 13, 2016. For these data, the temperature of the CCD detector ranged between 14.5 and 36.1 ^∘C. The observations were performed with a constant exposure time of 1 s. The data for 16 stars in the Sco I field[We have excluded five stars from these tests, one because it is a close visual double, the remaining because of insufficient data points.], 9 stars in the Cyg II field, 14 stars in the Cru/Car I field, and 5 stars in the Cyg/Lyr I field were reduced independently with the two pipelines described in Sect. <ref> and <ref>. The quality of the photometry is best characterized by its internal scatter which we evaluate using orbit samples. The reason for this is that the scatter in the raw photometry depends on four factors: (i) the sources of noise summarized by Eq. (<ref>) and described in Sect. <ref>, (ii) impulsive noise, not included in Eq. (<ref>), (iii) instrumental effects which occur as correlations of the measured flux with different parameters, see Appendix <ref>, (iv) intrinsic variability. In order to focus the comparison of the two pipelines on the influence of the impulsive noise, the raw photometry needs to be corrected for factors (iii) and (iv) first. The reasons for the occurrence of correlations and the procedures of decorrelation have been described e.g. by <cit.> and <cit.>, but for completeness, we summarize the parameters used in decorrelations in Appendix <ref>. The number of parameters for decorrelation has evolved with the subsequent data releases of BRITE photometry. As we mentioned earlier, presently there are four different data releases available. The format of the output files as well as the explanation of the parameters which are used for decorrelation is briefly described in Appendices <ref> and <ref>, respectively. Each data release provides a different number of parameters for decorrelation.The test data mentioned above, obtained with the two pipelines, were independently corrected for factors (iii) and (iv) leaving a slightly different number of data points. The correction for the intrinsic variability, i.e. factor (iv), was made in two steps. Using Fourier frequency spectra, all periodic signals were identified, fitted by least-squares and subsequently subtracted from the data. Then, all residual non-periodic signals (if present) were removed by means of detrending. Detrending was based on the interpolation between averages calculated in time intervals of 1 d with Akima interpolation <cit.>. In order to make the final photometry directly comparable, we subsequently used only those data points which had times common to the two resulting light curves. The examined parameter was the mean scatter (standard deviation) per orbit. Figure <ref> shows the ratio of median standard deviations, R_σ={σ_ chop,i} / {σ_ stare,i}, where σ_ chop,i and σ_ stare,i are standard deviations in the i-th orbit for data obtained with the stare and chopping pipelines, respectively. As can be seen from the figure, for the first two setups (Sco I, setup 1 and 2) there is a group of nine stars – the same for both setups – that have R_σ= 0.2 – 0.7. For these stars, the chopping pipeline provides much better photometry than the stare pipeline. These stars are located in the areas affected by CTI; see Sect. 6 of Paper II. In order to mitigate the problem, starting from mid-July 2015 all BRITE satellites were set to slower read-out, which effectively solved the CTI problem (Paper II)[It seems that switching to lower read-out solved only one problem related to the CTI, i.e. smearing of images. We have recently discovered that signal for stars located in regions of strong CTI is partly lost due to the traps in the detector. The problem needs to be investigated in detail. At the time of writing it is not known yet how severely BRITE data are affected.]. The remaining four setups of UBr observations were obtained with the slower read-out. For the Sco I setup 4 observations, R_σ= 1.03 ± 0.03, i.e. is close to unity showing no advantage of chopping pipeline.For the next three setups the mean R_σ gradually decreases to 0.92 ± 0.04 for Cyg II, 0.75 ± 0.04 for Cru/Car I, and 0.62 ± 0.04 for Cyg/Lyr I, see the dashed lines in Fig. <ref>. This clearly shows that the longer time elapsed since the launch (the setups are presented chronologically), the better the results are obtained with the chopping pipeline in comparison with the stare pipeline. The decrease of R_σ with time is related to the increasing number of hot pixels in BRITE detectors. Switching all BRITE satellites to the chopping mode of observing is therefore fully justified and may even allow to extend the mission beyond the point where standard photometry would not be possible.We also expected that the advantage of using the chopping pipeline will be better enhanced at higher CCD temperatures because of the larger number of hot pixels. To make an appropriate comparison, we used the same data as in the previous test, but confined to the four setups not affected by CTI. For each star, the standard deviation as a function of CCD temperature was evaluated and then normalized to the scatter at 25^∘C (using data with T= 25 ± 1^∘C). The ratios of the resulting normalized standard deviations are shown in Fig. <ref>. Surprisingly, the mean values of these ratios calculated from the dependencies for all 44 stars show practically no dependence on temperature (black squares in Fig. <ref>). There are, however, strong differences between different stars. The ratio of standard deviations can either drop with temperature as for HD 142669 = ρ Sco (green dots) or HD 105937 = ρ Cen (red crosses) or increase as for HD 194093 = γ Cyg (blue circles).This shows that the benefits of the chopping pipeline compared to the stare pipeline do not neccessarily increase for higher temperatures. This can be explained as follows. First, the frequency of RTS increases with higher temperature; see Paper II and <cit.> for more details. Therefore, the probability that a hot pixel shows the same intensity in two successive images decreases. This introduces unwanted impulsive noise appearing in difference images, which is not corrected by the current chopping pipeline. Second, the dark current noise and readout noise are larger by a factor of √(2) in difference images, cf. Eqs. (<ref>) and (<ref>). This difference may also become more prominent at higher temperatures. §.§ Comparison between theoretical expectations and real performanceIt is also interesting to see how the real scatter in BRITE photometry compares with the calculations presented in Sect. <ref>. A comparison has been done for the same test data for 44 stars as those presented in Sect. <ref>. The results are shown in Fig. <ref>. One can see that for both observation modes the scatter of the photometry is larger than predicted by theory. For most stars the ratio σ_ orb/σ_ orb^ theo ranges between 1 and 4 with a small increase towards brighter stars. Stars brighter than ∼1.5 mag are affected by known nonlinearities (see Sect. 3.2.3 in Paper II), which combined with variable blurring and intra-pixel sensitivity variations results in a noise floor of approximately 0.7 – 1.0 mmag for the brightest objects. Therefore, the scatter in the photometry of such stars is much larger than predicted.The main source of the degradation of BRITE photometry in comparison with theoretical expectations is the presence of high impulsive noise which has a non-stationary character. This is why σ_ orb/σ_ orb^ theo values (lower panel of Fig. <ref>) are on average higher for stars in Cru/Car field in comparison with Sco field observed about a year earlier. The other important factor that contributes to the increase of the real scatter is a non-perfect stability of the satellites resulting in image blurring. Decorrelations applied to the data cannot fully account for these effects. With consecutive data releases (Appendix <ref>) more parameters that can be used for decorrelations are provided. This means that we can believe that the later the data release the more effectively instrumental effects can be removed from the photometry. Below, we summarize in detail the possible factors that can lead to the degradation of the photometry in both observing modes. The items are marked with `[S]' or/and `[Ch]' indicating whether a given item is applicable to stare or chopping mode of observing, respectively.∙[S] Interpolations introduce uncertainties in the estimation of the intensity of the replaced pixels. Their quality is decreased by the changes of the temperature of CCD, which induce subtle variations of stellar profiles. ∙[S] An optimized selection of the hot pixels in master dark frames is always a trade-off. Too many pixels classified as hot would decrease the photometric quality, but leaving too many hot pixels would increase RTS-related noise. ∙[S & Ch] The precision of the centroid position, based on the centre of gravity, suffers from dark current noise and interpolation errors. ∙[S] The shapes of stellar profiles are usually highly non-circular (see Fig. <ref>) and change due to the changes of the temperature of the optics. The optimal circular aperture does not always cover the whole stellar profile. In consequence, small differences in the position of the centroid can lead to a non-negligible change of the flux measured in the aperture. ∙[S & Ch] To some extent, all images are blurred. Since we use a constant aperture, a different part of the total flux is measured, depending on the amount of blurring. This effect is largely compensated by decorrelation with blurring parameters, but some excess of scatter due to this effect can be expected. ∙[Ch] The background may change between consecutive difference images, especially when scattered light from either the Moon or Earth changes rapidly in time. Starting from Data Release 5 this effect can be compensated for by a decorrelation with respect to parameter APER0; see Appendix <ref>. ∙[Ch] The RTS-related noise increases with temperature because transitions between many meta-stable states happen more frequently, reducing the effectiveness of impulsive noise cancellation in difference images. § CONCLUSIONS AND FUTURE WORKThe main goal of BRITE-Constellation is to provide precise photometry of bright stars. Unfortunately, radiation-induced defects in CCDs complicated the mission and made the image processing a difficult task. These defects degraded the quality of the images introducing impulsive noise and possibly also affected the performance of star trackers, which are crucial for the stability of pointing of the spacecraft. The chopping mode of observing was introduced at the beginning of 2015 to mitigate the effects of the increasing CCD radiation damage and the influence of CTI on the photometry (see Paper II, Sect. 6.2.3). The light curves for stars affected by CTI provided by the normal pipeline were contaminated by distinctive artefacts and therefore were scientifically almost useless. A switch to the chopping mode of observing improved the situation significantly, which we have discussed in Sect. <ref> and have demonstrated using six setups of the UBr data as examples (Fig. <ref>). After changing the CCD read-out rate in July 2015, the CTI artefacts practically disappeared. The comparison of results from both pipelines shows the advantage of the chopping pipeline for almost all observations, particularly for those affected by CTI. In addition, it was shown that the advantage of using the chopping pipeline has become greater with time. This is strictly related to the increase of a number of hot pixels generated by radiation in orbit with time. The chopping mode offers much better perspectives for improvements. One possibility is a reduction of the impulsive noise in difference images. There are many techniques that address the problem <cit.> and we will search for one that will be optimal for BRITE data. Other planned changes to the photometric pipeline are discussed below. In the context of the unavoidable increase of radiation damage of CCDs with time in all BRITE satellites, it is clear that the chopping mode will allow to extend considerably the length of the mission. The development plan of the reduction pipelines is as follows: ∙A new pipeline for images obtained in the stare mode of observing will be developed. The images will be re-reduced using thresholded apertures obtained in a similar way as is presently done in the chopping pipeline (see Sect. <ref>). In addition, the same parameters for decorrelation which are now provided with DR5, but are lacking in previous data releases, will be provided. All this should improve the quality of the photometry obtained from these data obtained from the first two years of observations and allow for a better correction for instrumental effects.∙The positions of stars in subrasters in a given frame are not independent because a satellite is a rigid body. We plan to take advantage of this fact to improve precision of the estimation of the positions of stellar centroids.∙Profile fitting photometry will be investigated. This will include temperature dependent variability of intrinsic stellar profiles and the usage of a model of satellite movement during exposures. This movement is the primary cause of smearing of stellar images. Again, the rigidity of a satellite will be used to account for the effect of smearing.∙Several impulsive noise reduction techniques for initial hot pixel replacement will be tested. Currently, a simple median filter algorithm is employed and the hot pixels are assumed to be stable between two consecutive exposures, which is not always true. At present, the BRITE satellites provide photometric data in two passbands at the 1 mmag per orbit level. At the time of writing, the data for over 400 stars have been acquired, often during multiple runs. For about 90% of these stars images were reduced and extracted photometric data sent to the users.At the same time, decorrelation techniques have also been developed to account for all identified instrumental effects. The detection threshold for periodic variability in the best datasets reaches 0.15 – 0.2 mmag. This means that BRITEs provide the best photometry ever made for many of the brightest stars in the sky. The additional advantage of BRITE photometry is good frequency resolution and small aliasing in the most interesting frequency region (below 10 d^-1) for stars having observations obtained by two or more satellites. Good frequency resolution is a consequence of the long observing runs lasting 5 – 6 months and the fact that some fields are re-observed (see Fig. <ref> and Table <ref>). All this shows that precise photometry from low-Earth orbits with the use of nano-satellites, despite the difficulties and the occurrence of instrumental effects, is a scientifically viable undertaking.This reasearch was supported by the following Polish National Science Centre grants: 2016/21/D/ ST9/00656 (A. Popowicz), 2016/21/B/ST9/01126 (A. Pigulski), 2015/17/N/ST7/03720 (K. Bernacki), and 2015/18/A/ST9/00578 (E. Zocłońska, G. Handler). A. F. J. Moffat, S. M. Rucinski, and G. A. Wade are grateful for financial aid from NSERC (Canada). In addition, A. F. J. Moffat acknowledges support from FQRNT (Quebec). O. Koudelka, R. Kuschnig, and W. Weiss acknowledge support by the Austrian Research Agency (FFG) as part of the Austrian Space Application Programme (ASAP). K. Zwintz acknowledges support by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF, project V431-NBL). The Polish contribution to the BRITE project is funded by the NCN PMN grant 2011/01/M/ST9/05914. We thank Radek Smolec for his comments made upon reading the manuscript and Monica Chaumont and Daniel Kekez (UTIAS-SFL) for making test images from the integrated UBr satellite available. We thank anonymous referee for the useful comments.aa § DATA RELEASES AND SETUPSFollowing one year of proprietary period, and a possible six month extension for justified cases, BRITE data are made public through the BRITE Public Data Archive[https://brite.camk.edu.pl/pub/index.html] (BPDA). Presently, the data are available as four data releases, Data Release 2 (DR2[DR1 (Data Release 1) was later superseded by DR2 in which additional flag indicating if the aperture extends over the subraster edge, has been added. Therefore, DR1 data are no longer in use.]) to Data Release 5 (DR5); see Table <ref>. The information on 15 released fields up to now is summarized in Table <ref>. BRITE-Constellation data are made available as text files with several columns preceded by a header in which some general information on the satellite, data length, data format, etc, is included. The data format for each data release is shown in Table <ref>; the meaning of the parameters included in text files is explained in the table's footnote and Appendix <ref>. §.§ SetupsData obtained with a given BRITE satellite during its run in a given field are split into parts, so-called setups, during the reduction process. The reason for this splitting is usually a change in observing conditions. When observations of a new field start, stars are positioned only in preliminary subrasters. After a short time, typically one or two days since the onset of observations, the subrasters need to be readjusted because some stars are not properly placed (e.g. too close to a subraster edge). The other reason for an adjustment could be a change of exposure time. Consequently, the first setup of a given run is usually short. The subsequent changes to the observing procedure also result in new setups. The changes can be the following: ∙ Changes to the list of observed stars. Sometimes, especially if there were data transmission issues, the list of observed stars was shortened to allow complete transmission of the data. Next, some stars were dropped because of their faintness. On the other hand, it might happen that the list of observed stars was extended to include more objects, usually faint, because both data transmission and the overall performance of a satellite warranted good photometry for them. This kind of change usually did not affect stars that were observed throughout the run and therefore for such stars the two setups, before and after the change, can be merged without applying offsets. ∙ Long term offsets in `absolute' pointing drove stars close to the edge of the subrasters. The reason for this small loss of collimation between tracker and the main camera is not known. This required adjustments of positions of all subrasters. ∙ Due to the limitations of the hardware used for reduction, images from the longest setups were split in two parts. This means that for these two parts there was no change in the observing conditions, but because the data were reduced independently, the two parts may show slightly different dependencies on parameters as a consequence of a possible difference in the shape and size of the optimal apertures. In such situations, the two parts have the same number of the setup, but are marked as `part1', and `part2'. § PARAMETERS FOR DECORRELATIONThe raw BRITE photometry is subject to many instrumental effects. These effects occur as a consequence of: (i) lack of flat-fielding, (ii) lack of cooling of the detector, (iii) temperature effects on telescope optics, (iv) non-perfect stability of a satellite during exposure, (v) other effects. Fortunately, most of these effects can be to a large extent mitigated by decorrelating raw magnitudes with a number of parameters which are provided together with the raw fluxes. Examples of decorrelation applicable for BRITE data were presented e.g. by <cit.> and <cit.>. In general, the presented methods iteratively determine the dependencies between decorrelation parameters and raw magnitudes. If large-amplitude intrinsic variability is present, the variability is modelled by a superposition of sinusoidal terms and subtracted. Then, decorrelations are performed using residuals from the fit, but applied to the original light curve. The fitting of the model is a part of the iterative decorrelation sequence. For more details the interested reader is kindly referred to the original papers.Decorrelations are now regarded as the most important preliminary step in the analysis of BRITE data. In most cases only after this step do the data become scientifically useful. Depending on the data release, different sets of parameters are provided (Table <ref>). They are described in the following.The XCEN, YCEN and CCDT parameters are provided in all data releases. The XCEN and YCEN are, respectively, the x and y coordinates of the centroid of the stellar profile in the coordinate system of a subraster. The way in which they were obtained is described in detail in Sects <ref> and <ref>. CCDT is the average temperature of the CCD, measured by four temperature sensors; see Fig. 3 in Paper II.The FLAG parameter, provided first in DR2, indicates if the optimal aperture is fully rendered within a subraster (FLAG = 1) or not (FLAG = 0). This parameter was abandoned in DR3 and DR4 because images which would have FLAG = 0 were eliminated at the classification step in the reduction of chopping data. This was changed back in the chopping pipeline for DR5. The images with the aperture extending beyond the subraster edge are not used in the calculation of the optimal aperture, but the photometry is done and it is the decision of the user whether to reject flagged data points or not. For this purpose, the data points obtained from such images are flagged with FLAG = 0.It was recognized very early that due to the imperfect stability of the satellites, BRITE images are slightly blurred. The amount of blurring has a direct consequence on the flux measured through a constant aperture. Therefore, starting from DR3, i.e. the first data release for chopping data, the first parameter which is a good measure of blurring, PSFC, was introduced. Later on, since DR4, another blurring parameter, PSFC2, was added. These two parameters are strongly correlated but using both in decorrelations allows for a better correction for the effects of blurring. The first parameter, PSFC (in DR3) or PSFC1 (in DR4 and DR5), is defined as follows: PSFC =PSFC1 = ∑_ p∈ M_+(I_ p/I_ total)^2, I_ total = ∑_ p∈ M_+I_ p,where I_ p is the signal in pixel p belonging to mask M_+ (see Sect. <ref> for the explanation of the concept of masks),and I_ total is the sum of signals from all pixels within the mask M_+. Since the signal in each pixel is normalized by I_ total, PSFC is independent of variations of stellar flux (variability of a star). When the stability of a satellite is good, the relative intensities of pixels reach their maximum values, and therefore their squares are maximized. When blurring occurs, the charge is spread more evenly among pixels and the value of PSFC is reduced. The idea of this parameter is based on the image energy (the sum of the squared intensities over all pixels), which is largest when the image is sharp and lower when the flux is spread over a larger CCD area.The other blurring parameter, PSFC2, introduced with DR4, is defined asPSFC2 =I_R^2∑_k=1^4 ∑_i=1^L I_i J_i,k, I_R = ∑_ p∈ R I_p,where I_R is the sum of signals from all pixels in a subraster R, J_i,k is the signal in the i-th pixel in the image that has been shifted by one pixel in one of the four (k) directions: up, down, left and right. The summation is performed over L pixels, which the image I has in common with its shifted copy J. This smearing parameter is based on the correlation of the image with its shifted copies. The correlation increases when smearing is larger. Thus, sharp images will have low values of PSFC2. The PSFC1 and PSFC2 parameters are the same as parameters A and B described by <cit.> in Appendix A of their paper.Starting from DR3, a parameter related to the RTS phenomenon was added to the data files. The RTSC parameter is the indicator of a possible RTS in a subraster column. The dark current RTS occurs rarely in a single pixel within the optimal aperture and if this is the case, we get a significant outlier in the final light curve, which can be easily removed. An RTS that affects a whole column results in a relatively smaller photometric offset, due to the much lower dark charge in columns. If the RTS in a column occurs, the difference image contains dimmer or brighter vertical line (depending on the direction of the transition). For example, in the upper right image in Fig. <ref> a slightly a dimmer line runs through the negative stellar profile. As such a disturbance may not be properly identified and rejected, the RTSC parameter was introduced to deal with it. If a frame is not affected by a column RTS, the RTSC parameter is equal to several ADUs. The RTS phenomena change it to several tens of ADUs. In order to calculate the RTSC parameter, the median intensity is computed in each column of a difference image, excluding pixels identified as belonging to stellar profile. In DR3 the RTSC was defined as the maximum of the absolute values of these medians. Since DR4 the sign is preserved, so that RTSC can be either positive or negative. In order to indicate this small change in the definition of RTSC, the RTSC in DR3 is marked as RTSC(+) in Table <ref>, denoting that only non-negative values are possible, while for DR4 and DR5, it is marked as RTSC(+/-), showing that both negative and positive values are allowed.Finally, with DR5 we have introduced two new parameters, RTSP, and APER0 (see Table <ref>). RTSP is related to the RTS phenomenon too, but this time, the signals in all pixels within the optimal aperture are compared. The analysis for a given i-th image is made using two neighbouring raw images, (i-1)-th and (i+1)-th. Since in these two images the star is located at the opposite side of a subraster compared to the i-th image, it is easy to find the differences in intensity between the pixels in the i-1-th and i+1-th images that are confined in the stellar profile in the i-th image. The RTSP is defined as the maximum of the absolute values of these differences. Similarly to RTSC, its sign is preserved, so that RTSC can be either positive or negative. The APER0 parameter is the median value of all pixels located outside both apertures in a difference image. It was introduced to account for small differences in background which may occur between two positions in the chopping mode of observing. The differences can be introduced by the bright Moon and/or Earth if not far from the observed field. | http://arxiv.org/abs/1705.09712v1 | {
"authors": [
"Adam Popowicz",
"Andrzej Pigulski",
"Krzysztof Bernacki",
"Rainer Kuschnig",
"Herbert Pablo",
"Tahina Ramiaramanantsoa",
"Elzbieta Zoclonska",
"Dietrich Baade",
"Gerald Handler",
"Anthony F. Moffat",
"Gregg A. Wade",
"Carolie Neiner",
"Slavek M. Rucinski",
"Werner W. Weiss",
"Otto Koudelka",
"Piotr Orleanski",
"Alexander Schwarzenberg-Czerny",
"Konstanze Zwintz"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170526204824",
"title": "BRITE-Constellation: Data processing and photometry"
} |
1Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA Contact: [email protected] We present analysis of NuSTAR X-ray observations in the 3–79 keV energy band of the Seyfert 2 galaxy NGC 4388, taken in 2013. The broadband sensitivity of , covering the Fe Kα line and Compton reflection hump, enables tight constraints to be placed on reflection features in AGN X-ray spectra, thereby providing insight into the geometry of the circumnuclear material. In this observation, we found the X-ray spectrum of NGC 4388 to be well described by a moderately absorbed power law with non-relativistic reflection. We fit the spectrum with phenomenological reflection models and a physical torus model, and find the source to be absorbed by Compton-thin material (N_H = (6.5±0.8)×10^23 cm^-2) with a very weak Compton reflection hump (R < 0.09) and an exceptionally large Fe Kα line (EW = 368^+56_-53 eV) for a source with weak or no reflection. Calculations using a thin-shell approximation for the expected Fe Kα EW indicate that an Fe Kα line originating from Compton-thin material presents a possible explanation. § INTRODUCTIONIt is well established that Active Galactic Nuclei (AGN) are powered by accreting supermassive black holes (SMBH), with a significant fraction of AGN prone to moderate to Compton-thick obscuration due to circumnuclear gas and dust. Under the unified model of AGN, much of the obscuration is produced by a uniform, dusty torus, with the observed differences between Seyfert 1s and Seyfert 2s simply arising from different viewing angles of the torus <cit.>. Several torus spectral models have been developed which self-consistently model toroidal reprocessing of the hard X-ray spectrum, such as the XSPEC models<cit.> and<cit.>. The picture of a uniform dusty torus is likely an oversimplification of AGN geometry, with a more realistic description being a clumpy torus composed of many optically thick clouds <cit.>. Cloud obscuration along the line of sight of the observer smears the rigid border between type 1 and type 2 and thus can lead to misclassification of some objects <cit.>. Evidence for a clumpy or filamentary dust structure has been found in mid-infrared observations of the Circinus AGN using the Very Large Telescope Interferometer (VLTI) <cit.>. Such clumpy torus models can also better explain transitions between Sy2 and Sy1 type spectra observed in a number of sources <cit.>. Analysis of the X-ray spectra of Seyfert 2 AGN can provide detailed insight into the geometry of the circumnuclear material. The absorption of soft X-ray photons allows us to measure the column density of the material in the line of sight () while features such as the Compton reflection hump (CRH) near 20-30 keV and Fe Kα line at 6.4 keV produced from reprocessing of the continuum X-ray emission, indicate the global amount of Compton-thick and Compton-thin material, respectively <cit.>. NGC 4388 is one of the brightest Seyfert 2s at hard X-ray energies <cit.> with an intrinsic 2–10 keV luminosity L_2-10 of ∼4.5×10^42 erg s^-1, measured by <cit.>. The galaxy is viewed at an inclination angle i ≃ 72 and redshift z =0.0084 <cit.>, with a SMBH mass of M_BH = (8.5±0.2)×10^6 M_⊙ determined from water maser measurements <cit.>. Several broad-band observations of this source have been performed with X-ray missions such as BeppoSAX <cit.>, INTEGRAL <cit.> and <cit.>. INTEGRAL observations conducted from 2003 to 2009 revealed strong variations in hard X-ray emission in the 20–60 keV band on 3–6 month timescales, while RXTE observations have shown rapid (hour-timescale) variability in the column density of the absorbing medium <cit.>. Past observations have shown the source to be moderately absorbed, with column densities in the range 10^23 < <10^24 cm^-2, thus leading to its classification as a Compton-thin Seyfert 2 galaxy <cit.>. Broadband X-ray studies with BeppoSAX and INTEGRAL have been unable to constrain the reflection component in NGC 4388 due to the low sensitivity at energies above 10 keV <cit.>. The observatory, with its hard X-ray focusing optics, has enabled sensitive broadband observations to be performed and detailed modelling of the CRH and Fe K bandpass, thereby placing tight constraints on reflection features <cit.>. <cit.> studied X-ray absorption in NGC 4388 and other water maser AGN by performing torus model fits to the data, however these authors did not perform detailed modeling of the reflection and Fe line features.In this paper we present an analysis of the hard X-ray spectrum of NGC 4388 from observations made in 2013 in the 3–79 keV energy range. We investigate both physically motivated torus models and phenomenological ones. We compare our results with previous INTEGRAL, Swift, and RXTE measurements. In this work, all uncertainties were calculated at the 90% confidence level and standard values of the cosmological parameters (h_0 = 0.7, Ω_Λ = 0.7,Ω_m = 0.3) were used to calculate distances. § OBSERVATION AND DATA REDUCTION The NuSTAR satellite observed NGC 4388 twice on 2013 December 27 with a total exposure time of 22.8 ks. Reduction of raw event data from both modules, FPMA and FPMB <cit.> was performed using the NuSTAR Data Analysis Software (NuSTARDAS, version 1.2.1), distributed by the NASA High Energy Astrophysics Archive Research Center (HEASARC) within the HEASOFT package, version 6.16. We took instrumental responses from the NuSTAR calibration database (CALDB), version 20150316. Raw event data were cleaned and filtered for South Atlantic Anomaly (SAA) passages using themodule. We then extracted source energy spectra from the calibrated and cleaned event files using themodule. Detailed information on these data reduction procedures can be found in the NuSTAR Data Analysis Software Guide <cit.>. An extraction radius of 60 was used for both the source and background regions. We extracted the background spectrum from source-free regions of the image, and away from the outer edges of the field of view, which have systematically higher background. The spectral files were rebinned using the HEASOFT taskto give a minimum of 20 photon counts per bin. We did not include NuSTAR data below 3 keV or above 79 keV. In all our modeling we include a cross-correlation constant of ∼ 1 between FPMA and FPMB to account for slight differences in calibration <cit.>. § SPECTRAL MODELINGWe performed spectral modeling of NGC 4388 using XSPEC v12.8.2 <cit.>. We used cross sections from <cit.> and solar abundances from <cit.>. Fitting a simple power law model showed evidence of a strong Fe Kα line and soft absorption, but relatively little Compton reflection (see figure 1 (b)). Adding both an absorbed component with partial covering and a redshifted iron line (XSPEC model ) significantly improved the fit (χ^2/dof = 578/535) and showed no evidence of a reflection feature, as seen in the residuals in figure 1 (c). We then applied both a phenomenological slab reflection model and a physically-motivated torus model to the data, as described below: * XSPEC model : Models an absorbed power law with a Gaussian Fe Kα line and a cold Compton reflection component.models absorption with a variable covering factor.<cit.> models reflection off a slab of infinite extent and optical depth covering between 0 and 2π Sr of the sky relative to the illuminating source, corresponding to R between 0 and 1. * XSPEC model<cit.>: Obscuring material is arranged uniformly in a toroidal structure around the central AGN, with a fixed opening angle of 60. Provides self-consistent modeling of the scattered power law, Fe Kα line, and Compton reflection features. Phenomenological models have some limitations in that the reflection component, e.g. modelled via , is produced from a slab of infinite extent and optical depth.provides a more realistic description, by modeling the obscuring material in a toroidal geometry with a finite optical depth. However,is limited in that it assumes a uniform density torus, with a sharp change in the line of sight absorption at the edges of the torus. For themodel, we set iron and light element abundances to solar, the cutoff energy E_c was fixed to 1000 keV and the inclination angle of the plane of reflecting material was fixed to the inclination of the galaxy (72). Inclination values were found to be unconstrained when left free, thus justifying using a fixed value for this parameter. The normalisation and photon index of the incident power law were tied to those of the reflected continuum power law. For themodel, we performed fits with the torus inclination fixed at 72, matching the observed inclination of the galaxy. Column densities, photon indices and inclinations of the scattered continuum and emission line were tied to those of the zeroth-order continuum. The normalisation of the scattered continuum was tied to the zeroth-order continuum whilst the line normalisation was left free. We found that coupling the continuum, reflection and Fe line regions in this manner provided the best fit to the data. Table 1 shows the best fit parameter values for both the phenomenologicalmodel and physically motivatedmodel. Figure 1 shows the corresponding data and residuals for themodel fit; the unfolded energy spectrum is shown with model components and residuals for themodel in figure 2.If we assume reflection off a centrally illuminated, Compton-thick disk with solar abundances, then the expected EW of the Fe line with respect to the flux of the CRH is <10 eV, as detailed in <cit.>.We found NGC 4388 to be Compton-thin with a very weak CRH (R < 0.09), yet it exhibits a large Fe Kα line EW. To explain such a high Fe line EW through a large Fe abundance would require unphysically high supersolar iron abundance values, thus we rule out such a scenario. The best fit Fe Kα line width is consistent with zero (upper limit of < 70 eV), indicating an absence of line broadening and implying that the fluorescent material is located far from the central source. We found a covering factor of 0.9 for the absorbing material, pointing to a small opening angle of the torus. Values of Γ, , F_2-10 and EW_Fe are consistent between the two models applied, with both models producing equally good fits to the data, indicating that physically motivated torus models provide a statistically equivalent description of the hard X-ray spectra compared to phenomenological slab models, and are suitable for characterising this Compton-thin source. As a check that our spectral modeling succesfully recovers intrinsic AGN parameters, we calculated the Eddington ratio λ_Edd = L_Bol/L_Edd using a bolometric correction of 10 to the intrinsic 2–10 keV luminosity. Given a black hole mass of 8.5×10^6 M_⊙, L_Edd≃ 1.1×10^45 erg s^-1. We found the intrinsic 2–10 keV luminosity to be ∼ 8.9×10^41 erg s^-1. The expected value of Γ can be calculated from the known relationship between Γ and λ_Edd, as detailed in <cit.>, and was found to be ∼ 1.60, which is in good agreement with the best fit Γ values from our spectral modeling. § DISCUSSION Overall, the NuSTAR data shows that the spectrum of NGC 4388 is well characterised by both phenomenological models and physically motivated torus models. We found the source to be moderately obscured by Compton-thin material, with a very weak reflection component and a strong Fe line. The best fit results for Γ and were in fairly good agreement with past observations by INTEGRAL <cit.> and Swift <cit.>. The constraint on the reflection component found from themodel of R < 0.09 is consistent with that obtained from archival RXTE data <cit.> and combined long-term Swift & INTEGRAL observations <cit.>.The lack of Compton reflection in NGC 4388 is not unusual. <cit.> found that 15% of Seyfert-like AGN in the RXTE archive show no significant contribution from a CRH (R < 0.1), however three of the five sources (Cen A, Cyg A, and 3C 111) are actually radio loud AGN and all show weak Fe lines as well.A handful of other Compton-thin Seyfert 2's in the sample had large Fe lines with little contribution from Compton reflection, but lacked the statistics to place good constraints on R. One of these, NGC 2110, was observed by Suzaku and analyzed by <cit.>, whichconfirmed the lack of reflection, but found much lower values for the Fe line EW as well (∼ 50 eV).One possibility that is investigated by <cit.> is a scenario in which there is a large global amount of material that is not Compton-thick. To calculate the expected amount of Fe emission from Compton-thin material, we can use a thin-shell approximation and the following equation based on <cit.>: EW_ Kα = f_ c ωf_ KαA_ abundN_H ∫_E_ K-edge^∞ P(E) σ_ ph(E)dE/P(E_ line) with f_ c the covering fraction of the absorber, ω the fluorescent yield, f_ Kα the fraction of photons contributing to the Fe line production, A_ abund the Fe abundance relative to hydrogen, P(E) the continuum power law, and σ_ ph(E) the K-shell absorption cross section as a function of energy. Assuming solar abundances for A_ abund and using values for the fluorescent yield and cross section from <cit.>, we can calculate the contribution to the Fe line EW from a uniform shell of material with the column density given in Table 1 for the line of sight absorption, that is centred on a continuum emission source which is assumed to be an isotropic, point-source emitter.We find a maximum EW of 548 eV assuming a 100% sky-covering fraction, which is consistent with our measured EW, indicating that an Fe line originating from Compton-thin material could be a plausible scenario. Previous studies of NGC 4388 <cit.> have also suggested that the absence of a CRH in the hard X-ray region may point to non-isotropic emission of radiation that fails to illuminate the disk. Another possibility is a poorly illuminated torus with a very large opening angle (that is, a very flat torus), or a more complex geometry of the circumnuclear material <cit.>. It is worth noting that our best-fitmodel physically corresponds to a Compton-thin torus intersecting the line of sight and also providing the necessary Fe line flux.While our analysis of the observation of NGC 4388 found the CRH to be absent in this source, the detection of a CRH has been reported in a past observation of NGC 4388 with . <cit.> analyzed a 100 ks observation of NGC 4388 from 2005 December, and for the first time were able to detect the CRH (R = 1.40^+0.29_-0.36). They postulated that since the source had decreased in luminosity compared to previous observations reported in <cit.>, they could be seeing a delay in the decrease of the CRH due to a light echo in Compton-thin material that is light years away from the central source. The Fe line was resolved in the XIS data with a width of 45^+5_-6 eV from simultaneous broadband fits of XIS and HXD data, corresponding to a radial distance of ∼ 0.01 pc. However, the paper concluded that the Fe line broadening was attributed to the Compton shoulder rather than intrinsic broadening from material close to the central source. Furthermore, the observation revealed short term flux variablility (half-day timescales) wherein the reflection fraction R changes but the reflection flux and Fe line flux did not vary significantly. Thus the fluorescent iron line emission appears to be decoupled from direct emission and likely originates in distant reflecting material located several light years from the continuum source, consistent with past INTEGRAL observations <cit.>. The 2 - 10 keV flux observed by was found to be 2.0×10^-11 erg cm ^-2 s^-1, which is the same order of magnitude as that observed with (∼8.0×10^-12 erg cm^-2 s^-1), indicating it is likely that the source remained in a relatively low flux state over long timescales and thus the CRH disappeared from the data in the 7 years between the and observations. § SUMMARY Our spectroscopic analysis of the moderately obscured Seyfert 2 galaxy NGC 4388 from observations revealed the hard X-ray spectrum to be well represented by both phenomenological reflection models and physically motivated torus models. One possible explanation for the exceptionally large EW of the Fe line and weak CRH in this source is the presence of a large global amount of material which is capable of producing the observed EW but is not sufficiently thick to produce a distinguishable CRH feature. The detection of a CRH in the X-ray spectrum of NGC 4388 from observations performed in 2005 can be explained by a light echo in Compton-thin material located light years from the central source, resulting from the source being in a low flux state several years prior to the observation. However, further multi-epoch hard X-ray monitoring will be needed to conclude whether this is a likely explanation. We have made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the NuSTAR Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA). M. Baloković acknowledges support from NASA Headquarters under the NASA Earth and Space Science Fellowship Program, grant NNX14AQ07H. Facility: apj The NuSTAR View of the Seyfert 2 Galaxy NGC 4388 N. Kamraj1, E. Rivers1, F. A. Harrison1, M. Brightman1, M. Baloković1 December 30, 2023 ========================================================================= | http://arxiv.org/abs/1705.09260v1 | {
"authors": [
"N. Kamraj",
"E. Rivers",
"F. A. Harrison",
"M. Brightman",
"M. Balokovic"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170525170119",
"title": "The NuSTAR View of the Seyfert 2 Galaxy NGC 4388"
} |
=1= 24truecm = 16truecm = -1.3truecm = -2truecmtheoremTheorem propositionProposition StatementStatement LemmaLemma proofProofCorCorollary[theorem] remarkRemark #1to0ptto 0pt1em2cm10000#1to8pt | http://arxiv.org/abs/1705.09633v3 | {
"authors": [
"Zi-Yu Tang",
"Yen Chin Ong",
"Bin Wang"
],
"categories": [
"gr-qc",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170526160924",
"title": "Lux in obscuro II: Photon Orbits of Extremal AdS Black Holes Revisited"
} |
firstpage–lastpage Hairy black-hole solutions in generalized Proca theoriesLavinia Heisenberg^1,Ryotaro Kase^2, Masato Minamitsuji^3, andShinji Tsujikawa^2 December 30, 2023 ========================================================================================= Nuclear starbursts and AGN activity are the main heating processes in luminous infrared galaxies (LIRGs) and their relationship is fundamental to understand galaxy evolution. In this paper,we study the star-formation and AGN activity of a sample of 11 local LIRGs imaged with subarcsecond angular resolution at radio (8.4 GHz) and near-infrared (2.2 μm) wavelengths.This allows us to characterize the central kpc of these galaxies with a spatial resolution of ≃100 pc. In general, we find a good spatial correlation between the radio and the near-IR emission, although radio emission tends to be more concentrated in the nuclear regions. Additionally, we use an MCMC code to model their multi-wavelength spectral energy distribution (SED) using template libraries of starburst, AGN and spheroidal/cirrus models, determining the luminosity contribution of each component, and finding that all sources in our sample are starburst-dominated, except for NGC 6926 with an AGN contribution of ≃64%. Our sources show high star formation rates (40 to 167 M_⊙ yr^-1), supernova rates (0.4 to 2.0 SN yr^-1), and similar starburst ages (13 to 29 Myr), except for the young starburst (9 Myr) in NGC 6926. A comparison of our derived star-forming parameters with estimates obtained from different IR and radio tracers shows an overall consistency among the different star formation tracers. AGN tracers based on mid-IR, high-ionization line ratios also show an overall agreement with our SED model fit estimates for the AGN. Finally, weuse our wide-band VLA observations to determine pixel-by-pixel radio spectral indices for all galaxies in our sample, finding a typical median value (α≃-0.8) for synchrotron-powered LIRGs. galaxies: interactions – galaxies: nuclei – galaxies: starburst – infrared: galaxies – radio continuum: galaxies§ INTRODUCTION Luminous Infrared Galaxies (LIRGs) are defined as those galaxies with an infrared luminosity L_IR[8–1000 μm] >10^11L_⊙. The majority of these sources, discovered in the early 1980s by the IRAS satellite, are actually galaxies undergoing a merging process.While most LIRGs are dominated by violent episodes of nuclear star formation <cit.>, many of them also contain an active galactic nucleus (AGN). The existence and link between these two processes <cit.> make LIRGs ideal laboratories in which to study the AGN-starburst connection.The parameters describing the star formation in a galaxy can be obtained by means of a number of indirect tracers and prescriptions based on different bands of the electromagnetic spectrum, including UV and far-IR continuum, as well as several recombination or forbidden lines<cit.>. Recently, near-IR observations have also proven useful to characterize the star formation properties of LIRGs through the study of super star clusters <cit.>, which are young (≃10 Myr) massive star clusters that preferentially form whenever there is strong ongoing starburst activity. Other tracers are used to infer the AGN type, its luminosity and its relative contribution to the bolometric luminosity of galaxies, such as X-ray emission <cit.>, mid-IR continuum <cit.> or optical <cit.> and IR spectral lines <cit.>.However, the large amounts of dust present in LIRGs, whose reprocessing of ultraviolet photons from massive stars is responsible for the high IR luminosities in these systems, impose a limitation to optical and near-IR tracers due to dust obscuration <cit.>. For this reason, observations at radio wavelengths, unaffected by dust extinction, are an alternative and powerful tool to trace starburst (SB) and AGN processes in the innermost regions of these systems <cit.>. There is a well-known tight linear correlation between the far-IR and radio (1.4 GHz) luminosities <cit.>, with no evident dependence with redshift <cit.>. To quantify this correlation, the so called q-factor <cit.> was defined as:q=log(FIR/3.72×10^12 Hz/S_1.49 GHz),where S_1.49 GHz is the flux density at 1.49 GHz in units of W m^-2 Hz^-1 and FIR is defined asFIR=1.26×10^-14(2.58 S_60 μm + S_100 μm),with S_60 μm and S_100 μm being the IRAS fluxes in Jy at 60 and 100 μm, respectively. The mean value of the q-factor in the IRAS Bright Galaxy Sample <cit.> is <q>=2.34 <cit.>.While a radio excess (q<2.34) is typically associated with a strong AGN contribution <cit.> and a FIR excess (q>2.34) is suggestive of intense star formation, this is not always the case <cit.>. In general, the q-factor cannot be used as a standalone tool to separate AGN from starburst galaxies <cit.>. The underlying physics for the FIR-radio correlation is usually associated with a star formation origin: dust reprocesses massive-star UV radiation into far infrared photons, while the explosions of those same stars as supernovae (SNe) accelerate the cosmic ray electrons, responsible for the radio non-thermal synchrotron radiation <cit.>. Thermal bremsstrahlung arising fromH ii regions (free-free emission) is also linked to this correlation <cit.>. Therefore, both thermal and non-thermal emission are correlated with the far-IR and are good tracers of star-formation. Correlations between radio emission and windows in the IR regime other than the far-IR exist <cit.>, but are not as clear nor that well studied. A problem of many of the starburst indicators mentioned above is the frequent contamination of the tracers by the effects of an AGN. Notable examples are the contamination of the radio continuum by putative jets <cit.> and the significant contribution to the far-IR through the heating of the narrow line region by the AGN <cit.>. A way to overcome this problem is to fit a multi-wavelength spectral energy distribution (SED) combining templates of both starbursts and AGN <cit.>. In this paper, we model the SED of a sample of 11 local LIRGs and compare our derived starburst and AGN properties with other models. We also present radio X-band (8.4 GHz) and near-IR K_S-band (2.2 μm; hereafter referred to as K-band) observations of this sample, comparing them and analyzing possible near-IR/radio correlations. The paper is structured as follows: in section <ref> we present the sample, together with a concise individual description of the sources, and in section <ref> we present our observations and details on the data reduction. Through the discussion (section <ref>) we describe our SED modeling in section <ref>, and compare it with other tracers of star formation and AGN activity. We then compare our radio and near-IR observations (section <ref>), analyze the special case of IRAS 16516-0948 (section <ref>) and discuss the radio spectral index of our sources (section <ref>). We summarize our results in section <ref>.Throughout this paper we adopt a cosmology with H_ 0 = 75 km s^-1 Mpc^-1, Ω_Λ = 0.7 and Ω_m = 0.3.§ THE SAMPLEThe sources in our LIRG sample are taken from the IRAS Revised Bright Galaxy Sample <cit.>, and fulfill the following criteria:D<110 Mpc, luminosity log(L_IR/L_⊙)>11.20, and declination δ>-35^∘. We chose those criteria so that our 8.4 GHz Karl G. Jansky Very Large Array (VLA) observations in A-configuration (angular resolution of ≃ 0.3^'') could image the central kpc region of all galaxies with spatial resolutions of ≃ 70-150 pc, which would allow us to disentangle the compact radio emission from a putative AGN from the diffuse, extended radio emission linked to a starburst, as well as potentially allowingthe detection of other compact sources, e.g., individual supernovae or supernova remnants (SNR). Furthermore, we excluded warm LIRGs with IRAS color f_25/f_60>0.2 to prevent contamination from obscured AGN activity <cit.>, except for Arp 299 (f_25/f_60=0.22), which is borderline, but was included in the study since it is one of the most luminous local LIRGs. A total of 54 out of 629 sources in the IRBGS satisfy the conditions above, from which ours is a representative sample. Finally, we also excluded galaxies with no nearby reference star to guide the adaptive optics (AO) observations with the VLT or theGemini telescopes.In Table <ref> we show our final sample, which consists of 11 LIRGs (although the two components of one of them, Arp 299, are treated separately), all of them included in the GOALS sample <cit.>, together with their IR luminosity and luminosity distance. We also include the merger stage as derived from IRAC 3.6 μm morphology <cit.>, and the q-factor values <cit.>, obtained using the IRAS fluxes from the IRBGS and the 1.4 GHz fluxes from the NRAO VLA Sky Survey <cit.>. An optical image of each source in our sample is shown in Figure <ref>. In Figure <ref> we show the correlation between L_IR and the radio luminosity at 1.4 GHz for the sources in our sample (plotted with a star symbol) compared with the resulting sources of a cross-match between the IRBGS and the New VLA Sky Survey by <cit.>. A short individual description of each source, some of which have been barely studied, is shown below. §.§ MCG +08-11-002This galaxy, also named IRAS 05368+4940, is a barred spiral galaxy (type SBab) with a complex morphology, in a late stage of merging <cit.>. Its mid-IR extended emission is clearly silicate dominated <cit.>. <cit.> found evidence of a possible preceding starburst episode, likely linked to the previous encounter of the galaxy nuclei.§.§ Arp 299Arp 299 is one of the most luminous LIRGs in the local Universe, and for that reason one of the most studied systems. It is in an early merger stage according to <cit.> or in a mid-stage according to <cit.> and <cit.>. Arp 299 is formed by two galaxies and exhibits two clear radio nuclei <cit.>, A and B, and two secondary components, C and C^'. The remaining compact structure, D, is believed to be a background quasar, unrelated to the system <cit.>. Each radio source is identified in Figure <ref>. Several supernovae have been recently discovered in the inter nuclear region <cit.> and in nucleus B <cit.>. However, it is the A nucleus that hosts a very rich supernova factory <cit.>, as well as a low luminosity AGN, suggested from X-ray observations <cit.>, and found by means of VLBI observations <cit.>. SED modeling for this system yielded a star formation rate (SFR) of 90 and 56 M_⊙ yr^-1 for the east and west components, respectively <cit.>.While there are different nomenclatures to name the two galactic components of Arp 299, some have produced confusion in the literature <cit.>. To avoid that, we use the unequivocal designations NGC 3690 East and NGC 3690 West for these components, which are treated individually throughout this paper. §.§ ESO 440-IG058This LIRG, also known as IRAS 12042-3140, consists of two merging galaxies separated by ≃6kpc. The northern component is very compact and has been classified as a LINER <cit.>. While the northern galaxy is dominated by star formation, the southern emission appears to be dominated by shocks <cit.>.Based on the IR luminosity, <cit.> estimated that ESO 440-IG058 has a star formation rate of 36 M_⊙ yr^-1 and an expected SN rate of 0.4 SN yr^-1. <cit.> derived an age of the stellar population of t≲6.5Myr for the currently star forming stellar population. §.§ IC 883IC 883, also known as UGC 8387, is a late-merger LIRG at a distance of 100 Mpc, showing a peculiar morphology, with extended perpendicular tidal tails visible in the optical and near-IR <cit.>. This source was classified as an AGN/SB composite <cit.>, which has been confirmed by the direct detection of a number of radio componentes that are consistent with an AGN and with SNe/SNRs <cit.> and with the detection, in the near-IR, of two SNe<cit.> within the innermost nuclear region of the galaxy. Through SED model fitting, <cit.> estimated a core collapse supernova (CCSN) rate of 1.1SN yr^-1 and a SFR of 185 M_⊙ yr^-1.IC 883 also presents strong PAH emission, silicate absorption and a steep spectrum beyond 20 μm <cit.>. Recently, using radio VLBI and X-ray data, <cit.> have reported unequivocal evidence of AGN activity, with the nucleus showing a core-jet structure and the jet having subluminal proper-motion.§.§ CGCG 049-057Also known as IRAS 15107+0724, this is the only LIRG in our sample classified as isolated. However, despite its apparent isolation <cit.>, it has a complex and dusty nuclear morphology, so some form of past interaction cannot be ruled out. It hosts an OH megamaser <cit.>. It is optically classified as a pure starburst, supported by Chandra X-ray observations <cit.>. However multi-band radio observations show evidence of a buried AGN within the SB <cit.>.§.§ NGC 6240This bright LIRG is a well-studied late-stage merger <cit.>. It hosts one of the few binary AGN detected so far using Chandra hard X-ray observations <cit.>, with a projected distance of ≃1 kpc. This was later supported by the detection of two compact unresolved sources at radio wavelengths with inverted spectral indices <cit.>, finding also a third component with a spectral index consistent with a radio supernova. Close to the southern nucleus, a water-vapor megamaser was found <cit.>. §.§ IRAS 16516-0948This unexplored LIRG was optically identified as a star forming galaxy <cit.>, and classified as a late merger from its infrared morphology <cit.>. IRAS 16516-0948 was part of the COLA project, for which high-resolution radio imaging did not detect any compact core <cit.>. Despite the scarce information available for this galaxy, we found it to be an interesting source, as discussed in section <ref>.§.§ IRAS 17138-1017This LIRG is a highly obscured starburst galaxy <cit.> in a late stage of interaction. An extremely extinguished supernova (A_V=15.7±0.8mag) was discovered in IRAS 17138-1017 using infrared K-band observations <cit.> Two more supernovae were also found: SN2002bw <cit.> and SN2004iq <cit.>.§.§ IRAS 17578-0400 This is a galaxy pair in an early stage of merging <cit.>. Ultra-hard X-ray (14–195 keV) observations with the Swift Burst Alert Telescope (BAT) searching for AGN did not detect any compact emission in this source <cit.>.There is abundant archival data in the X-ray, optical, IR, and millimeter bands for this source <cit.>, although no significant results for the purposes of this paper have been reported in the literature. §.§ IRAS 18293-3413 This source was classified as an H ii galaxy based on its optical spectrum <cit.>. It was detected in hard X-ray data (2–10 keV), obtained with ASCA, at a 5σ level by <cit.>, finding no evidence of any AGN contribution to the X-ray spectrum. There are discrepancies on the merger stage determination. <cit.> classified it as a mid-stage merger from the visual inspection of Spitzer-IRAC 3.6 μm images; on the other hand, <cit.> classified it as a very early merger, with canonical disks and no tidal tails based on its HST morphology. However, <cit.>, using high-resolution near-IR K-band adaptive optics imaging with VLT/NACO, showed the galaxy to have a very complex morphology, strongly suggesting a late stage interaction. The system includes a rare un-evolved elliptical companion as well. Using SED modeling, <cit.> estimated a CCSN rate of 1 SN yr^-1 for IRAS 18293-3413.Supernovae SN2004ip <cit.> and AT2013if (Kool et al. in prep.) were discovered within the nuclear regions of this galaxy. §.§ NGC 6926This relatively low luminosity LIRG is a spiral galaxy in a very early phase of interaction with the dwarf elliptical NGC 6929, located 4^' to the east. Optically identified as a Seyfert 2 <cit.>, NGC 6926 has a powerful water-vapor megamaser <cit.>, typically found in heavily obscured AGN <cit.>. Its X-ray hardness ratio <cit.> is also suggestive of an AGN. This is the only source for which we lack near-IR observations.§ OBSERVATIONS AND DATA REDUCTION The observational data used in this study comes from (1) multi-wavelength archival data obtained through the VizieR photometry tool (used for the SED model fits) and (2) from our own observations at both radio and near-IR wavelengths. Table <ref> shows a summary of these observations. The images are shown in Figure <ref> and discussed in section <ref>. | http://arxiv.org/abs/1705.09663v2 | {
"authors": [
"Rubén Herrero-Illana",
"Miguel Á. Pérez-Torres",
"Zara Randriamanakoto",
"Antxon Alberdi",
"Andreas Efstathiou",
"Petri Väisänen",
"Erkki Kankare",
"Erik Kool",
"Seppo Mattila",
"Rajin Ramphul",
"Stuart Ryder"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170526180001",
"title": "Star formation and AGN activity in a sample of local Luminous Infrared Galaxies through multi-wavelength characterization"
} |
Effective Sampling: Fast Segmentation Using Robust Geometric Model Fitting Ruwan Tennakoon, Alireza Sadri, Reza Hoseinnezhad, and Alireza Bab-Hadiashar, Senior Member, IEEE R.B. Tennakoon, A. Sadri, R. Hoseinnezhad and A. Bab-Hadiashar are with the School of Engineering, RMIT University, Melbourne, Australia.E-mail: [email protected] 30, 2023 ================================================================================================================================================================================================================================================================================================================= Trajectory Prediction of dynamic objects is a widely studied topic in the field of artificial intelligence. Thanks to a large number of applications like predicting abnormal events, navigation system for the blind, etc. there have been many approaches to attempt learning patterns of motion directly from data using a wide variety of techniques ranging from hand-crafted features to sophisticated deep learning models for unsupervised feature learning. All these approaches have been limited by problems like inefficient features in the case of hand crafted features, large error propagation across the predicted trajectory and no information of static artefacts around the dynamic moving objects. We propose an end to end deep learning model to learn the motion patterns of humans using different navigational modes directly from data using the much popular sequence to sequence model coupled with a soft attention mechanism. We also propose a novel approach to model the static artefacts in a scene and using these to predict the dynamic trajectories. The proposed method, tested on trajectories of pedestrians, consistently outperforms previously proposed state of the art approaches on a variety of large scale data sets. We also show how our architecture can be naturally extended to handle multiple modes of movement (say pedestrians, skaters, bikers and buses) simultaneously. § INTRODUCTION Learning and inference from visual data have gained tremendous prominence inArtificial Intelligence research in recent times. Much of this has been due to the breakthrough advances in AI and Deep Learning that have enabled vision and image processing systems to achieve near human precision on many complex visual recognition tasks. In this paper we present a method to learn and predict the dynamic spatio-temporal behaviour of people moving using multiple navigational modes in crowded scenes. As humans we possess the ability to effortlessly navigate ourselves in crowded areas while walking or driving a vehicle. Here we propose an end-to-end Deep Learning system that can learn such constrained navigational behaviour by considering multiple influencing factors such as the neighbouring dynamic subjects and also the spatial context in which the subject is. We also show how our architecture can be naturally extended to handle multiple modes (say pedestrians, skaters, bikers and buses) simultaneously. § PROBLEM STATEMENT We formulate the problem as a learning-cum-inferencing task. The question we seek to answer is “having observed the trajectories [x_i^(1),…,x_i^(T)], 1≤ i≤ N of N moving subjects for T time units, where are each of these subjects likely to be at times T_1,…,T_n after the initial observation time T?” where x_i^(t) denotes the 2-dimensional spatial co-ordinate vector of the i^th target subject at time t. In other words, we need to predict values of [x_i^(T_1),…,x_i^(T_n)]. We assume we have annotations of the spatial tracks of each unique subject that is participating in the scene.§ RELATED WORK§.§ RNNs for Sequence to Sequence modeling Encoder Decoder Models <cit.> were initially introduced for machine translation tasks in <cit.> followed by automated question answering in <cit.>. These architectures encode the incoming sequential data into a fixed size hidden representation using a Recurrent Neural Network(RNN) and then decode this hidden representation using another RNN to produce a sequentially temporal output.These networks have also been modified to introduce an attention mechanism into them. These networks are inspired by the attention mechanism that humans possess visually. We as humans, adjust our focal point over time to focus more at a specific region of our sight to a higher resolution and the surrounding area to a lower resolution. Attention Networks have been used very successfully for automatic fine-grained image description/annotation <cit.>.§.§ Object-Object Interaction modeling Helbing and Molnar's social force model<cit.> was the first to learn interaction patterns between different objects such as attractive and repulsive forces. Since then, several variants such as (i) agent based modeling<cit.> to use human attributes as model priors to learn behavioral patterns, and (ii) feature engineered approaches like that of Alahi et.al.<cit.> which extract social affinity features to learn such patterns, have been explored.Other approaches include finding the most common object behaviour by means of clustering<cit.>. Giannotti et. al.<cit.> analyzed GPS traces of several fleets of buses and extracted patterns in trajectories as concise descriptions of frequent behaviour, both temporally and spatially.Most recently, Alahi et.al.<cit.> captured the interactions between pedestrians using multiple Long Short Term Memory Networks(LSTMs)<cit.> and a social pooling mechanism to capture human-human interactions. While they captured this dynamic interaction, their model failed to understand the static spatial semantics of the scene. Such spatial modeling exists in <cit.> but these do not include the dynamic modeling of the crowd.None of the above approaches naturally extend to multiple classes of moving subjects. §.§ Spatial Context modeling Earlier works include matching-based approaches<cit.> which rely on keypoint feature matching techniques. These are slow to compute since they need to match each test image with a database of images. Also, since it is a direct matching approach there is no semantic understanding of the scene. Most of the conventional approaches tend to be brittle since they rely heavily on hand-crafted features.A deep learning approach was also used for spatial context modeling in <cit.>. This work hypothesizes that dynamic objects and static objects can be matched semantically based on the interaction they have between each other. The authors assume that a random image patch of the scene contains enough evidence, based on which the discrimination between likely patches and unlikely patches can be made for a particular object. They did not explore the possibility of adding additional static context around the patches to augment the model. We build on this distinction later on in section 4.1. § PROPOSED SYSTEMThis section describes our solution in detail and is organized as follows. Firstly, we describe our proposed Spatially Static Context Network(SSCN) to model the static spatial context around the subject of interest. Next, we define the Pooling Mechanism which captures the influence, the neighboring subjects and the nearby static artefacts have on a target subject. Next, we describe our complete model which uses an attention mechanism with LSTMs to learn patterns from spatial co-ordinates of subjects while preserving the spatial and dynamic context around the subjects of interest. Though LSTMs have traditionally been used to model (typically short-term) temporal dependencies in sequential data, their use along with an appropriate attention mechanism enables us to use them for tasks like trajectory planning that requires long-term dependency modeling. We also show how this extends in principle to multiple classes of moving subjects. §.§ Spatial Context Matching Modeling the spatial context in a given scene is a challenging task since it should be semanticallyrepresentative and also highly discriminative. It is also very important for the model to generalize to a variety of complex scenes and enable inferences about human-space interactions in them.Our proposed architecture is composed of Convolutional Neural Networks(CNNs)<cit.> and is inspired by the Spatial Matching Network introduced in <cit.>. Although, our architecture is very similar to the one proposed in this work, we differ significantly from this approach in two ways. First, it is redundant to build an input branch which takes image patches of different objects because an object belonging to the same semantic class (car, pedestrian, bicyclist) should ideally have the same spatial matching score with any particular random patch of the image. For example, any car should have the same matching score with a patch of road and hence we do not need to differentiate between different cars.Second, it might be difficult for a network to look at a small scene patch and infer the matching score from it. For instance, a trained CNN might have learned different textures of different static artefacts like that of a road or a pavement, but such textures could occur anywhere in the image. For example, the texture of the roof in a particular scene could match the texture of the pavement in a different scene image. So it is important for the network to have information about the larger context or the region surrounding the input patch.This will help the network to generalize better across different scenes and have a better semantic understanding of different complex scenes.We incorporate our hypothesis in our proposed network shown in figure 1. We call this network as the Spatially Static Context Network(SSCN). The network has three input streams - Subject stream, Patch stream and Context stream. For the subject input stream, we input a class label s∈{0,1,2,…} to indicate whichever semantic class the dynamic subject belongs to. This input is passed through an embedding layer followed by a dense fully connected layer.For the patch stream the input image is that of the grid cell of interest ρ. To incorporate a local context around that patch, we also take the part of the image surrounding that cell. Hence, we take the size of the grid cell of interest as g_ρ× g_ρ and add an annulus of width g_ρ around it (we appropriately pad the original image in order to add such a context for the cells on the boundary of the image). Thus, the final size of input patch becomes 3g_ρ× 3g_ρ. This is later re-sized to d_P× d_P for a fixed d_P. We pass this input patch through a stack of convolutional layers along with pooling and local response normalization after each layer followed by a dense fully connected layer as shown in figure 1.The third stream, that captures the overall image context, is in line with our second hypothesis. This stream of the network is incorporated to introduce a global context around the image patch of interest. The input to this stream ζ is the whole scene image re-sized to a fixed dimension size of d_I× d_I. We again use another stack of convolutional layers along with pooling and local response normalization after each layer followed by a dense fully connected layer to extract hierarchical features of the scene image.To train such a network, we merge all the input streams by concatenating the outputs of their respective dense fully connected layers and add a stack of 2 fully connected layers followed by a fully-connected output layer. The output layer consists of just a single sigmoid neuron, giving the likelihood of a subject of type sstepping on the given patch ρ. The ground truth likelihood we use for training is the value associated with the grid cell of interest (not the complete input patch).We train the model over all triplets (s,ρ,ζ)∈𝒟 of subject type, patch and image, by minimizing the cross entropy between the actual likelihood μ_sρζ and the predicted likelihood μ̂_sρζ - H_𝒟 = -1/|𝒟|∑_(s,ρ,ζ)∈𝒟(μ_sρζlog(μ̂_sρζ)+(1-μ_sρζ)log(1-μ̂_sρζ))The ground truth likelihood value for each patch is computed by counting the frequency of unique subjects of type s occupying the patch at some time during the course of their individual trajectories and dividing it by the total number of unique subjects of the same type in the scene. Note that 𝒟 is constructed from the annotated videos by tracking and counting the incidence of each subject in the video with every patch in a subsample of video frames. §.§ Pooling MechanismOur model was trained on a subsample (one in every ten frames) of frames from the videos that formed our training dataset. We describe how we pool the static and dynamic contexts for a given frame ℱ from the sample along with the representation of the historical trajectory. We use the subscript ℱ to indicate the fact that the pooling being done is specific to the frame.§.§.§ Dynamic Context Pooling Humans moving in a crowded area adapt their motion based on the behaviour of the people around them. For example, pedestrians often completely alter their paths when they see someone else or a group of people approaching them. Such behaviour cannot be predicted by observing a pedestrian in isolation without considering the surrounding dynamic and static context. This behaviour motivated the pooling mechanism of the Social LSTM model<cit.>.We borrow the same pooling mechanism to capture such influences from neighbouring subjects for our model.We use LSTMs to learn an efficient hidden representation of the temporal behaviour of subjects as part of the encoder. Since these hidden representations would capture each subject's behaviour until the observed time step, we can use these representations to capture the influence that the neighbouring subjects would have on a target subject.We consider a spatial neighbourhood of size (d_s× d_s) around each moving subject (again with appropriate padding for the boundary cells) which in turn is subdivided into a (g_s× g_s) grid with each grid cell of size (d_s/g_s×d_s/g_s). Let h_jsℱ^(t) denote the LSTM encoded hidden representation (a vector of dimension d_H) of j^th subject of type s at time t (when the current frame is ℱ). Also let C be the number of subject classes. We construct social tensors<cit.> 𝒮_iℱ^(t) each of size (g_s× g_s× d_H× C) that capture the social context in a structured way 𝒮_iℱ^(t)(r,c,:,:) = [ ∑_j=1^N_s𝕀_ircℱ^(t)[s,j].h_jsℱ^(t-1)]_s=1^Cwhere N_s is the total number of subjects of type s and 𝕀_ircℱ^(t)[s,j] is an indicator function which denotes whether the j^th subject of type s is in the (r,c)^th grid cell of the spatial neighbourhood of the i^th subject at time t.§.§.§ Static Context Pooling As described earlier, the SSCN model is designed to predict the likelihood of a subject like a pedestrian stepping on a specific input image patch, given the larger context around the patch and the scene itself. We use this to provide a surrounding context for each subject from its current position which in turn influences the next position of the subject.We first build a spatial map for each subject class and location in a frame of the video, with the probabilities of a subject of that class ever visiting that location. This map is built offline using the pretrained SSCN network described in Section <ref>. Given a subsample of frames, for every frame ℱ in the sample, patch ρ and subject class s we build the mapℳ_sρ^ℱ = SSCN(s,ρ, ℱ).We extract a Reachability Tensor R^(t)_iℱ for the i^th subject (say of class s) at time t from the static context map ℳ_sρ^ℱ. Given its current position x_i^(t), let Φ_i^(t) be the collection of all patches of size (3g_p× 3g_p) that are centered at (g_p× g_p) patches at most (d_R/2) away on each axis from x_i^(t). We basically construct a reachability context tensor accounting for a patch of size (d_R× d_R) with x_i^(t) at the center. The extracted reachability context tensor is thereforeR^(t)_iℱ = [SSCN(s,ρ, ℱ)]_ρ∈Φ_i^(t) §.§ Spatio - Temporal Attention Model In our work, we take the encoder-decoder architecture as the base model and apply a soft attention mechanism on top of it. The motivation for applying an attention mechanism is straight-forward. Subjects often change their pre-panned trajectories suddenly when the 'context' changes. Imagine a pedestrian in an airport walking towards the security, suddenly realizing that he/she needs to pick up the baggage tag and as a result making a sharp course-correction to move towards the check-in counter. Since, the model proposed by Alahi et. al.<cit.> only takes the last time step hidden representation of the pedestrian of interest, the model will be responsive to immediate instincts like collision avoidance but not be very useful in long term path planning. Moreover, once the model starts its predictions, even a small error in the prediction could mean that the erroneous hidden representations are propagated to future time steps.The full architecture for our spatio-temporal attention model is shown in figure 2.We first embed the spatial coordinate input x_iℱ^(t), the dynamic context pooled tensor S_iℱ^(t) and static reachability tensor R_iℱ^(t) to fixed dimensions using three separate sigmoid embedding layers.p_iℱ^t=ϕ(x_iℱ^(t),W_e), a_iℱ^t=ϕ(S_iℱ^t,W_a), c_iℱ^t=ϕ(R_iℱ^t,W_c)where W_e,W_a and W_c are the embedding matrices and ϕ is the sigmoid function. Next, the three embeddings p_iℱ^t, a_iℱ^t and c_iℱ^t are concatenated to form the input to the encoder. The encoder outputs a fixed size hidden state representation h_iℱ^t at each time step t, h_iℱ^t=LSTM(h_iℱ^t-1,p_iℱ^t,a_iℱ^t,c_iℱ^t,W_enc)where h_iℱ^t-1 is the hidden state representation output of the decoder at the last timestep (t-1) and LSTM is the encoding function for the encoder with weights W_enc.We use a context vector C_iℱ^(t)=∑_j=t-k+1^tα^(j)_i.h^(j)_iℱ that depends on the encoder hidden states occuring in a fixed size temporal attention window of size k — [h_iℱ^(t-k+1),….,h_iℱ^(t-1),h_iℱ^(t)]. This makes it possible for the network to dynamically generate different context vectors at different timesteps with the knowledge of k encoded states back in time. We follow the previously proposed attention mechanism like in Bahdanau et. al.<cit.> to compute the attention weights α^(j)_i. This context vector is generated by the attention mechanism which puts an emphasis over the encoder states and generates a 'behaviour context' for the subject of interest.The context vector C_iℱ^(t) feeds into the decoder unit to generate the decoder states. The decoder in turn generates a 5-tuple as its output representing the parameters of a bivariate Gaussian distribution over the predicted position of the subject at the next time step. Denoting the decoder state at time t (for the frame ℱ) as s^(t)_iℱ, the decoder state and the predicted bivariate Gaussian model θ_iℱ^(t)≡𝒩(μ_i^(t),σ_i^(t),ρ_i^(t)) for the position of the subject x_iℱ^(t) at time t are computed ass^(t)_iℱ = LSTM(s^(t-1)_iℱ,x_iℱ^(t-1), C_iℱ^(t),W_dec), θ_iℱ^(t)=ReLU(s^(t)_iℱ,W_o)Note that x_iℱ^(t-1), the position at time (t-1), is taken asthe actual ground truth value at time (t-1) during training and the model predicted value during inference. The above model can be trained for multiple classes of subjects with a separate model for each individual class.Our model therefore accounts the current spatial coordinates, social context that incorporates all the moving subjects in the scene and the static context that accounts for the reachability of the subjects across the patches in the scene.During inference, the final predicted position x̂_(t)^i is sampled from the predicted distribution θ_i^(t). §.§ Cost Formulation We train the network by maximizing the likelihood of the ground truth position being generated from the predicted distribution. Hence, we jointly learn all the parameters by minimizing the negative log-Likelihood loss L_i=-∑_t=T_1^T_nlog(P(x_i^(t)|μ_i^(t),σ_i^(t),ρ_i^(t)) for the i^th trajectory. An important aspect of the training phase is that, since the LSTM layers of the encoder and decoder units are shared between all the subjects of a particular type, all parameters of the models have to be learned jointly. Thus, we back-propagate the loss for each trajectory i of each subject type at every time step t.§ EXPERIMENTS§.§ DatasetWe use three large scale multi-object tracking datasets - ETH <cit.> , UCY <cit.> and the Stanford Drone Dataset <cit.>. The ETH and UCY datasets consist of 5 scenes with 1536 unique pedestrians entering and exiting the scenes. It includes challenging scenarios like groups of people walking together, 2 different groups of people crossing each other and also behaviour such as a pedestrian deviating completely from it's followed path almost instantaneously.On the other hand, the Stanford Drone Dataset<cit.> consists of multiple aerial imagery comprising of 8 different locations around the Stanford campus and objects belonging to 6 different classes moving around. We use only the trajectories of pedestrians to train and test our models. §.§ Setup We set the hyperparameters of our model using a cross validation strategy following a leave one out approach. For the SSCN model we take the size of the grid g_ρ in the patch stream as 60 and the re-sized input dimension as 227 x 227. The input dimension of the context stream is set to 512 x 512. The overall network is trained using a learning rate of 0.002, with Gradient Descent Optimizer and a batch size of 32. To compare our results with that of previous state of the art model - S-LSTM<cit.>, we limit the subject type to only pedestrians. We set the common hyperparameters of the Spatio-Temporal Attention Model to be the same as that of theirs. Trajectories are downsampled so as to retain one in every ten frames. We observe the trajectories for a period of 8 time steps (T) with an attentional window length k of 5 time steps and predict for future 12 time steps (n). We set the reachability distance d_R to 60. We also limit the number of pedestrians in each frame to 40. The model is trained using a learning rate of 0.003 with RMSProp as the optimizer. §.§.§ Evaluation Metrics We use two evaluation metrics as proposed in Alahi et.al.<cit.> -(i) Average Displacement error - The euclidean distance between the predicted trajectory and the actual trajectory averaged over all time-steps for all pedestrians and(ii) Final Displacement error - The average euclidean distance between the predicted trajectory point and the actual trajectory point at the end of n time steps. §.§ Quantitative Results We build two separate models for the complete problem statement - one with only dynamic context pooling coupled with the attention mechanism which we denote as - D-ATT Model and the second with static context pooling added to the D-ATT model which we denote as - SD-ATT. We cannot test the SD-ATT model on ETH and UCY datasets as the resolution of the videos in these datasets is too low which makes it impossible to use the SSCN model for static context pooling. Still, we consistently outperform the S-LSTM<cit.> and O-LSTM<cit.> models on both the evaluation metrics on all three datasets as shown in table 1. We also show that the results of the SD-ATT model on the Stanford Drone Dataset are much better than the Social LSTM<cit.> model. §.§ Qualitative ResultsWe demonstrate scenarios where our models perform better than the S-LSTM<cit.> model. Firstly, in Figure 3 we show the results for SSCN model. The second column of the figure shows an example of the constructed saliency map with the shade of blue denoting the likelihood value of the corresponding patch. We can see that the likelihood values are low near the roundabout, car and trees and high for areas such as road and pavement. In rest of the figures, each unique pedestrian is depicted by a unique colour. In Figure 4, the first column shows that the SD-ATT model has learned to predict non linear trajectories as well. The next two columns depict the collision avoidance property learned by the model. In both the examples, the model either decelerates one of the pedestrian or it diverts it to avoid collision. Figure 5(a) compares the predictions of SD-ATT model against the Social LSTM<cit.> model. Since, the S-LSTM model only considers the last time step's hidden representation, it thinks that the pedestrian wants to take a turn and hence follows the curved path. On the other hand, our SD-ATT model interprets this as a sharp turn since it has only seen this change in the behaviour over the last two time steps and hence takes a gradual turn. This demonstrates the advantage of using an attention mechanism. Figure 5(b) demonstrates the advantage of static spatial pooling. In both the examples shown, the pedestrian walks straight for T time steps because of which S-LSTM<cit.> and D-ATT models predict a straight path. On the other hand, SD-ATT model captures a static obstacle in front of the pedestrian and hence takes a diversion from the followed path.§ CONCLUSIONWe propose a novel deep learning approach to the problem of human trajectory prediction. Our model successfully extracts motion patterns in an unsupervised manner. Compared to the previous state of the art works, our approach models both dynamic and spatial context around any type of subject of interest which results in better prediction of trajectories. Our proposed method outperforms previous state of the art method on three large scale datasets. In addition to this, we also propose a novel CNN based SSCN architecture which helps in better semantic understanding of the scene. Future work includes evaluating the proposed models for multiple classes of objects. abbrv 30 key-1Ilya Sutskever, Oriol Vinyals, and Quoc VV Le. Sequence to sequence learning with neural networks. 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"authors": [
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"published": "20170526053736",
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http://arxiv.org/abs/1705.09866v4 | {
"authors": [
"Manuel Valera",
"Zhengyang Guo",
"Priscilla Kelly",
"Sean Matz",
"Vito Adrian Cantu",
"Allon G. Percus",
"Jeffrey D. Hyman",
"Gowri Srinivasan",
"Hari S. Viswanathan"
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^1National physical laboratory, Hampton Road, Teddington, TW11 0LW, UK ^2Laboratoire de Physique Theorique et Hautes Energies, CNRS UMR 7589, Universites Paris 6 et 7, place Jussieu 75252 Paris, Cedex 05 France ^3L. D. Landau Institute for Theoretical Physics, Chernogolovka, 142432, Moscow region, Russia ^4Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96 Göteborg, Sweden ^5Royal Holloway, University of London, Egham, TW20 0EX, UK ^†[email protected] Noise and decoherence due to spurious two-level systems (TLS) located at material interfaces is a long-standing issuein solid state quantum technologies. Efforts to mitigate the effects of TLS have been hampered by a lack of surface analysis tools sensitive enough to identify their chemical and physical nature.Here we measure the dielectric loss, frequency noise and electron spin resonance (ESR) spectrum in superconducting resonators and demonstrate that desorption of surface spins is accompanied by an almost tenfold reduction in the frequency noise.We provide experimental evidence that simultaneously reveals the chemical signatures of adsorbed magnetic moments anddemonstrates their coupling via theelectric-field degree of freedom to the resonator, causing dielectric (charge) noise in solid state quantum devices. Suppression of 1/f noise in solid state quantum devices by surface spin desorption A. V. Danilov^4==================================================================================As the complexity of solid state quantum circuits continues to increase, so do the challengesto both fabrication technology and materials science<cit.>. Improved device and systems engineering has lead tomaterial imperfectionsbeing a dominant source of noise and decoherence, and further improvements in material properties and a better understanding of the underlying materials physics are needed to make technologies such as large scale solid state quantum computing feasible<cit.>. The enhanced sensitivity of superconducting qubits and resonators has revealed that materials once considered to be near-perfect crystals, actually contain sufficient imperfections to behave as disordered systems.One unexpected consequence of the enhanced sensitivity to disorder of quantum devices was their ability to verify detailed predictions of theStandard Tunnelling Model (STM)<cit.>.The STM, originally developed to model the low-temperature acoustical and electromagnetic properties of glasses, assumes the presence of a large ensemble of two-level systems (TLS) which can absorb energy via their electric dipole moments, leading to dissipation via subsequent phonon decay. TLSaffect the performance of a many different solid state devices including superconducting resonators and qubits<cit.>, field-effect transistors<cit.>, single charge devices<cit.> and ion traps<cit.>. Understanding and removing TLS is therefore important for a wide range of applications in solid state physics, materials science and chemistry. While the origin of these TLS remains elusive, engineering advances have reduced TLS loss to a level wheremost remainingTLS arelocated at or in thin surface oxide layers<cit.>. In this regime the STM fails<cit.>. Remarkably, measurements of TLS-induced 1/f noise at low temperatures show an increasing noise∝ T^-(1+μ), with μ∼ 0.3found in both resonators<cit.> and qubits<cit.>. This dependence, clearly different from the vanishing T^3 dependence of the STM, is a signature of strong long-range TLS interactions. Furthermore, high quality (high-Q) resonators typically show a much weaker power dependence of the quality factor than what is predicted by the STM<cit.>. This prompted the development of a Generalised Tunnelling Model (GTM)<cit.> which takes into account strong dipole-dipole interactions between TLS<cit.>, successfully capturing the observed physics. Despite this success existing models do not give information about the chemical nature of surface TLS; something that is clearly needed for theirmitigation. Directly studying the chemical nature of TLS using established surface analysis techniques remains extremely challenging. One reason is that the density of TLS is very small, <1% of surface sites, and likely comprised of very light elements<cit.>, weakly adsorbed molecules<cit.> or electronic defect states<cit.>. These are easily introduced by exposing devices to ambient conditions<cit.>, inhibiting the use of many surface analysis techniques<cit.>.In constrast to charge TLS, magnetic dipoles as sources of flux noiseoriginate from a bath of paramagnetic surface spins<cit.>, and can therefore be identified by electron spin resonance (ESR) techniques usingsensitive tools derived from solid state quantum technologies<cit.>. IdentifyingTLS thatcouple through their charge degree of freedom is much more challenging due to the lack of direct identification methods that can reveal chemical fingerprints.In this work we show that changes observed in noise and loss measurements of superconducting resonatorsdirectly correlate with ESR data, which reveals important new clues about the chemical and physical nature of surface TLS.We further show that desorbing spins with a simple annealing treatment leads to a reduction of the frequency noise by almost an order of magnitude (see Figure <ref>a and <ref>b).This also allows us to directly identify the origin of the TLS responsible for noise as atomic scale electric dipoles; some of which are comprised of physisorbed atomic hydrogen<cit.>, while others are associated with free radicals. Our results suggest that these paramagnetic species not only cause a fluctuating magnetic environment<cit.>, but also are responsible for dielectric (charge) noise.§ EXPERIMENTSWe simultaneously measure the 1/f frequency noise and dielectric losses as a function of temperature and driving power (average photon number ⟨ n⟩)of two NbN superconducting resonators (with frequencies ν_0=4.6 GHz and 5.0 GHz)<cit.> patterned on the same c-cut Al_2O_3 substrate. The full high sensitivity ESR spectrum is subsequently obtained at T=10 mKby measuring the quality factor of the resonator as a function of applied magnetic field and the zero field loss is subtracted to obtain the magnetic field induced loss Q_b^-1<cit.>.We then anneal the device at moderate temperature (300^∘C), a technique that has shown to remove some of the spins native to the surface of the device<cit.>. The same noise and loss measurement protocol is repeated in a second measurement and finally the ESR spectrum is measured again, confirming the successful removal of some of the spins. Throughout this paper we refer to these two consecutive measurements as 'before' and 'after' spin desorption respectively.The frequency noise is measured in two resonatorsusing a high precision dual Pound locking technique adapted from frequency metrology<cit.> that continuously monitors the centre frequency of the resonators. Values for the dielectric loss tangent tanδ_0 before and after annealing are extracted from quality factor measurements at low power, and from an independent measurement of the temperature-dependent frequency shift of the resonators we find the intrinsic loss tangent tanδ_i. For further details see Methods. § RESULTSThe main result of this work is shown in Figure <ref>. In summary, after annealing anddesorption ofsurface spins we observe almost an order of magnitude reduction (on average 9.1 and 8.4 times for the two resonators respectively) in the frequency noise spectral density (Figure <ref>a-c) for both measured resonators at the lowest temperatures.The reduction in noise is observed together with a reduction in number of surface spins. Figures <ref>d and <ref>e display the ESR spectrum measured in-situ after collecting all the noise data, before and after annealing.The measured ESR spectrum reveals the presence of atomic hydrogen on the Al_2O_3 surface originating from water dissociation<cit.> and electronic charge states (with a g-factor of 2.0), likely due to absorption of oxygen radicals on the surface in accordance with previous findings<cit.>.An initial density of n_H = 2· 10^17 m^-2 hydrogen spins is completely removed and we extract a reduction in spin density due to the central peak from n_e = 0.91 · 10^17 m^-2 to ñ_e = 0.17 · 10^17 m^-2 spins/m^2, a factor of 5.3. The wide background plateau remained unchanged.Intriguingly, in contrast to the tenfold reduction in noise, we find that the intrinsic loss tangent tanδ_i is only reduced by 30% after surface spin desorption.For each resonator we also measured the power and temperature dependence of the quality factor, from which we also see only a very small reduction in the loss (see Table <ref> for exact values).§ DISCUSSIONThis small reduction in loss but large reduction in noise can be explained within the framework ofstrongly interacting TLS and the GTM, which naturally partitions the TLS as two distinct entities, one predominantly responsible for loss and one for noise.The microscopic picture is the following. Associated with each TLS there is a fluctuating dipoled_0 that couples to the applied microwave electric field E from the resonator.Among the TLS we can distinguish between coherent (quantum)electrical dipoles (cTLS) that are characterized by fast transitions between their states and relatively small decoherence rates, and slow classical fluctuators (from now on referred to as TLF) that are characterized by decoherence times shorter than the typical time between the transitions.The picture is sketched in Figure <ref>a with typical distances between thermally activated (excited) cTLS and TLF as inferred from our measurements.At low temperatures, slow fluctuators weakly coupled to cTLS mainly contribute to the dephasing of the high energycTLS and are responsible for their line-width Γ_2<cit.>. Slow fluctuators that are located close to the cTLS, and therefore are strongly coupled, shift the cTLS energy by an amount larger than Γ_2. These fluctuators create highly non-Gaussian noise that cannot be regarded as a contribution to the line-width.For resonant cTLS, having an energy splitting E ≈ħν_0, the interaction with a few strongly coupled TLF translates to the energy of the cTLS drifting in time, as illustrated in Figure <ref>b and c; it is this drift that ultimately generates 1/f noise in the resonator<cit.>.The intrinsic loss(at low fields) on the other hand arises from direct phonon relaxation from the resonantly coupled cTLS, and depends only on the number of cTLS, as shown in Figure <ref>d.Within the framework of the GTM our experimental findings of a small reduction in loss and a dramatic reduction in noise imply that desorption of surface spins did not affect the density of cTLS, instead the surface spins can be attributed to the TLF.The conceptual picture of these two separate TLS communities is further supported by additional experimental findings: for a homogeneous bath of non-interacting TLS (STM)we expect Q_i(⟨ n⟩)∼⟨ n ⟩^α with α = 0.5.The observed dependence is much weaker: a fit to a power law returns α≈ 0.2 for both resonators before and after desorption (see Supplementary). On the other hand, for interacting TLS we do expect a weak logarithmic dependence of the microwave absorption on stored energy in the resonator<cit.>1/Q_i(⟨ n⟩) = P_γ F tanδ_i ln(C √(|n_c|/|⟨ n⟩|)+c_0 ).Here C is a constant, c_0 accounts for power-independent losses, F is a geometric filling factor andP_γ is a normalization factor that depends on the spectral density of TLF switching rates. In Figure <ref> we show that our data fits very well to this logarithmic power dependence.Interestingly, we find that P_γ increasesafter spins were removed. This implies that the remaining slow fluctuators have a narrower range of switching rates and are likely different in nature than the spins that were desorbed. Independently, another important indication of the applicability of our model is given by the analysis of the temperature dependence of the1/f noise spectrum. The interaction gives a vanishing density of states for cTLS at low energies, P(E) ∝ E^μ with 0<μ<1, and this results in a scaling of the noise spectrum with temperature S_y(T) ∝ T^-(1+ 2 μ) (for T < hν_0/k_B). In agreement with previous studies<cit.> we findμ≈ 0.3 (see Supplemental material), both before and after spin desorption. This is further evidence that desorption only affects the number of slow TLF present on the sample.We now combine all available data to produce a qualitative picture (as sketched in Figure <ref>) of the microscopic properties of the cTLS and TLF, by taking the GTM beyond the original assumptions of identical densities and dipole moments of cTLS and TLF<cit.>. The details of this theory and analysis can be found in the Supplemental material, here we only summarize the results. Assuming the dipole moment for resonant cTLS to be on the atomic scale,d_0 = 1eÅ∼ 5 D (i.e. similar to what was previously deduced from spectroscopy measurements<cit.>), we arrive at dipole-dipole interaction strength U_0 ≈ 15Knm^3. Before spin desorption, we find from the intrinsic loss tangent the cTLS line-width Γ_2 ∼ 20MHz at T=60mK (see Supplemental material), which translates into the density of resonant cTLS ρ_TLS≈ 15GHz^-1μm^-2 in agreement with Ref. <cit.>, where the authors found ∼ 50 resonant cTLS per μm^2 in the frequency range 3-6 GHz, i.e. resonant cTLS arelocated at a typical distance r_cTLS∼ 1 μm from each other, similar to the densities found in qubit tunnel junctions<cit.>. Next, the measured amplitude of the noise A_0 can be related to the density of thermally activated (fluctuating) TLF and their dipole moment d_F. We find d_F/d_0ρ_F ≈ 5±4 · 10^-3 nm^-2.The thermally activated TLF constitutes a fraction T/W of the total number of TLF, where W is the bandwidth of the distribution of TLF energy level splittings. For weakly absorbed spins it is reasonable to expect that W∼ 100 K, limited by the observed desorption energy. From the total spin density measured by ESR we have n_e+n_H ≈ 3 · 10^-1 nm^-2. Combining these estimates, assuming all the TLFs are the observed spins, we have for the density of thermally activated TLF ρ_F = (n_e+n_H)(T/W)∼ 2 · 10^-4 nm^-2 (i.e. thermally activated TLFs are separated by an average distance r_F ∼ 100nm) and d_F/d_0 ∼ 30±25. The large uncertainty in d_F/d_0stems from its strong dependence on the filling factor (∝ F^3) and the volume where the TLF are situated, which both cannot be accurately estimated. However, the message of this order of magnitude estimation is that the assumption that all TLS are the observed spins is indeed plausible. Furthermore, the dipole moment of a surface TLF is likely larger compared to that of TLS in the bulk, as would be expected since the physisorbed and easily desorbed spins are likely to move larger distances. After spin desorption the noise amplitude decreases by a factor ∼ 10, the loss is only reduced by ∼ 30 % and the normalization constant P_γ increases ∼ 65% due to lower TLF switching rates. From this we can finally find a corresponding change in the density of TLF before and after spin desorptionρ_F(T)/ρ̃_F(T)= A_0P̃_γtanδ_i/Ã_0P_γtanδ̃_̃ĩ =15.23 and17.7,for the two resonators. Here we denote quantities for the 'after' measurement by the tilde symbol. These values correlate remarkably well with the change in the total number of spins in the three ESRpeaks (n_e+n_H)/ñ_e=17.1 (4.6 GHz resonator), and again indicates that spins contribute to the frequency noise in our high-Q superconducting resonators and take on roles as slow (mobile<cit.>) fluctuators.§ CONCLUSIONSBased on the experimental evidence from the loss, noise and ESR spectrum, all obtained on the same device, we have found that surface spins that are known to give rise to magnetic noise in quantum circuits<cit.>are also responsible for the low frequency dielectric noise of the resonator. These spins, remarkably present in densities also inferred to be responsible for flux noise in SQUIDs and qubits<cit.>,take on roles as slow classical fluctuators that cause an energy drift of resonant coherent TLS. Removing a majority of these spins gives an almost tenfold reduction in dielectric noise.In our device the observed surface spins constituteweakly physisorbed atomic hydrogen together with free radicals (g=2). We note that the nature of the g=2 spins is still not entirely clear. A large portion can be associated with surface adsorbates, likely oxygen radicals<cit.>, or other light molecular adsorbents<cit.>. The remaining fraction of free radicals may be a result of insufficient annealing or they may be of a different chemical or physical origin with much higher desorption barriers. Another possibility is that the remaining more robust localised charges and cTLS are intrinsic to the Al_2O_3 surface itself<cit.>, more resembling "bulk" defects<cit.>. Nevertheless, our approach reveals a new aspect of the noise in solid state quantum devices as we show that observed magnetic dipoles, with their fingerprint revealed through state-of-the-art surface analysis using in-situ micro-ESR, couple via the electric field degree of freedom and give rise to dielectric noise. Similar physics is expected for a wide range of oxide surfaces relevant for quantum technologies. The importance of magnetic moments has previously been widely overlooked in resonators sinceelectrical dipoles have been considered the dominating mechanism for dielectric noise. Our results instead indicate that while having a small influence on power loss, these spins (and their associated electric dipoles) constitute a major source of noise and dephasing in modern high coherence solid state devices by their proximity to coherently coupled resonant cTLS, and our results hint at a connection between the similar densities found for sources of flux<cit.>, charge<cit.> and dielectric noise in quantum circuits.§ ACKNOWLEDGEMENTSThis work was supported by the UK government's Department for Business, Energy and Industrial Strategy. The authors would like to thank S. Lara-Avila for assistance with fabrication. LF acknowledges support by ARO grant W911NF-13-1-0431 and by the Russian Science Foundation grant#14-42-00044. AD acknowledges support from VR grant 2016-04828. § AUTHOR CONTRIBUTIONSSdG, SK, AD, TL and AT conceived the experiment. AA prepared and treated the samples. SdG performed the noise and ESR measurements with assistance from TL. SdG analysed the data and LF articulated the theory, both with inputs from JB, AT and TL. All authors discussed the results. SdG, LF and TL wrote the manuscript with input from all authors. § METHODSSample preparation.Sapphire substrates were annealed in situ at high temperature, 800^∘C, for20 minutes prior to deposition of 2 nm NbN. After cooling down to 20^∘C, an additional 140 nm NbN was sputtered.Resonators were patterned using electron beam lithography (UV60 resist, MF-CD-26 developer, DI water rinse) and subsequent reactive ion etching in a NF_3 plasma. Resist was removed in 1165 remover followed by oxygen plasma treatment. Resonator designs were identical to those reported in Ref. <cit.>. After the first round of noise measurement the same sample was warmed up, shipped from UK to Sweden, and heated in vacuum to ∼ 300^∘C for 15 minutes to desorb surface spins, then shipped back to the UK, and mounted in the same cryostat with the same noise measurement setup ∼ 72 hours later. Remarkably, the detrimental surface spins are not re-introduced even after this time. Measurement setup.We used a cryogen-free dilution refrigerator with a base temperature of 10 mK and a 3-axis superconducting vector magnet for noise and ESR measurements. The cryostat was equipped with heavily attenuated coaxial lines, cryogenic isolators and a low noise high electron mobility transistor (HEMT) amplifier with a noise temperature of ∼ 4 K. All noise measurements were performed with the leads to the vector magnet completely disconnected. Only after completion of noise measurements the magnet was connected to measure the ESR spectrum. The plane of the superconductor thin-film was found to high precision (<0.1^∘) by applying a small field and carefully tilting the angle of the applied field while finding the maximum of the resonance frequency of the resonators.ESR measurements were performed by sweeping the magnetic field and measuring the characteristics of the resonators using a vector network analyser. Noise measurements were performed using a Pound locking technique<cit.> that tracks the resonance frequency (and its fluctuations) in real-time. For a detailed explanation of the technique, see supplemental material. 999 devoret2013 Devoret, M. H. and Schoelkopf, R. J. Superconducting circuits for quantum information: an outlook. http://dx.doi.org/10.1126/science.1231930Science 339, 1169 (2013). paladino2014 Paladino, E.,Galperin, Y. M., Falci, G. and Altshhuler, B. L.1/f noise: Implications for solid-state quantum information. http://dx.doi.org/10.1103/RevModPhys.86.361Rev. Mod. 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The chemistry of water on alumina surfaces: Reaction dynamics from first principles. http://dx.doi.org/10.1126/science.282.5387.265Science 282, 265-268 (1998). lisenfeld2016 Lisenfeld, J. et al. Decoherence spectroscopy with individual two-level tunneling defects. http://dx.doi.org/10.1038/srep23786Scientific Reports 6, 23786 (2016). martinis2004 Martinis, J. M. et al. decoherence in Josephson qubits from dielectric loss. http://dx.doi.org/10.1103/PhysRevLett.95.210503Phys. Rev. Lett. 95, 210503 (2005). sarabi2016 Sarabi, B.,Ramanayaka, A. N., Burin,A. L.,Wellstood, F. C. andOsborn, K. D. Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics. http://dx.doi.org/10.1103/PhysRevLett.116.167002Phys. Rev. Lett. 116, 167002 (2016). bedilo2014Bedilo, A. F.,Shuvarakova, E. I.,Rybinskaya, A. A., and Medvedev, D. A. Characterization of electron donor and electron acceptor sites on the surface of sulfated alumina using spin probes. http://dx.doi.org/10.1021/jp503523kJ. Phys. Chem. C 118, 15779 (2014). astafiev2004 Astafiev, O.,Pashkin, Yu. A., Nakamura, Y., Yamamoto, T. and Tsai, J. S. Quantum noise in the Josephson charge qubit. http://dx.doi.org/10.1103/PhysRevLett.93.267007 Phys. Rev. Lett. 93, 267007 (2004). § SUPPLEMENTAL MATERIAL §.§ Model for interacting TLS Our experiments on noise and loss indicate that interactions between TLS are important. They also demonstrate that while the spin desorption procedure significantly affects the magnitude of the noise it has only a minor effect on the intrinsic loss tangent. In this section we discuss the full microscopic model of interacting TLS and their physical origin that follows from the data. The fact that surface spin removal has a small effect on the loss tangent implies that the spins do not contribute significantly to the loss at high frequency and thus are not a part of the resonant cTLS ensemble. However, the reduced noise implies that the spins are a significant host of the bath of slow fluctuating dipoles (TLF). The ESR spectrum also indicates that the desorbed spins (both H and free electron states) can be highly mobileand can tunnel a long distance, i.e. they can easily serve as the dominant fraction of slow fluctuating dipoles.Coherent cTLS are associated with fluctuating dipoles d_0 and are described by the Hamiltonian H_TLS=Δ/2σ^z+Δ_0/2σ^x characterized by an asymmetry Δ, tunneling matrix element Δ_0, and σ^a, a=x,y,z are the Pauli matrices. In the rotated basis, the Hamiltonian is simply H_TLS=E S^z, where E=√(Δ^2+Δ_0^2) is the TLS energy splitting and S^z=1/2 (cosθσ^z+sinθσ^x) with tanθ =Δ_0/Δ. The interaction strength is set by the dipole-dipole interaction scale U_0=d_0^2/ε_h, where ε_h=10 is the dielectric constant of the host medium. As a consequence of this interaction, the TLS density at low energies is P_TLS(E,sinθ) = P_0/cosθsinθ (E/E_ max )^μ, where μ<1 is a small positive parameter.Among coherent TLS we distinguish high, E ≫ k_BT, and low E≤ k_BT (thermally activated) energy TLS. In addition, some TLS can be (near) resonant with the resonator, E∼ hν_0.The slow fluctuators are represented by classical fluctuating dipoles with moment d_F, characterized by switching rates γ with aprobability distribution P_F(E,γ)=P_0^F/γ and γ_min≪γ≪γ_max. Such a distribution for the switching rates appears naturally for thermally activated tunneling. The loss in a high quality resonator is caused by fluctuating dipoles with energies close to the resonator frequency ν_0. In the regime of low temperature, k_BT ≪ hν_0, the resonant cTLS have a small dephasing width due to their interaction with thermally activated TLS and TLF. This width is given by<cit.>Γ_2 = ln ( Γ_1^max/Γ_1^min ) χT^1+μ/ν_0^μ,where χ=P_0U_0(ν_0/E_max )^μ≈tanδ_iis a dimensionless parameter, obtained directly from loss tangent measurements, that controls the effect of the interaction on the resonant cTLS. Γ_1^max and Γ_1^min arethe minimum and maximum relaxation rates of these cTLS respectively. Direct measurements give Γ_1^max≈ 10^4s^-1 for the thermally activated cTLS at T ≈ 35mK<cit.>. The precise value of Γ_1^min for thermally activated cTLS is not known. However, the electrical noise data shows that 1/f noise generated by these cTLS extends to very low frequencies f≤ 1mHz beyond which the dependence changes. This implies that Γ_1^min≈ 10^-3 s^-1, such that ln ( Γ_1^max/Γ_1^min ) ≈ 20. The total number of resonant cTLS in a volume V_h (=2.4 and 2.2 · 10^-16 m^3 for the two resonators respectively) of host material can then be estimated from the measured loss tangent as N_res = χ/U_0 V_h Γ_2; their average distance is r_cTLS∼ (χΓ_2/U_0)^-1/3 in bulk material andr_cTLS∼ (d χΓ_2/U_0)^-1/2 in a thin film of thickness d ≪ r_cTLS. The noise in the resonator is due to the slow TLF that interact strongly with these resonant cTLS and create highly non-Gaussian noise, that cannot be regarded as a contribution to Γ_2. These TLF are located at distance r< R_0, where R_0^3= d_F/d_0U_0/Γ_2.Their switchings bring the cTLS in and out of resonance with the resonator leading to 1/f frequency noise. The number of thermally activated TLF strongly coupled to a resonant cTLS is N_F(T)=P_0^F 4 π/3 R_0^3 T and their average distance is r_F ∼ (P_0^F T)^-1/3. If the total number of such fluctuators, N_F^tot(T) = N_res N_F(T) ≫ 1, the frequency noise spectrum of the resonator can be expressed as a superposition of Lorenztians generated by the switching of the TLF strongly coupled to the resonant cTLS. In the limit of weak electric field E⃗we find that the noise is given by<cit.>S_δν/ν_0^2= 8/15⟨ d_0^4 ⟩χ/U_0 Γ_2 F(E⃗)N_F(T)∫_γ_min^γ_maxγ P(γ)/γ^2+ω^2 dγ .Here P(γ)=P_γ/γ is the normalized distribution function of slow fluctuators with P_γ=ln^-1[γ_max/γ_min]andF(E⃗)=∫_V_h |E⃗|^4 dV/4 (∫_Vε_0 |E⃗|^2 dV)^2≈F^2/ε_h^2 V_h where we introduced the filling factor F= ∫_V_hε_h|E⃗|^2 dV/ 2 ∫_Vε_0|E⃗|^2 dV∼ 0.01-0.02 <cit.> which accounts for the fact that the TLS host material volume V_h may only partially fill the resonator mode volume V. We note that the uncertainty in accurately determining F gives a large range for the possible dipole moment ratio d_F/d_0. The ranges given for the quantities in the discussion on d_F in the main manuscript are the values obtained for the estimated range of the filling factor. Notice that in this limit the noise spectrum scales with temperature as ∝ T^-(1+2 μ). Eq. (<ref>) gives the 1/f noise spectrum S_δν/ν_0^2=A_0/2π f. The amplitude A_0 can be expressed through the total number ofthermally activated fluctuators N^tot_F(T) asA_0 ≈πd_F/d_0 N^tot_F(T) [χ F^2 P_γ ](U_0/Γ_2V_h )^2 . §.§ Number of photonsFigure <ref>a shows the number of photons in the 4.6 GHz resonator vs internal loss (Q_i^-1) for the two measurements at two temperatures. Each measurement was made in the same sample cell using the same microwave setup, and the initial assumption is that in the two separate measurements the attenuation in the cryostat was the same. Each data point corresponds to a 2 dB increment in the applied power, both datasets starting at the same low applied power. Therefore, the range of microwave powers applied to the sample is expected to be the same across both measurements. This is further validated by the measurement of white noise levels that are the same (within a factor 2). The white noise level in these measurements is dominated by the microwave power incident on the cryogenic amplifier.The number of photons within the resonator scales with the loaded quality factor, and therefore also with the internal quality factor. As discussed in the main text, the spin desorption leads to an increase in Q_i, meaning that for the same applied microwave power, the number of photons in the resonator is different between the "before" and "after" measurements. Importantly, the noisescales with both the number of photons within the resonator and with Q_i. As is consistent with the literature, we calibrate the applied power such that we compare noise for the same number of photons within the resonator.§.§ Power dependence of Q_iFor consistency we here also provide the analysis of the quality factor data within the framework of the STM. Here we expect at strong fields ⟨ n⟩≫ n_c Q_i^-1 = F tanδ_i /(1+⟨ n⟩/n_c)^α + Q_i,0^-1,where the constant Q_i,0 accounts for power-independent loss and n_c is a critical photon number for saturation, α=0.5 and F is the filling factor of the TLS hosting medium in the resonator. By fitting the measured Q_i(⟨ n⟩) to this power law we find α∼ 0.2 both before and after spin desorption. Typical fits can be seen in Figure <ref>.§.§ ESR-spectrum and spin densityThe ESR-spectrum in Figure <ref> is obtained by measuring the transmitted microwave signal, S_21, around resonance as a function of applied magnetic field using a vector network analyser.All noise measurements were performed first, making sure the resonator was not poisoned by vortices. Once noise measurements were completed, the superconducting magnet leads were connected and a magnetic field applied in the plane of the superconducting film. The measured microwave transmission was fitted to<cit.>S_21 = 1-(1-S_21,min)e^iφ/1+2iQν_0-f/ν_0,to extract the internal quality factor Q_i = Q/S_21,min. The parameter φ accounts for the asymmetry in the resonance line-shape accounting for possible impedance mismatch.The spin-induced loss is then calculated as Q_b^-1(B)= Q_i^-1(B)-Q_i^-1(B=0).We fit the ESR-spectrum to a model of two coupled oscillators to extract the collective coupling, Ω, and line width γ_2 (=1/T_2 for a Lorentzian ESR peak) of the spin system. S_21(ω) = 1+κ_c/i(ω-ω_0)-κ+Ω^2/i(ω-ω_s)-γ_2/2,Eq. <ref> here describes the central g=2 peak only and ω_0=2πν_0 and ω_s = gμ_BB/ħ is the angular resonance frequency and induced Zeeman splitting of the spins respectively, and κ_(c) = ω_0/Q_(c). From the collective coupling Ω we can evaluate the surface spin density based on the geometry of the resonator<cit.>. Comparing the same resonator before and after annealing also gives a direct measure of the relative reduction in spin density independent of resonator geometry via the observed reduction in collective coupling of the spins, Ω∝√(n). In the 'After' measurement we have removed ∼ 2· 10^17 Hydrogen spins/m^2 and the density of g=2 spins was reduced 5.3 times to ∼ 0.17· 10^17 spins/m^2. Figure <ref>e shows the good agreement of the ESR data to theory.We note that the reduction of 5.3 times is larger than previously observed<cit.>, suggesting that the g=2 spins have a larger desorption energy than the hydrogen.§.§ CW power saturation measurements: T_1When evaluating the spin density it is essential to ensure that the spin ensemble is not saturated by the microwave signal in the resonator. To verify this we measure the ESR-spectrum at a wide set of applied powers and extract the dissipation into the spin system at the g=2 peak as a function of circulating power in the resonator. The result for one such measurement (after annealing, evaluated for the g=2 peak) is shown in Figure <ref>a for three different temperatures.The method to evaluate Q_s is described in Ref. <cit.> together with the methodology used to extract the spin relaxation time T_1 plotted versus temperature in Figure <ref>b. Interestingly we find a T^-1 dependence of the relaxation time, a signature of direct spin-lattice relaxation as the dominant mechanism for spin energy relaxation<cit.>. Direct phonon relaxation and a T_1∝ T^-1 dependence is also the dominant mechanism for TLS relaxation in amorphous glasses at low temperatures, well captured by both the STM and the GTM<cit.>, predicting a similar dominating phonon relaxation time in the ms range.We note that spin desorption does not change T_1, while the electron spin dephasing time T_2 inferred from the transition line-width increases marginally (table <ref>), an indication of reduced spin-spin induced decoherence, alternatively the remaining spins could be of a different nature. §.§ Noise measurement setup: Dual Pound lockingThe measurement setup we use is a further development of the Pound locking technique<cit.> for microwave resonators. This modification allows us to simultaneously measure the frequency noise in two different resonators, increasing the amount of data collected and allowing for measurement of correlated noise. Pound locking is a highly accurate technique to directly measure frequency noise of microwave oscillators.This as opposed to measuring the phase noise S_φ using a homo/heterodyne technique<cit.>. The advantage is that we gain in sensitivity and the measurement does not suffer from additional complications such as the Leeson effect, and it is especially useful in cryogenic environments, where homo- or heterodyne techniques suffer from a wide range of fluctuations, such as in electrical length in each of the two measurement paths (signal and reference), and thermal fluctuations. The Pound locking technique instead sends the signal and reference through the same physical transmission line, where the reference takes the form of a phase modulated spectrum on top of the signal, making the measurement insensitive to first order in any variations in electrical length. The phase modulation frequency is recovered by a non-linear detector (here a diode) and any deviations in the signal frequency from the resonator frequency causes a beating at the phase modulation frequency. This beating is nulled using a lock-in in series with a PID controller which adjusts the signal frequency sent out by the microwave generator to match the instant resonance frequency.Instead of a single Pound loop we here run two loops in parallel, as shown in Figure <ref>. Each loop, A andB, works in the same way as described in detail in Ref. <cit.>, locked to the 4.6 and 5.0 GHz resonator respectively. The microwave signals from each loop are combined and sent through the same transmission line in the cryostat, and later selectively split to each arm using 7th order tunable YIG filters with a bandwidth of 40 MHz tuned to each respective resonance frequency. This type of multiplexed setup in principle allows for an arbitrary number of Pound loops in parallel without introducing any cross-coupling and errors in frequency measurement, as long as the YIG filters can selectively isolate the phase modulation spectrum from each measured resonator.The power applied in each Pound loop was carefully verified using a spectrum analyser and adjusted to be equal in the two measurements, both for power incident on the resonators and power incident on the detector diode. §.§ Loss tangent measurements To obtain the loss tangent we measure the frequency shift of each resonator while slowly ramping up the temperature of the cryostat over the course of ∼ 120 minutes. The frequency is measured using the Pound-loop. The loss tangent tanδ_i is then extracted from fits of the ν_0(T) data to the STM (and GTM).δν(T)=F tanδ[ Re(Ψ(1/2+ν_0h/2π i k_BT) +Ψ(1/2+ν_0h/2π i k_BT_0)-lnT/T_0) ].Here δν(T) = (ν_0-ν(T))/ν_0, Ψ is the di-gamma function and T_0 is a reference temperature. The measured data and fits to Eq. (<ref>) are shown in Figure <ref>. Extracted parameters are shown in Table <ref>. §.§ Noise analysis The sampled frequency vs time signal recorded from the Pound loop is converted to frequency noise spectral density S_y by calculating the overlapping Allan-variance σ_y^2(τ) (AVAR) for M discrete samplings f_k(nτ) at multiples n of the sampling rate τ. σ_y^2(nτ) = 1/2(M-1)∑_k=1^M-1(f_k+1-f_k)^2For 1/f noise thespectral density S_y(f) = h_-1/f relates to the Allan variance σ_y^2 = 2ln(2)h_-1 via the coefficient h_-1 <cit.>. The AVAR is evaluated at several time-scales t=nτ ranging from 20 to 80 seconds, well within the 1/f noise limit, and the average value for h_-1 is obtained with high statistical significance. Error bars are calculated from the standard deviation of the multiple evaluations of the AVAR in the same time interval. Each datapoint in Figure <ref> is the result of a 2.8 hours long measurement, collecting 10^5 samples without interruption at a rate τ^-1=10 Hz. Such long measurement times are required to obtain statistically significant results for h_-1 since the 1/f noise in these high-Q resonators is only exceeding the system white noise level at frequencies below ∼ 1-0.1 Hz, in particular at high temperatures and low applied powers and especially in the 'After' measurement where the 1/f noise level is significantly lower. Full data for both resonators measured is shown in Figure <ref>.§.§ Temperature dependence of S_yTo extract μ we measure the temperature dependence of S_y, which in the low power and low temperature limit is expected to scale as S_y(T) ∝ A_μ T^-(1+2μ). The low temperature limit is given by T < hν/k_B≈ 220 mK for our ν = 4.6 GHz resonator. This measurement and fits to extract μ are shown in Figure <ref>. Confidence intervals given for μ are propagated error bars from the calculation of S_y. Indeed we find μ>0 for both measurements and whilst error bars are relatively large, we conclude that interaction is still present and μ has not changed by a significant amount.We do not have data at low enough photon numbers to accurately evaluate μ for the 5 GHz resonator.§.§ Correlated noiseTo rule out external factors, such as system noise, magnetic field or thermal fluctuations, vibrations, and vortices influencing the results we verify that the measured 1/f noise is local to each resonator by measuring their correlated noise. We evaluate the correlated noise as the coherence function from the spectral densitiesC = |S_AB|^2/S_AAS_BB.Here S_XY is the cross-power spectral density S_XY = ∫_-∞^∞ dt e^-iω t∫_-∞^∞ dτν_X(τ) ν_Y(t+τ)of frequency fluctuations ν_A(t) and ν_B(t), where A and B denote the two different resonators. Figure <ref> shows the measured coherence C(0.1Hz) as a function of temperature and for the two extreme powers applied to each resonator, obtained from the same data as in Figure <ref>. We observe no correlations at the time-scale of 0.1 Hz that is relevant for the 1/f noise analysis performed in this work. As another control experiment we measured the coherence while applying a weak (0.02 mT) external magnetic field perpendicular to the superconducting thin-film plane of the sample at a frequency of 0.2 Hz. The measured coherence is very strong at this particular frequency and its higher harmonics, as shown in Figure <ref>. These measurements clearly verify that we have successfully eliminated any common sources of noise and the dominating contribution to the 1/f noise originates from noise sources local to each resonator within the entire measurement space presented in this work.99 supp_faoro2015 Faoro, L. and Ioffe, L. B. Interacting tunneling model for two-level systems in amorphous materials and its predictions for their dephasing and noise in superconducting microresonators. http://dx.doi.org/10.1103/PhysRevB.91.014201Phys. Rev. B 91, 014201 (2015). supp_lisenfeld2016 Lisenfeld, J. et al. Decoherence spectroscopy with individual two-level tunneling defects. http://dx.doi.org/10.1038/srep23786Scientific Reports 6, 23786 (2016). supp_burnett2016 Burnett, J., Faoro, L. and Lindström, T.Analysis of high quality superconducting resonators: consequaneces for TLS properties in amorphous oxides. http://dx.doi.org/10.1088/0953-2048/29/4/044008Supercond. Sci. Technol. 29, 044008 (2016). supp_khalil2013 Khalil, M. S. et al. Evidence for hydrogen two-level systems in atomic layer deposition oxides. http://dx.doi.org/10.1063/1.4826253Appl. Phys. Lett. 103, 162601 (2013). supp_degraaf2017 de Graaf, S. E.,Adamyan, A. A., Lindström, T., Erts, D., Kubatkin, S. E., Tzalenchuk,A. Ya. and Danilov, A. V. Direct identification of dilute surface spins on Al2O3: origin of flux noise in quantum circuits. http://dx.doi.org/Phys. Rev. Lett. 118, 057703 (2017). supp_schweiger A. Schweiger, G. Jeschke, Principles of pulse electron paramagnetic resonance, Oxford University Press (New York 2001). supp_tobiaspound Lindström, T., Burnett, J., Oxborrow, M. andTzalenchuk, A. Ya. Pound-locking for characterization of superconducting microresonators. http://dx.doi.org/10.1063/1.3648134Rev. Sci. Instrum. 82, 104706 (2011). supp_barendsnoisepaper Barends, R., Hortensius, H. L., Zijlstra, T., Baselmans, J. J. A., Yates, S. J. C., Gao, R. J. and Klapwijk, T. M. Noise in NbTiN, Al, and Ta superconducting resonators on silicon and sapphire substrates. http://dx.doi.org/10.1109/TASC.2009.2018086IEEE Trans. Appl. Supercond. 19, 936 (2009). rubiola Rubiola, E. Phase noise and frequency stability in oscillators(Cambridge University Press, 2009). | http://arxiv.org/abs/1705.09158v1 | {
"authors": [
"S. E. de Graaf",
"L. Faoro",
"J. Burnett",
"A. A. Adamyan",
"A. Ya. Tzalenchuk",
"S. E. Kubatkin",
"T. Lindström",
"A. V. Danilov"
],
"categories": [
"cond-mat.mtrl-sci",
"quant-ph"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170525131342",
"title": "Suppression of 1/f noise in solid state quantum devices by surface spin desorption"
} |
[email protected] Department of Electrical, Electronic and Computer Engineering, Gifu University, Gifu 501-1193, Japan Using the tight-binding approach, we analyze the effect of a one-dimensional superlattice (1DSL) potential on the electronic structure of black phosphorene and transition metal dichalcogenides. We observe that the 1DSL potential results in a decrease of the energy band gap of the two-dimensional (2D) materials. An analytical model is presented to relate the decrease in the direct-band gap to the different orbital characters between the valence band top and conduction band bottom of the 2D materials. The direct-to-indirect gap transition, which occurs under a 1DSL potential with an unequal barrier width, is also discussed. Effect of one-dimensional superlattice potentials on the band gap of two-dimensional materials Shota Ono December 30, 2023 ==============================================================================================§ INTRODUCTIONRecently, a wide variety of graphene sister materials, such as black phosphorene (BP) <cit.> and monolayer transition metal dichalcogenides MX_2 (M=Mo, W; X=S, Se, Te) <cit.>, have been synthesized. Furthermore, a recent search on the Materials Project database yielded more than 600 stable two-dimensional (2D) materials that can be synthesized by exfoliation.<cit.> Since some of these 2D materials possess a finite band gap, they are expected to be used in a host of next-generation electronic and optoelectronic devices. Strain engineering has attracted a lot of attention in the context of providing a tunable band gap of 2D materials.<cit.> For example, the early studies, based on the density-functional theory, have shown that the strain can yield a semiconductor-metal transition in BP.<cit.> An analytical study has shown that the band gap increases (decreases) by a few hundred meV when the tensile (compressive) strain lies in the 2D plane.<cit.> In view of the tight-binding (TB) description, the strain application can be interpreted as the hopping parameter modification. How the locally modified on-site potential parameter influences the magnitude of the band gap in 2D materials should then be determined. Thus far, monolayer graphene in a one-dimensional superlattice (1DSL) potential has been extensively investigated based on the Dirac-type <cit.> and TB Hamiltonian.<cit.> The presence of a 1DSL potential has been shown to be able to drastically change the energy band structure around the Dirac cone, yielding electron supercollimation, extra Dirac cones, and an opening of the band gap. As the graphene superlattices have been fabricated by several experiments, where the period of the superlattice structure is from a few nm up to several hundreds of nm,<cit.> a theoretical study for the 1DSL potential effect on the band gap of 2D materials would be important for future applications. Herein, we study the electronic structure of the BP and MX_2 in 1DSL potentials. By using the TB models developed by Rudenko and Katsnelson <cit.> and Liu et al,<cit.> we show that the band gap of the 2D materials is lowered by the 1DSL potential. Through a simple model analysis, the decrease in the direct band gap is shown to be attributed to the different orbital characters between the valence band (VB) top and conduction band (CB) bottom. We also show the direct-to-indirect gap transition that occurs under a 1DSL potential with an unequal barrier width. The models examined below can be applied to situations in which the 1DSL potential is controlled by two (top and back) gates,<cit.> or in which the 2D layer is placed on a substrate with an appropriate lattice mismatch.<cit.> § 2D MATERIAL SUPERLATTICE To study the 1DSL potential effect on the electronic band structure of BP and MX_2, we first consider a conventional unit cell that consists of N atoms. The area of the unit cell is defined as S_1∈ [0,a_x)⊗[0,a_y), as shown in Fig. <ref>(a) and (b) for BP and MX_2, respectively. For example, a_x=4.37 and a_y=3.32 for BP and a_x=3.19 and a_y=√(3)a_x for MoS_2 in units of Å. Next, we create a 1× Q supercell, that is, S_Q ∈ [0,a_x)⊗[0,Qa_y) with a positive integer Q. The position of each atom can be defined as R^s = (R_x^s,R_y^s) with s= 1,⋯, NQ. The 1DSL potential that changes periodically along the y-direction is applied to the 2D materials as v_s =V_0 for0≤ R^s_y < Λ_y2-V_0 forΛ_y2≤ R^s_y < Λ_y,where Λ_y = Qa_y [see Fig. <ref>(f)]. In the following, we set Q=20, unless noted otherwise. The use of a larger Q (and Λ_y) does not change the V_0-dependence of the band gap size, as demonstrated below. The first Brillouin zone (BZ) has a rectangular shape surrounded by four lines k_x = ±π / a_x and k_y = ±π / (Qa_y), as shown in Fig. <ref>(g). For the pristine BP and MX_2 (i.e., the case of V_0 = 0 and Q=1), the minima of the band gaps (i.e., both the CB bottom and VB top) are located at the Γ point and (2π/(3a_x), 0) in the first BZ, respectively. Since these are located along the k_y=0 line, the application of the 1DSL potential, given by Eq. (<ref>) with Q 1, does not change the location of the band gap minimum even if zone foldings occur. Below, we will consider only the electronic band structure along the k_x-direction.For the case of the 1DSL potential that changes periodically along the x-direction, the location of the band gap minimum is also determined with consideration of the zone-folding concept. Similar to the case above, the band gap minimum is still located at the Γ point for the BP superlattices. In contrast, the location of the band gap minimum moves along the k_x-direction with Q for the MX_2 superlattices. By using the zone-folding concept, one can observe that the band gap minimum is located at the Γ point when Q=3i and (± 2π/(3Qa_x), 0) when Q3i with a positive integer i. The most important observation is that the direction of the 1DSL potential does not alter the main result in this work, that is, the band gap reduction arising from the 1DSL potential.It should be noted that when the period of the 1DSL potential, given by Eq. (<ref>), is large (Q≫ 1), the group velocity along the k_y-direction is negligibly small compared with that along the k_x-direction. This is true when we consider the 1DSL potential that changes periodically along the x-direction; given a large period along the x-direction, the group velocity along the k_x-direction is, in turn, negligibly small compared with that along the k_y-direction.For the study of the MX_2 superlattices below, we assume that the 1DSL potential is independent of the orbitals d_z^2, d_xy, and d_x^2-y^2, for simplicity. The application of the orbital-dependent 1DSL potential does not change the main result of this work.§.§ Black Phosphorene First, we study the electronic structure of the BP superlattice. We use the TB model developed by Rudenko and Katsnelson.<cit.> They showed that the BP has a direct band gap at the Γ point, where the VB top and CB bottom consist of a mixture of the s, p_x, and p_z orbitals and have different orbital characters. The TB Hamiltonian of the BP superlattice is given by H_ BP = ∑_s( ϵ_s + v_s ) c_s^† c_s+∑_s s' t_ss' c_s'^† c_s,where the first and second terms in the right hand side are the on-site potential and kinetic energies. c_s and c_s^† are the electron destruction and creation operators at the sth atom site, respectively. t_ss' is the hopping integral between the sites s and s'. These are given by t_14 = -1.220, t_12 = 3.665, t_18 = -0.205, t_13 = -0.105, and t_25 = -0.055 in units of eV [see also Fig. <ref>(a) and (c) for the atom positions].<cit.> The 1DSL potential, v_s defined by Eq. (<ref>), is added to the TB Hamiltonian. Figures <ref>(a)-(c) show 16 dispersion curves around the band edge for V_0 =0, 0.15, and 0.3 eV, respectively. The zero energy is located at the middle of the CB bottom and VB top at the Γ point by adding the on-site potential of ϵ_s = 0.42 eV. When V_0 is increased from 0 to 0.3 eV, the band gap decreases from 1.52 to 0.97 eV. This arises from the charge redistribution within the 1DSL potential period, which will be explained in Secs. <ref> and <ref>. §.§ Transition metal dichalcogenides Next, we compute the band gap of MX_2 superlattice. We use the three-band TB model developed by Liu et al.<cit.> This model is constructed by considering the d-d hoppings between M-d_z^2, d_xy, and d_x^2-y^2 orbitals, where d_z^2 (d_xy, d_x^2-y^2) is the basis of the irreducible representation of A'_1 (E') for the point group D_3h. The pristine MX_2 exhibits a direct band gap at K and K' points. The VB top mainly consists of the d_xy and d_x^2-y^2 orbitals, while the CB bottom consists of the d_z^2 orbital. By using γ and γ' to denote the three atomic orbitals, the Hamiltonian of the MX_2 superlattice is given byH_MX_2 = ∑_s∑_γ(ϵ_γ + v_s) c_s,γ^† c_s,γ+ ∑_s s'∑_γ,γ' T_γ,γ'(R^s' - R^s) c_s',γ'^† c_s,γ,where c_s,γ and c_s,γ^† are the electron destruction and creation operators at the sth atom site with the energy ϵ_γ, respectively. T_γ,γ'(R) is the hopping integral between the orbital γ at the site 0 and the orbital γ' at the site R. The summation of s and s' in Eq. (<ref>) is taken up to the third nearest-neighbor (NN) sites; R=R_i for the first NN sites, R=R'_i = R_i + R_i+1 for the second NN sites, and R=R”_i = 2R_i for the third NN sites, where i=1,2,3,4,5, and 6, with R_7=R_1 [see also Fig. <ref>(d) for the translation vectors R_i]. The expressions of 3×3 matrix T_γ,γ' (R) for 18 NN sites are provided in the Appendix <ref>. Among MX_2, we consider, as an example, the monolayer MoS_2 superlattice. We use the TB parameters of MoS_2 obtained by the density-functional theory calculations within the local-density approximation.<cit.> Then, the values of ϵ_γ with γ = d_z^2 and γ = d_xy, d_x^2-y^2 are set to 0.820 eV and 1.931 eV, respectively. Figures <ref>(a)-(c) show the energy dispersion curves along the k_x-direction of the MoS_2 superlattice for various V_0s. 16 dispersion curves around the band edge are shown. Similar to the case of the BP superlattice, the band gap at k_xa_x = 2π /3 (vertical dotted line) drastically decreases with increasing V_0.§.§ Gap variation Figure <ref> shows the energy gap of the MoS_2 superlattice as a function of the barrier height 2V_0 for various Qs. For comparison, the 2V_0-dependence of the band gap of the BP superlattice is also shown. The energy gap E_g (V_0) decreases monotonically with increasing 2V_0, while the decrease in E_g (V_0) is moderate for small Q. For larger Q, the band gap difference is approximately expressed by the linear relation E_g(V_0) - E_g(0) ≃ - 2V_0.This reflects the fact that the charges are strongly localized to the region with v_s = V_0 and -V_0 at the VB top and CB bottom, respectively. Figures <ref>(a) and <ref>(b) show the R_y-dependence of the charge density at the VB top and CB bottom, respectively. The contributions from d_z^2, d_xy, and d_x^2-y^2 orbitals are shown when Q=20 and V_0=0.5 eV (solid). For comparison, the case of V_0 = 0 eV is also shown (dashed). The charge density at the VB top consists of d_xy and d_x^2-y^2 and is distributed around R_y/Λ_y ≤ 1/2, that is, the region with v_s = V_0 [Fig. <ref>(a)], while that at the CB bottom consists of d_z^2 dominantly and is distributed around R_y/Λ_y > 1/2, that is, the region with v_s = - V_0 [Fig. <ref>(b)]. In such a charge distribution, the eigenenergy would be linearly proportional to the on-site potential energy, resulting in the relation of Eq. (<ref>). Figures <ref>(c) and <ref>(d) show the R_y-dependence of the charge density of the three orbitals at the VB top and CB bottom, respectively, for Q=4. For such a small Q, the charge density is finite even around R_y/Λ_y > 1/2 (R_y/Λ_y ≤ 1/2) at the VB top (the CB bottom). In this case, the magnitude of the upward and downward shifts of the bands is small for the VB and CB, respectively, yielding the slight decrease in the band gap.It should be noted that the semiconductor-metal transition would be observed when E_g(0) = 2V_0 is satisfied in Eq. (<ref>) for large Qs. For example, V_0 ≃ 0.9 and 0.75 eV is needed to observe the transition in MoS_2 and BP, respectively.§ DISCUSSION §.§ Origin of the band gap decrease In Sec. <ref>, it has been shown that the 1DSL potential reduces the magnitude of the direct band gap. We attribute such a reduction to the different orbital characters between the VB top and CB bottom. As mentioned in Secs. <ref> and <ref>, this condition is satisfied in BP and MX_2. To show how the decrease in the direct band gap is explained in terms of the orbital characters at the band edges, we consider, as a simple example, a 2D square net that consists of the atoms shown in Fig. <ref>(e). There is an atom in the unit cell whose size is a_x × a_y. Each atom has an atomic energy of ϵ_γ. We assume that there are two energy levels γ=a and b satisfying the relation ϵ_a< ϵ_b. The sth atom position is denoted by R^s = (R_x^s,R_y^s) = (na_x,ma_y) with integers n and m. The different potentials are added to study the superlattice potential effect: v_s = V_0(>0) and -V_0 for m = 2l and m=2l+1 with an integer l, respectively. The real-space TB Hamiltonian for the 2D square net is given byH_ SQ = ∑_s∑_γ=a^b( ϵ_γ + v_s )c_s,γ^† c_s,γ+ ∑_s s'∑_γ=a^b t_γc_s',γ^†c_s,γ,where t_γ is the electron hopping integrals between the energy levels ϵ_γ at the nearest-neighbor sites. The electron hopping between the energy levels γ=a and γ = b is assumed to be negligible. By imposing the periodic boundary condition to form the energy bands, the TB Hamiltonian in a reciprocal-space becomes a 4×4 matrix:H̃_ SQ(k)= [ h_a^+ 2t_a cosβ 0 0; 2t_a cosβ h_a^- 0 0; 0 0 h_b^+ 2t_b cosβ; 0 0 2t_b cosβ h_b^- ],where, for γ=a and b,h_γ^+ = ϵ_γ + V_0 +2t_γcosα, h_γ^- = ϵ_γ - V_0 +2t_γcosα,with α = k_x a_x and β = k_y a_y being the wavevector k = (k_x,k_y). The energy eigenvalues are given byE_γ^± (k)= ϵ_γ +2t_γcosα±√(V_0^2 + 4t_γ^2cos^2 β).Below, we use the indexes (γ,±) to denote the energy band. We assume t_a > 0 and t_b<0, since we study the band structure with semiconducting properties. Then, both the energy maximum of the band (a,+) and the energy minimum of the band (b,-) are located at the Γ point. The energy difference Δ E is explicitly given byΔ E=E_b^- (k=0) - E_a^+ (k=0)= ϵ_b - ϵ_a+ 2(t_b - t_a)- √(V_0^2 + 4t_b^2) - √(V_0^2 + 4t_a^2).When Δ E >0, the bands (a,+) and (b,-) serve as the VB and CB, respectively. From Eq. (<ref>), it is clear that the energy gap Δ E decreased with increasing V_0. This is because the charges are redistributed to obtain the energy gain. The eigenvectors of the VB top [i.e., at the Γ point in the (a,+) band] and the CB bottom [i.e., at the Γ point in the (b,-) band] are respectively given by[ C_a^+; C_a^-; C_b^+; C_b^- ] = 2t_a√(w_a,-^2 + 4t_a^2)[1; w_a,-/(2t_a);0;0 ] and [ C_a^+; C_a^-; C_b^+; C_b^- ] = 2t_b√(w_b,+^2 + 4t_b^2)[0;0;1; - w_b,+/(2t_b) ], where w_γ,p = √(V_0^2 + 4t_γ^2) +p V_0 with p=± and γ = a,b. C_γ^+ and C_γ^- are the probability amplitude of the orbital γ at the sites m=2l and m=2l+1, respectively. Since w_a,-/(2t_a) <1 and - w_b,+/(2t_b) >1 for finite V_0, the charges at the VB top and CB bottom are mainly distributed at the sites m=2l (v_s = V_0) and m=2l+1 (v_s=-V_0), respectively. As a result, the (a,+) and (b,-) bands shift to higher and lower energies, respectively, which leads to the decrease in the band gap. Thus, we determined the relationship between the band gap and the orbital difference at the band edge. Although the 2D square net model above is quite simple, it captures the main physics behind the band gap reduction observed in Figs. <ref> and <ref>.It is possible to construct a TB model that includes more than two atoms in a unit cell, for example, v_s = V_0 for m=4l and 4l+1, and v_s= - V_0 for m=4l+2 and 4l+3 with an integer l. In such a case, the magnitude of the band gap decreases more significantly because more charges are redistributed within the unit cell. This is also consistent with the observation in Fig. <ref>. §.§ Direct-to-indirect gap transition While we have also studied other types of 1DSL potentials, such as a cosine-type potential, where the net total potential per unit cell vanishes, the results in this work are qualitatively the same. When the net total potential per unit cell is finite, the system exhibits an indirect band gap. To show the latter, we study the MoS_2 under a 1DSL potential with unequal barrier width. The 1DSL potential used is given by v_s =V_0 for0≤ R^s_y < Λ_y10-V_0 forΛ_y10≤ R^s_y < Λ_y.Figure <ref> shows the band structure of the MoS_2 under a 1DSL potential given by Eq. (<ref>) with Λ_y = 20a_y for various V_0s. As V_0 increases, the band gap decreases, similar to the cases of the 1DSL potential with an equal barrier width shown in Fig. <ref>. Interestingly, when V_0 ≥ 0.5 eV, the energy of the VB top at k_x a_x=0 overcomes that at k_x a_x=2π/3, yielding the indirect band gap. This originates from the different orbital characters of the VB top between k_xa_x = 0 and k_xa_x = 2π /3 in the pristine MoS_2, where the former mainly consists of a Mo d_z^2 orbital, while the latter consists of Mo d_xy and d_x^2-y^2 orbitals.<cit.> In addition, since the charge of the CB bottom at k_xa_x = 2π /3 is distributed around the region with v_s = -V_0, as shown in Fig. <ref>(b), the charge of the VB top at k_x a_x=0 tends to be, in turn, distributed around the region with v_s = +V_0 (i.e., R^s_y < Λ_y/10). This leads to an increase in the energy of the VB top at k_xa_x = 0, compared with that at k_xa_x = 2π /3. § CONCLUSIONSThrough the TB calculations for the BP and MX_2 superlattices, we have demonstrated that the presence of a 1DSL potential yields a decrease in the band gap of 2D materials. An analytical investigation shows that this also holds if a pristine 2D material has (i) a direct-band gap and (ii) different orbital characters between the VB top and the CB bottom. It has also been found that the band gap experiences a direct-to-indirect gap transition when a 1DSL potential with an unequal barrier width is applied. We expect that various 2D material superlattices will be created in future experiments, as the graphene superlattices are fabricated by several experiments.<cit.> SO would like to thank M. Aoki for fruitful discussions and G.-B. Liu and D. Xiao for providing useful information about the TB model of MX_2. This study was supported by a Grant-in-Aid for Young Scientists B (No. 15K17435) from JSPS. § HOPPING INTEGRALS The matrix elements T_γ,γ'(R) in Eq. (<ref>) can be expressed by 17 parameters.<cit.> By using the notation (d_z^2, d_xy, d_x^2 -y^2) = (1,2,3), the 17 hopping parameters for the first NN sites are explicitly given as: t_0=T_1,1(R_1),t_1=T_1,2(R_1),t_2=T_1,3(R_1),t_11=T_2,2(R_1),t_12=T_2,3(R_1),t_22=T_3,3(R_1),for the second NN sites:r_0=T_1,1(R'_1),r_1=T_1,2(R'_1),r_2=T_1,2(R'_4),r_11=T_2,2(R'_1),r_12=T_2,3(R'_1),and for the third NN sites:u_0=T_1,1(R”_1), u_1=T_1,2(R”_1), u_2=T_1,3(R”_1),u_11=T_2,2(R”_1), u_12=T_2,3(R”_1), u_22=T_3,3(R”_1).The other matrix elements are obtained with the aid of the symmetry property. Those are explicitly written as, for the first nearest-neighbor (NN) sites R_i,T (R_1) =[t_0t_1t_2; -t_1 t_11 t_12;t_2 - t_12 t_22 ], T (R_2) = [ t_02c t_1 - 2dt_2 - 2dt_1 -2 ct_2;- 2c t_1 -2 dt_2ct_11 + 3ct_22 -dt_11 - t_12 + dt_22; 2dt_1 - 2ct_2 -dt_11 + t_12 + dt_223ct_11 + ct_22 ], T (R_3) = [t_0 - 2c t_1 + 2dt_2- 2dt_1 -2 ct_2;2c t_1 + 2 dt_2 ct_11 + 3ct_22 dt_11 + t_12 - dt_22; 2 dt_1 -2 ct_2 dt_11 - t_12 - dt_22 3ct_11 + ct_22 ], T (R_4) =[t_0- t_1t_2;t_1 t_11 - t_12;t_2 t_12 t_22 ], T (R_5) = [t_0 - 2c t_1 - 2dt_22dt_1 -2 ct_2; 2c t_1 - 2dt_2 ct_11 + 3ct_22 - dt_11 + t_12 + dt_22;- 2dt_1 -2 ct_2 - dt_11 - t_12 + dt_22 3ct_11 + ct_22 ], T (R_6) = [t_0 2c t_1 + 2dt_22dt_1 -2 ct_2; - 2c t_1 + 2dt_2 ct_11 + 3ct_22 dt_11 - t_12 - dt_22;- 2dt_1 -2 ct_2 dt_11 + t_12 - dt_22 3ct_11 + ct_22 ],and, for the next NN site R'_i = R_i + R_i+1 with R_7=R_1: T (R'_1) = [r_0r_1 -e r_1;r_2 r_11 r_12;- e r_2 r_12 r_11 + 2e r_12 ],T (R'_2) = [ r_0 02e r_1; 0 r_11 + r_12/e 0;2e r_1 0 r_11 - e r_12 ],T (R'_3) = [r_0- r_1 -e r_1;- r_2 r_11 - r_12;- e r_2 - r_12 r_11 + 2e r_12 ],T (R'_4) = [ r_0 r_2 - e r_2; r_1r_11r_12; - e r_1r_12 r_11 + 2 e r_12 ],T (R'_5) = [ r_0 0 2 e r_1; 0 r_11 + r_12/e 0; 2 e r_2 0 r_11 - e r_12 ],T (R'_6) = [ r_0 - r_2 - e r_2; - r_1r_11- r_12; - e r_1- r_12 r_11 + 2 e r_12 ], with c=1/4, d=√(3)/4 and e= 1/√(3). 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"authors": [
"Shota Ono"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170526093140",
"title": "Effect of one-dimensional superlattice potentials on the band gap of two-dimensional materials"
} |
http://arxiv.org/abs/1705.09317v1 | {
"authors": [
"G. Puglisi",
"D. De Tommasi",
"M. F. Pantano",
"N. Pugno",
"G. Saccomandi"
],
"categories": [
"physics.bio-ph"
],
"primary_category": "physics.bio-ph",
"published": "20170525183109",
"title": "A predictive model for protein materials: from macromolecules to macroscopic fibers"
} |
|
Magnetic Evolution and the Disappearance of Sun-like Activity Cycles [ December 30, 2023 ====================================================================A community's identity defines and shapes its internal dynamics. Our current understanding of thisinterplayis mostly limited to glimpses gathered from isolated studies of individual communities. In this work we provide a systematic exploration of the nature of this relation across a wide variety of online communities. To this end we introduce a quantitative, language-based typology reflecting two key aspects of a community's identity: how distinctive, and how temporallydynamic it is. By mapping almost 300 Reddit communitiesinto the landscape induced by this typology, we reveal regularities in how patterns of user engagement vary with the characteristics of a community. Our results suggest thatthe waynew and existing users engage with a community depends strongly and systematically on the nature of the collective identity it fosters, in ways that are highly consequential to community maintainers.For example, communities with distinctive and highly dynamic identities are more likely to retain their users.However, such niche communities also exhibit much larger acculturation gaps between existing users and newcomers, which potentially hinder the integration of the latter. More generally, our methodology reveals differences in how various social phenomena manifest across communities, and shows that structuring the multi-community landscape can lead to a better understanding of the systematic nature of this diversity.§ INTRODUCTION“If each city is like a game of chess, the day when I have learned the rules, I shall finally possess my empire, even if I shall never succeed in knowing all the cities it contains.”— Italo Calvino,Invisible Cities A community's identity—defined through the common interests and shared experiences of its users—shapes various facets of the social dynamics within it <cit.>.Numerous instances of this interplay between a community's identity and social dynamics have been extensively studied in the context of individual online communities <cit.>.However, the sheer variety of online platforms complicates the task of generalizing insights beyond these isolated, single-community glimpses.A new way to reason about the variation across multiple communities is needed in order to systematically characterize the relationship betweenproperties of a community and the dynamics taking place within. One especially important component of community dynamics is userengagement.We can aim tounderstand why users join certain communities <cit.>, what factors influence user retention <cit.>, and how users react to innovation <cit.>. While striking patterns of user engagement have been uncovered in prior case studies ofindividualcommunities <cit.>,we do not know whether these observations hold beyond these cases, or when we can draw analogies between different communities.Are there certain types of communities where we can expect similar or contrasting engagement patterns?To address such questions quantitativelywe need to providestructure tothe diverse and complexspace of online communities. Organizing the multi-community landscape would allow us to bothcharacterizeindividual points within this space, and reason aboutsystematic variations in patternsof user engagementacross the space. Present work: Structuring the multi-community space In order to systematically understand the relationship between community identity[We use “community identity” and “collective identity" interchangeably to refer to the shared definition of a group, derived from members' common interests and shared experiences. We are not directly concerned with the more sociopolitical and psychological connotations of these terms <cit.>.] and user engagement we introduce a quantitative typology of online communities. Our typology is based on two key aspects of :how distinctive—or niche—a community's interests are relative to other communities, and how dynamic—or volatile—these interests are over time. These axes aim to capturethe salience of a community's identity and dynamics of its temporal evolution. Our main insight in implementing this typology automatically and at scale is that the language used within a community cansimultaneously capture howdistinctive and dynamic its interests are.This language-basedapproach draws on a wealth of literature characterizing linguistic variation in online communities and its relationship to community and user identity <cit.>.Basing our typology on language is also convenient since it renders our framework immediately applicable to a wide variety of online communities, where communication is primarily recorded in a textual format.Using our framework, we map almost 300 Reddit communities onto the landscape defined by the two axes of our typology (Section <ref>).We find that this mapping induces conceptually sound categorizations that effectively capture key aspects of community-level social dynamics.In particular, we quantitatively validate the effectiveness of ourmappingby showing that our two-dimensional typologyencodes signals that are predictive of community-level rates of user retention, complementing strong activity-based features. Engagement and community identity We apply our framework to understand how two important aspects of user engagement in a community—the community's propensity to retain its users(Section <ref>), and its permeability to new members (Section <ref>)—vary according to the type of collective identity it fosters.We find that communities that are characterized by specialized, constantly-updatingcontent have higher user retention rates,but also exhibitlarger linguistic gaps that separate newcomers from established members. More closely examining factors that could contribute to this linguistic gap, we find that especially within distinctive communities, established users have an increased propensity to engage with the community's specialized content, compared to newcomers (Section <ref>).Interestingly, while established members of distinctive communities more avidly respond to temporal updates than newcomers, in more generic communities it is the outsiders who engage more with volatile content, perhaps suggesting that such content may serve as an entry-point to the community (but not necessarily a reason to stay). Such insights into the relation between collective identity and user engagement can be informative to community maintainers seeking to better understand growth patterns within their online communities.More generally, our methodology stands as an example of how sociological questions can be addressed in a multi-community setting. In performing our analyses across a rich variety of communities, we reveal both the diversity of phenomena that can occur,as well as the systematic nature of this diversity.§ A TYPOLOGY OF A community's identity derives from its members' common interests and shared experiences <cit.>. In this work, we structure the multi-community landscape along thesetwo key dimensions of community identity: how distinctive a community's interests are, and how dynamic the community is over time. We now proceed to outline our quantitative typology, which maps communities along these two dimensions. We start by providing an intuition through inspecting a few example communities. We then introduce a generalizable language-based methodology and use it to map a large set of Reddit communitiesonto the landscape defined by our typology of community identity.§.§ Overview and intuition In order to illustrate the diversity within the multi-community space, andto provide anintuition for the underlyingstructure captured by the proposed typology, we first examine a few example communities and draw attention to some key social dynamics that occur within them.We consider four communities from Reddit: in Seahawks, fans of the Seahawks football team gather to discuss games and players; in BabyBumps, expecting mothers trade advice and updates on their pregnancy; Cooking consists of recipe ideas and general discussion about cooking; while in pics, users share various images of random things (like eels and hornets). We note that these communities are topically contrasting and foster fairly disjoint user bases. Additionally, these communities exhibit varied patterns of user engagement. While Seahawks maintains a devoted set of users from month to month, pics is dominated by transient users who post a few times and then depart. Discussions within these communities also span varied sets of interests. Some of these interestsare more specific to the community than others: risotto, for example, is seldom a discussion point beyond Cooking. Additionally, some interests consistently recur, while others are specific to a particular time: kitchens are a consistent focus point for cooking, but mint is only in season during spring.Coupling specificity and consistency we find interests such as easter, which isn't particularly specific to BabyBumps but gains prominence in that community around Easter (see Figure <ref>.A for further examples).These specific interests provide a window into the nature of the communities' interests as a whole, and by extension their community identities.Overall, discussions in Cooking focus on topics which are highly distinctive and consistently recur (like risotto). In contrast, discussions in Seahawks are highly dynamic,rapidly shifting over time as new games occur and players are traded in and out.In the remainder of this section we formally introduce a methodology for mapping communities in this space defined by their distinctiveness and dynamicity (examples in Figure <ref>.B).§.§ Language-based formalizationOur approach follows the intuition that a distinctive community will use language that is particularly specific, or unique, to that community.Similarly, a dynamic community will use volatile language that rapidly changes across successive windows of time. To capture this intuition automatically,we start by defining word-level measures of specificity and volatility. We then extend these word-level primitives to characterize entire comments, and the community itself. Our characterizations of words in a community are motivated by methodology from prior literature that compares the frequency of a word in a particular setting to its frequency in some background distribution, in order to identify instances of linguistic variation <cit.>. Our particular framework makes this comparison by way of pointwise mutual information (PMI).In the following, we use c to denote one community within aset𝒞 of communities, and t to denote one time period within the entire history T of 𝒞. We account for temporal as well as inter-community variationby computingword-level measures for each time period of each community's history, c_t. Given a word w used within a particular community c at time t, we define two word-level measures: Specificity We quantify the specificity _c (w) of w to c by calculating the PMI of w and c, relative to 𝒞,_c (w)=log_c (w)/_𝒞 (w), where _c(w) is w's frequency in c. w is specific to c if it occurs more frequently in c than in the entire set 𝒞, hence distinguishing this community from the rest. A word w whose occurrence is decoupled from c, and thus has_c (w) close to 0, is said to begeneric.We compute values of_c_t (w)for each time period t in T; in the above description we drop the time-based subscripts for clarity.Volatility We quantify the volatility _c_t (w) of w to c_t as the PMI of w and c_t relative to c_T, the entire history of c:_c_t (w)=log_c_t (w)/_c_T (w).A word w is volatile at time t in c if it occurs more frequently at t than in the entire history T, behaving as a fad within a small window of time. A word that occurs with similar frequency across time, and hence hasclose to 0, is said to be stable. Extending to utterances Using our word-level primitives, we define the specificity of an utterance d in c, _c (d) as the average specificity of each word in the utterance.The volatility of utterances is defined analogously. §.§ Community-level measuresHaving described these word-level measures, we now proceed to establish the primary axes of our typology:DistinctivenessA community with a very distinctive identity will tend to have distinctive interests, expressed through specialized language.Formally, we define the distinctiveness of a community (c_t) as the average specificity of all utterances in c_t. We refer to a community with a less distinctive identity as being generic.DynamicityA highly dynamic community constantly shifts interests from one time window to another, and these temporal variations are reflected in its use of volatile language. Formally, we define the dynamicity of a community (c_t) as the average volatility of all utterances in c_t. We refer to a community whose language is relatively consistent throughout time as being stable. In our subsequent analyses, we focus mostly on examing the average distinctiveness and dynamicity of a community over time, denoted (c) and (c).§.§ Applying the typology to Reddit We now explain how our typology can beapplied to the particular setting of Reddit, and describe the overall behaviour of our linguistic axes in this context.Dataset description Reddit is a popular website where users form and participate in discussion-based communities called subreddits. Within these communities, users post content—such as images, URLs, or questions—which often spark vibrant lengthy discussions in thread-based comment sections.The website contains many highly active subreddits withthousands of active subscribers. These communities span an extremely rich variety of topical interests, as represented by the examples described earlier.They also vary along a rich multitude of structural dimensions, such as the number of users, the amount of conversation and social interaction,and thesocial norms determining which types of content become popular. The diversity and scope of Reddit's multicommunity ecosystem make it an ideal landscape in which to closely examine the relation between varying community identities and social dynamics.Our full dataset consists of all subreddits on Reddit from January 2013 to December 2014,[<https://archive.org/details/2015_reddit_comments_corpus>]for which there are at least 500 words in the vocabulary used to estimate our measures, in at least 4 months of the subreddit's history. We compute our measures over the comments written by users in a community in time windows of months, for each sufficiently active month, and manually remove communities where the bulk of the contributions are in a foreign language. This results in 283 communities (c), for a total of4,872community-months (c_t).[While we chose these cutoffs on the dataset to ensure robust estimates of the linguistic measures, we note that slight relaxations produce qualitatively similar results in the later analyses.] Estimating linguistic measures We estimate word frequencies_c_t(w),and by extension each downstream measure, in a carefully controlled manner in order to ensure we capture robust and meaningful linguistic behaviour.First, we only consider top-level comments which are initial responses to a post, as the content of lower-level responses might reflect conventions of dialogue more than a community's high-level interests. Next, in order to prevent a few highly active users from dominating our frequency estimates, we count each unique word once per user, ignoring successive uses of the same wordby the same user. This ensures that our word-level characterizations are notskewed by a small subset of highly active contributors.[Understanding the role that highly active users <cit.> playin shaping a community's dynamics is an interesting direction for future work.]In our subsequent analyses, we will only look at these measures computed over the nouns used in comments.In principle, our framework can be applied to any choice of vocabulary. However, in the case of Reddit using nouns provides a convenient degree of interpretability. We can easily understand the implication of a community preferentially mentioning a noun such as gamer or feminist, but interpreting the overuse of verbs or function words such as take or of is less straightforward. Additionally, in focusing on nouns we adopt the view emphasized in modern “third wave” accounts of sociolinguistic variation,that stylistic variation is inseparable from topical content <cit.>. In the case of online communities, the choice of what people choose to talk about serves as a primary signalofsocial identity. That said, a typology based on more purely stylistic differences is an interesting avenue for future work.Accounting for rare words One complication when using measures such as PMI, which are based off of ratios of frequencies, is that estimates for very infrequent words could be overemphasized <cit.>. Words that only appear a few times in a community tend to score at the extreme ends of our measures (e.g. as highly specific or highly generic), obfuscating the impact of more frequent words in the community.To address this issue, we discard the long tail of infrequent words in our analyses, using only the top 5th percentile of words, by frequency within each c_t, to score comments and communities.[For the purposes of the present analyses, this method produces reasonable output that is robust to small variations in our choice of parameters. However, it would be fruitful in future work to consider other methods, e.g., <cit.>, for capturing linguistic variation.]Typology output on RedditThe distribution ofandacross Reddit communities is shown in Figure <ref>.B, along with examples of communities at the extremes of our typology. We find thatinterpretable groupings of communities emerge at various points within our axes. For instance, highly distinctive and dynamic communities tend to focus on rapidly-updating interests like sports teams and games, while generic and consistent communities tend to be large “link-sharing” hubs where users generally post content with no clear dominating themes. More examples of communities at the extremes of our typology are shown in Table <ref>. We note that these groupings capture abstract properties of a community's content that go beyond its topic. For instance, our typology relates topically contrasting communities such as yugioh (which is about a popular trading card game) and Seahawks through the shared trait that their content is particularly distinctive. Additionally, the axes can clarify differences between topically similar communities: while startrek and thewalkingdead both focus on TV shows, startrek is less dynamic than the median community, while thewalkingdead is among the most dynamic communities, as the show was still airing during the years considered.§ COMMUNITY IDENTITY AND USER RETENTIONWe have seen that our typology produces qualitatively satisfying groupings of communities according to the nature of their collectiveidentity.This sectionshows that there is an informative and highly predictive relationship between a community's position in this typology andits user engagement patterns.We find that communities with distinctive and dynamic identities have higher rates of user engagement, and further show that a community's position in our identity-based landscape holds important predictive information that is complementary to a strong activity baseline.In particular user retention is one of the most crucial aspects of engagement and is critical to community maintenance <cit.>. We quantify how successful communities are at retaining users in terms of both short and long-term commitment.Our results indicate that rates of user retention vary drastically,yet systematically according to how distinctive and dynamic a community is (Figure <ref>). We find a strong,explanatory relationship between the temporal consistency of a community's identity and rates of user engagement: dynamic communities that continually update and renew their discussion content tend to have far higher rates of user engagement.The relationship between distinctiveness and engagement is less universal, but still highly informative: niche communities tend to engender strong, focused interest from users at one particular point in time, though this does not necessarily translate into long-term retention. §.§ Community-type and monthly retentionWe find that dynamic communities, such as Seahawks or starcraft, have substantially higher rates of monthly user retention than more stable communities(Spearman's ρ = 0.70, p<0.001, computed with community points averaged over months; Figure <ref>.A, left).Similarly, more distinctive communities, like Cooking and Naruto,exhibit moderately higher monthly retention rates than more generic communities (Spearman's ρ = 0.33, p<0.001; Figure <ref>.A, right).Monthly retention is formally defined as the proportion of users who contribute in month t and then return to contribute again in month t+1.Each monthly datapoint is treated as unique and the trends in Figure <ref> show 95% bootstrapped confidence intervals, cluster-resampled at the level of subreddit <cit.>, to account for differences in the number of months each subreddit contributes to the data. Importantly, we find that in the task of predicting community-level user retention our identity-based typology holds additional predictive valueon top of strong baseline features based on community-size (# contributing users) and activity levels (mean # contributions per user),which are commonly used for churn prediction <cit.>. We compared out-of-sample predictive performance via leave-one-community-out cross validation using random forest regressors with ensembles of size 100, and otherwise default hyperparameters <cit.>. A model predicting average monthly retention based on a community's average distinctiveness and dynamicity achieves an average mean squared error(MSE)of 0.0060 and R^2=0.37,[We measure out-of-sample R^2 relative to a baseline that predicts the mean of the training data <cit.>.]while an analogous model predicting based on a community's size and average activity level (both log-transformed) achieves MSE=0.0062 and R^2=0.35. The difference between the two models is not statistically significant(p=0.99, Wilcoxon signed-rank test). However, combining features from both models results in a large and statistically significant improvement over each independent model (MSE=0.0038,R^2=0.60, p<0.001 Bonferroni-corrected pairwise Wilcoxon tests). These results indicate that our typology can explain variance in community-level retention rates, and provides information beyond what is present instandard activity-based features. §.§ Community-type and user tenureAs withmonthlyretention, we find a strong positive relationship between a community's dynamicity and theaverage number of months that a user will stay in that community (Spearman's ρ = 0.41, p<0.001, computed over all community points; Figure <ref>.B, left). This verifies that the short-term trend observed for monthly retention translates into longer-term engagement and suggests that long-term user retentionmightbe strongly driven by the extent to which a community continually provides novel content.Interestingly, there is no significant relationship between distinctiveness and long-term engagement (Spearman's ρ = 0.03, p= 0.77; Figure <ref>.B, right). Thus, while highly distinctive communities like RandomActsOfMakeup may generate focused commitment from users over a short period of time, such communities are unlikely to retain long-term users unless they also have sufficiently dynamic content. To measure user tenures we focused on one slice of data (May, 2013) and measured how many months a user spends in each community, on average—the average number of months between a user's first and last comment in each community.[Analogous results hold for other reasonable choices of month.] We have activity data up until May 2015, so the maximum tenure is 24 months in this set-up, which is exceptionally long relative to the average community member (throughout our entire data less than 3% of users have tenures of more than 24 months in any community). § COMMUNITY IDENTITY AND ACCULTURATION The previous sectionshowsthat there is a strong connection between the nature of a community's identity andits basic user engagement patterns. In this section, we probe the relationship between a community's identity and how permeable, or accessible, it is to outsiders.We measure this phenomenon using what we call the acculturation gap, whichcomparesthe extent to which engaged vs. non-engaged users employ community-specific language.While previous work has found this gap to be large and predictive of future user engagement in two beer-review communities <cit.>, we find that the size of the acculturation gap depends strongly on the nature of a community's identity, with the gap being most pronounced in stable, highly distinctive communities (Figure <ref>).This finding has important implications for our understanding of online communities.Though many works have analyzed the dynamics of “linguistic belonging” in online communities <cit.>, our results suggest that the process of linguistically fitting in is highly contingent on the nature of a community's identity. At one extreme, in generic communities like pics or worldnews there is no distinctive, linguistic identity for users to adopt.To measure the acculturation gap for a community, we follow Danescu-Niculescu-Mizil et al danescu-niculescu-mizil_no_2013 and build “snapshot language models” (SLMs) for each community, which capture the linguistic state of a community at one point of time.[We use Katz-Backoff bigram language models with Good-Turing smoothing <cit.> and vocabularies of size 50,000.] Using these language models we can capture how linguistically close a particular utterance is to the community by measuring the cross-entropy of this utterance relative to the SLM:H(d, _c_t) = 1/|d|∑_b_i ∈ d_ c_t(b_i),where _c_t(b_i) is the probability assigned to bigram b_i from comment d in community-month c_t.We build the SLMs by randomly sampling 200 active users—defined as users with at least 5 comments in the respective community and month. For each of these 200 active users we select 5 random 10-word spans from 5 unique comments.[Using fixed-length spans controls for spurious length-effects <cit.>; the same controls are used in the cross-entropy calculations.] To ensure robustness and maximize data efficiency, we construct 100 SLMs for each community-month pair that has enough data, bootstrap-resampling from the set of active users.We compute a basic measure of the acculturation gap for a community-month c_t as the relative difference of the cross-entropy of comments by users active in c_t with that of singleton commentsby outsiders—i.e., users who only ever commented once in c, but who are still active[Users must comment at least 5 times in a month to be considered active in Reddit.] in Reddit in general:A(c_t) = 𝔼_d ∼𝒱_s[H(d,_c_t)]-𝔼_d ∼𝒱_a[H(d,_c_t)]/𝔼_d ∼𝒱_a[H(d,_c_t)].𝒱_s denotes the distribution over singleton comments, 𝒱_a denotes the distribution over comments from users active in c_t, and 𝔼 the expected values of the cross-entropy over these respective distributions.For each bootstrap-sampled SLM we compute the cross-entropy of 50 comments by active users (10 comments from 5 randomly sampled active users, who were not used to construct the SLM) and 50 comments from randomly-sampled outsiders.Figure <ref>.A shows that the acculturation gap varies substantially with how distinctive and dynamic a community is. Highly distinctive communities have far higher acculturation gaps, while dynamicity exhibits a non-linear relationship: relatively stable communities have a higher linguistic `entry barrier', as do very dynamic ones.Thus, in communities like IAmA (a general Q&A forum) that are very generic, with content that is highly, but not extremely dynamic, outsiders are at no disadvantagein matching the community's language. In contrast, the acculturation gap is large in stable, distinctive communities like Cooking that have consistent community-specific language. The gap is also large in extremely dynamic communities like Seahawks,which perhaps require more attention or interest on the part of active users to keep up-to-date with recent trends in content. These results show that phenomena like the acculturation gap,which were previously observed in individual communities <cit.>, cannot be easily generalized to a larger, heterogeneous set of communities. At the same time, we see that structuring the space of possible communities enables us to observe systematic patterns in how such phenomena vary. § COMMUNITY IDENTITY AND CONTENT AFFINITY Through the acculturation gap, we have shown that communities exhibit large yet systematic variations in their permeability to outsiders. We now turn to understanding the divide in commenting behaviour between outsiders and active community members at a finer granularity, by focusing on two particular ways in which such gaps might manifest among users: through different levels of engagement with specific content and with temporally volatile content. Echoing previous results, we find that community type mediates the extent and nature ofthe dividein content affinity.While in distinctive communities active members have a higher affinity for both community-specific content and for highly volatile content, the opposite is true for generic communities, where it is the outsiders who engage more with volatile content. We quantify these divides in content affinity by measuring differences in the language of the comments written by active users and outsiders.Concretely, for each community c, we define the specificity gap Δ_c as the relative difference between the average specificity of comments authored by active members, and by outsiders, where these measures are macroaveraged over users. Large, positive Δ_c then occur in communities where active users tend to engage with substantially more community-specific content than outsiders. We analogously define the volatility gap Δ_c as the relative difference in volatilities of active member and outsider comments. Large, positive values of Δ_c characterize communities where active users tend to have more volatile interests than outsiders, while negative values indicate communities where active users tend to have more stable interests. We find that in 94% of communities, Δ_c > 0, indicating (somewhat unsurprisingly) that in almost all communities, active users tend to engage with more community-specific content than outsiders. However, the magnitude of this divide can vary greatly:for instance, in Homebrewing, which is dedicated to brewing beer, the divide is very pronounced (Δ_c = 0.33) compared to funny, a large hub where users share humorous content (Δ_c = 0.011).The nature of the volatility gap is comparatively more varied. In Homebrewing (Δ_c = 0.16), as in 68% of communities, active users tend to write more volatile comments than outsiders (Δ_c > 0). However, communities like funny (Δ_c = -0.16), where active users contribute relatively stable comments compared to outsiders (Δ_c < 0), are also well-represented on Reddit. To understand whether these variations manifest systematically across communities, we examine the relationship betweendivides in content affinity and community type. In particular, following the intuition thatactive users have a relatively high affinity for a community's niche, weexpectthat the distinctiveness of a community will be a salient mediator of specificity and volatility gaps. Indeed, we find a strong correlation between a community's distinctiveness and its specificity gap (Spearman's ρ = 0.34, p < 0.001). We also find a strong correlation between distinctiveness and community volatility gaps (Spearman's ρ = 0.53, p < 0.001). In particular, we see that among the most distinctive communities (i.e., the top third of communities by distinctiveness),active users tend to write more volatile comments than outsiders (mean Δ_c = 0.098), while across the most generic communities (i.e., the bottom third), active users tend to write more stable comments (mean Δ_c = -0.047, Mann-Whitney U test p < 0.001). The relative affinity of outsiders for volatile content in these communitiesindicates that temporally ephemeral content might serveas an entry point into such a community, without necessarily engaging users in the long term. § FURTHER RELATED WORK Our language-based typology and analysis of user engagement draws on and contributes to several distinct research threads, in addition to the many foundational studies cited in the previous sections. Multicommunity studiesOur investigation of user engagement in multicommunity settings follows prior literature which has examined differences in user and community dynamics across various online groups, such as email listservs.Such studies have primarily related variations in user behaviour to structural features such as group size and volume of content <cit.>. In focusing on the linguistic content of communities, we extend this research by providing a content-based framework through which user engagement can be examined.Reddit has been a particularly useful setting for studying multiple communities in prior work. Such studies have mostly focused on characterizing how individual users engage across a multi-community platform <cit.>, or on specific user engagement patterns such as loyalty to particular communities <cit.>.We complement these studies by seeking to understand how features of communities can mediate a broad array of user engagement patterns within them. Typologies of online communities Prior attempts to typologize online communities have primarily been qualitative and based on hand-designed categories, making them difficult to apply at scale.These typologies often hinge on having some well-defined function the community serves, such as supporting a business or non-profit cause <cit.>, which can be difficult or impossible to identify in massive, anonymous multi-community settings. Other typologies emphasize differences in communication platforms and other functional requirements <cit.>, which are important but preclude analyzing differences between communities within the same multi-community platform. Similarly, previous computational methods of characterizing multiple communities have relied on the presence of markers such as affixes in community names <cit.>, or platform-specific affordances such as evaluation mechanisms <cit.>. Our typology is also distinguished from community detection techniques that rely on structural or functional categorizations <cit.>.While the focus of those studies is to identify and characterize sub-communities within a larger social network, our typology provides a characterization of pre-defined communities based on the nature of their identity.Broader work on collective identity Our focus on community identity dovetails with a long line of research on collective identity and user engagement, in both online and offline communities <cit.>. These studies focus on individual-level psychological manifestations of collective (or social) identity, and their relationship to user engagement <cit.>. In contrast, we seek to characterize community identities at an aggregate level and in an interpretable manner, with the goal of systematically organizing the diverse space of online communities. Typologies of this kind are critical to these broader, social-psychological studies of collective identity: they allow researchers to systematically analyze how the psychological manifestations and implications of collective identity vary across diverse sets of communities. § CONCLUSION AND FUTURE WORKOur current understanding of engagement patterns in online communities is patched up from glimpses offered by several disparate studies focusing on a few individual communities.This work calls into attention the need for a method to systematically reason about similarities and differences across communities.By proposing a way to structure the multi-community space, we find not only that radically contrasting engagement patterns emerge in different parts of this space, but also that this variation can be at least partly explained by the type of identity each community fosters. Our choice in this work is to structure the multi-community space according to a typology based on community identity, as reflected in language use.We show that this effectively explains cross-community variation of three different user engagement measures—retention, acculturation and content affinity—and complements measures based on activity and size with additional interpretable information. For example, we find that in niche communities established members are more likely to engage with volatile content than outsiders, while the opposite is true in generic communities.Such insights can be useful for community maintainers seeking to understand engagement patterns in their own communities.One main area of future research is to examine the temporal dynamicsin the multi-community landscape. By averaging our measures of distinctiveness and dynamicity across time, our present study treated community identity as a static property. However, as communities experience internal changes and respond to external events, we can expect the nature of their identity to shift as well. For instance, the relative consistency of harrypotter may be disrupted by the release of a new novel, while Seahawks may foster different identities during and between football seasons. Conversely, a community's type may also mediate the impact of new events. Moving beyond a static view of community identity could enable us to better understand how temporal phenomena such as linguistic change manifest across different communities, and also provide a more nuanced view of user engagement—for instance, are communities more welcoming to newcomers at certain points in their lifecycle?Another important avenue of future work is to explore other ways of mapping the landscape of online communities.For example, combining structural properties of communities <cit.> with topical information <cit.> and with our identity-based measures could further characterize and explain variations in user engagement patterns.Furthermore, extending the present analyses to even more diverse communities supported by different platforms (e.g.,GitHub, StackExchange, Wikipedia) could enable the characterization of more complex behavioral patterns such as collaboration and altruism, which become salient in different multicommunity landscapes. § ACKNOWLEDGEMENTS The authors thank Liye Fu, Jack Hessel, David Jurgens and Lillian Lee for their helpful comments. This research has been supported in part by a Discovery and Innovation Research Seed Award from the Office of the Vice Provost for Research at Cornell, NSF CNS-1010921,IIS-1149837, IIS-1514268 NIH BD2K, ARO MURI, DARPA XDATA, DARPA SIMPLEX, DARPA NGS2, Stanford Data Science Initiative, SAP Stanford Graduate Fellowship, NSERC PGS-D, Boeing, Lightspeed, and Volkswagen. aaai | http://arxiv.org/abs/1705.09665v1 | {
"authors": [
"Justine Zhang",
"William L. Hamilton",
"Cristian Danescu-Niculescu-Mizil",
"Dan Jurafsky",
"Jure Leskovec"
],
"categories": [
"cs.SI",
"cs.CL",
"cs.CY",
"physics.soc-ph"
],
"primary_category": "cs.SI",
"published": "20170526180002",
"title": "Community Identity and User Engagement in a Multi-Community Landscape"
} |
Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region,Russia Institut de Physique Nucléaire, IN2P3-CNRS, Université Paris-Sud, F-91406 Orsay Cédex, France;Laboratoire de Physique et Modélisation des Milieux Condensés, CNRS and Université Joseph Fourier, 25 avenue des Martyrs BP166, F-38042 Grenoble Cédex 9, FranceWith the Wigner Function Moments (WFM) method the scissors mode of the actinides and rare earth nuclei are investigated. The unexplained experimental fact thatin ^232Th a double hump structure is found finds a natural explanation within WFM. It is predicted that the lower peak corresponds to an isovector spin scissors mode whereas the higher lying states corresponds to the conventional isovector orbital scissors mode. The experimental situation is scrutinized in this respect concerning practically all results of M1 excitations.21.10.Hw, 21.60.Ev, 21.60.Jz, 24.30.CzExperimental status of the nuclear spin scissors modeP. Schuck Received/ Accepted====================================================== § INTRODUCTIONIn a recent paper <cit.> the Wigner Function Moments (WFM)or phase space moments methodwas applied for the first timeto solve the TDHF equations including spin dynamics. As a first step, only the spin orbit interaction was included in the consideration, as the most important one among all possible spin dependent interactions because it entersinto the mean field. The most remarkable result was the prediction of a new type of nuclear collective motion: rotational oscillations of "spin-up" nucleons with respect of "spin-down" nucleons (the spin scissors mode).It turns out that the experimentallyobserved group of peaks in the energy interval 2-4 MeV correspondsvery likely to two different types of motion: the orbital scissors mode and this new kindof mode, i.e. the spin scissors mode. The pictorial view of these two intermingled scissors is shown in Fig. <ref>. It just shows thegeneralization of theclassical picture forthe orbital scissors (see, for example, <cit.>) to include the spin scissors modes. In Ref. <cit.> the influence of the spin-spin interaction on the scissors modes was studied.It was found that such interaction does not push the predicted mode strongly up in energy.It turned out that the spin-spin interaction does not change the general picture of the positions of excitations described in <cit.> pushing all levels up proportionally to its strength without changing their order. The most interesting result concerns the B(M1) values of both scissors – the spin-spin interaction strongly redistributes M1 strength infavour of the spin scissors mode without changing their summed strength.A generalization of the WFM method which takes into account spin degrees of freedom and pair correlations simultaneously was outlined in<cit.>, where the rare earth nuclei were considered. As a result the agreement between theory and experiment in the description of nuclear scissors modes was improved considerably. The decisive role in the substantial improvement of results is played by theanti-aligned spins <cit.>.It was shown that the ground state nucleus consistsof two equal parts having nonzero angular momenta with opposite directions, which compensate each other resulting in the zero total angular momentum.This is graphically depicted in Fig. <ref>(a).On the other hand, when the opposite angular momenta become tilted, one excites the system and the oppositeangular momenta are vibrating with a tilting angle, see Fig. <ref>(b). It is rather obvious from Fig. <ref> that these tilted vibrationshappen separately in each of the neutron and proton lobes.These spin-up against spin-down motions certainly influence theexcitation of the spin scissors mode.The aim of the present paper is to find theexperimental confirmation of our prediction: the splitting of low lying (E<4 MeV) M1 excitations in two groups corresponding to spin and orbital scissors, the spin scissors being lower in energy and stronger intransition probability B(M1).It turns out that a similar phenomenon was observed and discussed by experimentalists already at the beginning of the "scissors era". For example, we cite from the paper ofC. Wesselborg et al. <cit.>: "The existence of the two groups poses the questionwhether they arise from one mode, namely from scissors mode, or whether we see evidence for two independent collective modes." We already touched this problem in the paper <cit.>, where we have found, that our theory explains quite naturally the experimental results of Oslo group <cit.> for ^232Th. We, therefore, will concentrate to a large extent on the explanation of the experiments in the actinides, see Sec. <ref>. In Sec. <ref>, we will try to see whether in the rare earth nuclei there are also signs of a double hump structure. In Sec. <ref> we will give our conclusions. The description of the WFM method and mathematical details are given in Appendix <ref>.§ THE SITUATION IN THE ACTINIDESGuttormsen et al <cit.> have studieddeuteron and ^3He-induced reactions on ^232Th and found in theresidual nuclei ^231,232,233Th and ^232,233Pa "an unexpectedly strong integrated strength of B(M1)=11-15 μ_N^2 in theE_γ=1.0-3.5 MeV region". The B(M1) force in most nuclei showsevident splitting into two Lorentzians. "Typically, the experimental splitting is Δω_M1∼ 0.7 MeV, and the ratio of thestrengths between the lower and upper resonance components is B_L/B_U∼ 2".Seeing this obvious splitting the question is raised: "What is the nature of the splitting?" Their attempt to explain thesplitting by a γ-deformation has failed. To describe the observed value of Δω_M1 the deformation γ∼ 15^∘ is required,that leads to the ratio B_L/B_U∼ 0.7 in an obvious contradictionwith experiment. The authors conclude that "the splitting may bedue to other mechanisms".Later <cit.> they reanalyzed their data for Th and Pa with the result shown in Fig. <ref> and presented the results of new experiments for ^237-239U (Fig. <ref>) with the conclusion: "The SR (Scissors Resonance) displays a double-hump structure that is theoretically not understood." When the paper <cit.> with our explanation of the two humps nature of SR was published, P. von Neumann-Cosel attracted our attention to the paper by A. S. Adekola et al <cit.> who have studied the scissors mode in the (γ,γ') reaction on the same nucleus ^232Th one year earlier. It turns out that these authors have also obtained the scissors mode splitting (see Fig. <ref>), but did not pay further attention to it. The energy and B(M1) values of the two humps of SR are shown in Table <ref>. As it is seen, two experiments <cit.> demonstrate very good agreement in the description of the splitting: Δ E and B_L/B_U. The big difference in B(M1) values is explained by the fact that in Oslo method one extracts the scissors mode fromstrongly excited (heated) nuclei, whereas in (γ,γ') reactions one deals with SR in cold (ground state) nuclei. WFM method describes smallamplitude deviations from the ground state, so our results must be closer to that of (γ,γ') experiment that is confirmed in Table <ref>.It is easy to see that the calculated energies E and B(M1) values of the spin and orbital scissors are in very good agreement with the experimental <cit.> energy centroidsand summarized B(M1) values of the lower and higher groups of levels respectively. Figure <ref> contains also the results of QRPA calculations of Kuliev et al <cit.> which are also very intreaguing. As in theexperimental works they find a bunch of low-lying states in the region 2.0-2.3 MeVand a second one in the region 2.7-3.4 MeV. They also find a third bunch close to 4 MeV.Those results are very similar to ours in what concerns the two low lying structures. It is very tempting to identify their low lying structure with the spin scissors and the second structure with the orbital one. However, the authors did not investigate their structures in those terms. Indeed in an QRPA calculation it is notevident to analyze what is spin and what orbital scissors mode. It is necessary to stress that the solution of the set of dynamical equations (<ref>) gives only two low lying eigenvalues, which are interpreted as the centroid energies of the spin scissors and the orbital scissors according to collective variables responsible for the generation of these eigenvalues (see, for example,Table I of our paper <cit.>). That is why we can compare the results of our calculations (two eigenvalues) only with the equivalent centroids of theexperimental scissors spectra.The experimental spectra of 1^+ excitations obtained in (γ,γ') reactions forthree actinide nuclei <cit.> are shown on Fig. <ref>.The spectrum of ^232Th can be divided in two groups with certainty.The division of spectra in two uranium nuclei is not so obvious. For example, there are four variants to divide the spectrum of ^236U in two groups: 1) to put the border (between two groups) into the energy interval1.80 ≤ E ≤ 2.04 MeV, 2) or into the interval 2.30 ≤ E ≤ 2.41 MeV, 3) or 2.51 ≤ E ≤ 2.69 MeV, 4) or 2.78 ≤ E ≤ 2.95 MeV. Which interval to choose? To exclude any arbitrariness in the choice of the proper interval, we apply a simple technical device. We folded theexperimental spectra with a Lorentzian of increasing width. The width was increased until only two humps remained (see Fig. <ref>).The blue arrows indicate the position of the minimum between the two humps. We want to stress the fact that the arrows are close to the positionwhere the minima at finite temperature occur for ^232Th and ^238U. This gives some credit to our method to divide the spectra even at zero temperatureinto two humps, since it can be expected that if there is a two hump structure at finite temperature, there should be a similar one also at zero temperature.Fig. <ref> demonstrates the results of folding of ^236U spectrum with smaller (a) and bigger (b) values of the width of the Lorentzian. It is seen that the spectrum of ^236U can eventually berepresented by a two-humps curve.Applying this procedure to ^232Th and ^238U we get the results presented in Table <ref>. The points of the spectra division are shown on Fig. <ref> by blue arrows. The Table <ref> demonstrates rather good agreement between the theory and experiment for both scissors in ^232Th; the agreement in ^236Ucan be characterized as acceptable. The situation is graphically displayed in Fig. <ref>. One observes an unexpectedly large value ofthe summed B(M1) for ^238U incomparison with that of ^236U and^232Th and with the theoretical result.The possible reason of this discrepancy was indicated by the authors of <cit.>: "M1 excitations are observed at approximately 2.0 MeV <E_γ< 3.5 MeV with a strong concentration of M1 states around 2.5 MeV. ...The observed M1 strength may include states from both the scissorsmode and the spin-flip mode, which are indistinguishable from each other based exclusively on the use of the NRF technique."The most reasonable (and quite natural) place for the boundary between the scissors mode and thespin-flip resonance is located in the spectrum gap between 2.5 MeV and 2.62 MeV. The summed M1 strengthof scissors in this case becomes B(M1)=4.38± 0.5 μ_N^2 in rather good agreementwith^236U and ^232Th. This value is also not so far from the theoretical result.After dividing in spin and orbital scissors it gives B(M1)_ or=1.92 μ_N^2 in good agreementwith the calculated value. We want to stress again that the folding of spectra with Lorentzians is an artifact to divide a given spectrum intoa lower lying and a higher lying group. The ensuing width of the two humps should not be interpreted as a true width.Also the specific choice of a Lorentzian has no significance. We could have chosen as well a Gaussian.Below we will apply exactly the same method for the case of Rare Earth nuclei to divide the spectra into two parts.As we will see, the case of Rare Earth nuclei is much harder but we will try and see.In conclusion, our method of separating the often pretty complex splitting patterns of the M1 strength into just two humps is an asumptionwhich derives from our present theoretical description where besides the orbital scissors part, a spin scissors part appears. It should be noted also that both scissors modes have an underlying orbitalnature, becauseboth are generated by the same type of collective variables – by the orbitalangular momenta (the variables Ł_λμ in (<ref>) with λ=μ=1).All the difference is that the "orbital" (conventional) scissors are generatedby the counter-oscillations of the orbital angular momentum of protons withrespect of the orbital angular momentum of neutrons, whereasthe "spin"scissorsare generated by the counter-oscillations of the orbital angular momentum of nucleons having thespin projection "up" with respect of the orbital angular momentumof nucleonshaving the spin projection "down".At the same time,both scissors are strongly sensitive to theinfluence of the spin part of the magnetic dipole operator (<ref>)Ô_11= μ_N/ħ√(3/4π)[g_sŜ_1 +g_ll̂_1].The Table <ref> demonstrates the sensitivity of both scissors to the spin-dependent part of nuclear forces:the moderate constructive interference of the orbital andspin contributionsin the case of the spin scissors mode and their very strongdestructive interference in the case of the orbital scissors mode.§ EXPERIMENTAL SITUATION IN RARE EARTHS We here will perform asystematic analysis of experimental data for rare earth nuclei, where the majority of nuclear scissors are found.We have studied practically all papers containing experimental data for lowlying M1 excitations in rare earth nuclei. For the sake of convenience we havecollected in Fig. <ref>all experimentally known spectra of low lying 1^+ excitations <cit.>. It should beemphasized, that only the levels with positive parities are displayed here. As a matter of fact, there exist many 1^π excitations in the energy interval 1 MeV- 4 MeV, whose parities π are not known up to now. One glance on Fig. <ref> is enough to understand that the situation with spectra in rare earth nuclei iscomplicated. Nevertheless, we will try to proceed in the same way as with the Actinides: we fold the spectra with a Lorentzian whose width is increaseduntil only two humps survive.The minimum between the two humps is indicated with an arrow in Fig. <ref>. The states in the low lying group are then identified as belonging to the spin scissors motion and the ones of high lying group as orbital scissors states. The values of B(M1) and energy centroids, corresponding to the described separation,are compared with theoretical results in the Table <ref>. Studying attentively theTable <ref> one canfind satisfactory agreement between theoretical and experimental results forthe orbital scissors in 13 nuclei: ^146,148Nd, ^154Gd, ^166,170Er, ^174,176Yb, ^176,178,180Hf, ^190Os and ^194,196Pt.The same degree of agreement can be found for the spin scissors in 15 nuclei: ^134Ba, ^146,148,150Nd, ^150,154Sm, ^154Gd, ^160Dy, ^172,174,176Yb, ^176,180Hf, ^192Os and ^194Pt.Therefore the satisfactory agreement between theoretical and experimentalresults for both, the orbital and spin scissors, is observed in 8 nuclei:^146,148Nd, ^154Gd, ^174,176Yb, ^176,180Hf and ^194Pt.Again, as in Fig. <ref> for the Actinides, in Fig. <ref> we compare for four rare earth nuclei the experimental centroids with our results.The agreement is nearly perfect. In the other four nuclei the agreement is less good but still acceptable.The harvest seems to be rather meager. However, we have to remember that spin and orbital scissors are surely mostly not quite separated.It is a lucky accident when they are clearly separated like in the Th and (eventually) in Pa isotopes and here as in Fig. <ref> .Therefore, an additional handful of nuclei in the rare earth region is a very welcome support of our theoreticalanalysis of the existence of two separate sissors modes: spin and orbital. This the more so as we get at leastsemi-quantitative agreement between experiment and theory, in the sense that for all those selected nuclei the transition probability for spin scissors is stronger than for orbital scissors.This finding gives support to our theoretical analysis. It is worth noting at the end of this section, that sums of experimental B(M1) values andthe respective energy centroids agree (with rare exceptions) very well withtheoretical predictions (see Fig. <ref>). § CONCLUSIONS AND OUTLOOK The aim of this paper was to find experimental indications about the existence of the "spin" scissors mode predicted in our previous publications<cit.>. To this end we have performed a detailed analysis ofexperimental data on 1^+ excitations for actinides and for Rare Earth nuclei of the N=82-126 major shell. First of all let us again comment on the very clear experimental situation in the actinidesfor heated nuclei of Th, Pa, U isotopes. In all cases a clear two humps structure has been revealed, see Figs. <ref> and <ref>. Unfortunately with our WFM technique, we are so farnot prepared to investigate nuclei at finite temperature.So we tried in this work to investigate scissors modes on top of the ground states.We tried to be as exhaustive in the presentation of published data in rare earth nuclei and theactinides as possible. For the actinides, there exists a very clear cut example given by ^232Th,see Figs. <ref> and <ref>, and our results are in good agreement with the experimentalvalues for position and B(M1) values.We stress again that this concerns also the feature that the B(M1)'s are stronger for the hypotheticalspin scissors than for the orbital one. For the Uranium isotopes the separation into two peaks of theM1 excitations is not so clear. So we tried to find signatures of splitting also in the rare earth nuclei.Despite of the fact that one can imagine that spin-scissorsand orbital ones are not wellseparated in nuclei, lighter than the actinides, we nevertheless found a handful of examples where our method ofseparation into a high lying and low lying group works and yields satisfying agreement with experiment.Here we again found thatB(M1)'s are stronger for low lying than for high lying part of levels. So, there are about half a dozenexamples which support our theoretical findings that there exist two groups of scissors modes,the spin scissors and orbital scissors modes. Actually we divided the experimental spectra of almost all rare earth nuclei shown in Fig. <ref>into low lying and higher lying parts indicated by the blue arrows. Without giving quantitative agreement with experiment besides for those 8 nuclei mentionedin the main text, we at least found most of thetimethat the low lying parts have stronger B(M1)'s than high lying groups again in qualitative agreementwith our theoretical investigation. In conclusion, there seems to exist as well some support of the rare earth nuclei for the existence of the spin scissors mode.Let us mention again that we obtained the two hump structures in folding the spectra with a Lorentzian (we also could have taken as well a Gaussian).This method of separating the often quite complex splitting patterns of the M1 strength in just two humps is an assumption that derives from the presenttheoretical description where besides the orbital scissors part, a spin-scissors part appears. A special mention may need the clear situation in ^232Th because thereexists a QRPAcalculation <cit.>. This QRPA calculation also reveals a two hump structure in close agreementwith experiment and with our findings. We may finish off with the remark that it does not seem evident to disentangle a QRPA spectrum into spin and orbital scissors states.For this it will be necessary to find the excitation operators which specifically excite the spin and orbital scissors individually.For the moment this task can only be achieved with our WFM method because the amplitudes have a clear connection to the physics of the obtained states.It will be a future project to make detailed comparison between the QRPA and our calculations.A further aim is to introduce finite temperature into the WFM formalism what eventually would allow to explain the strongly enhanced B(M1) transitions foundexperimentally in the Actinides. We wish to thank A. Richter for valuable remarks, M. Guttormsen and J. N. Wilson for fruitful discussions, A. A. Kuliev for sharing the results of his calculationsand P. von Neuman-Cosel who attracted our attention to the paper of . The work was supported by theCollaboration agreement. § WFM METHOD AND DETAILS OF CALCULATIONSThe basis of our method is theTime-Dependent Hartree–Fock–Bogoliubov (TDHFB) equation in matrix formulation <cit.>:iħ=[,̋]with=ρ̂-κ̂-κ̂^†1-ρ̂^*, =̋ĥΔ̂Δ̂^†-ĥ^*The normal density matrix ρ̂ and Hamiltonian ĥ are hermitian whereas the abnormal density κ̂ and the pairing gap Δ̂ are skew symmetric: κ̂^†=-κ̂^*,.The detailed form of the TDHFB equations isiħρ̇̂̇ =ĥρ̂-ρ̂ĥ -Δ̂κ̂^†+κ̂Δ̂^†,-iħρ̇̂̇^*=ĥ^*ρ̂^*-ρ̂^*ĥ^* -Δ̂^†κ̂+κ̂^†Δ̂,-iħκ̇̂̇ =-ĥκ̂-κ̂ĥ^*+Δ̂-Δ̂ρ̂^*-ρ̂Δ̂,-iħκ̇̂̇^†=ĥ^*κ̂^† +κ̂^†ĥ-Δ̂^† +Δ̂^†ρ̂+ρ̂^*Δ̂^†. We do not specify the isospin indices in order to make formulae more transparent. Let us consider matrix form of (<ref>) in coordinatespace keeping spin indices s, s' with compact notation . Then the set of TDHFB equations with specified spin indices reads <cit.>: iħρ̇_rr”^↑↑ = ∫d^3r'(h_rr'^↑↑ρ_r'r”^↑↑-ρ_rr'^↑↑ h_r'r”^↑↑ +ĥ_rr'^↑↓ρ_r'r”^↓↑-ρ_rr'^↑↓ h_r'r”^↓↑ -Δ_rr'^↑↓κ^†_r'r”^↓↑ +κ_rr'^↑↓Δ^†_r'r”^↓↑),iħρ̇_rr”^↑↓ = ∫d^3r'(h_rr'^↑↑ρ_r'r”^↑↓-ρ_rr'^↑↑ h_r'r”^↑↓ +ĥ_rr'^↑↓ρ_r'r”^↓↓-ρ_rr'^↑↓ h_r'r”^↓↓),iħρ̇_rr”^↓↑ = ∫d^3r'(h_rr'^↓↑ρ_r'r”^↑↑-ρ_rr'^↓↑ h_r'r”^↑↑ +ĥ_rr'^↓↓ρ_r'r”^↓↑-ρ_rr'^↓↓ h_r'r”^↓↑),iħρ̇_rr”^↓↓ = ∫d^3r'(h_rr'^↓↑ρ_r'r”^↑↓-ρ_rr'^↓↑ h_r'r”^↑↓ +ĥ_rr'^↓↓ρ_r'r”^↓↓-ρ_rr'^↓↓ h_r'r”^↓↓ -Δ_rr'^↓↑κ^†_r'r”^↑↓ +κ_rr'^↓↑Δ^†_r'r”^↑↓),iħκ̇_rr”^↑↓ = -Δ̂_rr”^↑↓ +∫d^3r'(h_rr'^↑↑κ_r'r”^↑↓+κ_rr'^↑↓h^*_r'r”^↓↓ +Δ_rr'^↑↓ρ^*_r'r”^↓↓+ρ_rr'^↑↑Δ_r'r”^↑↓),iħκ̇_rr”^↓↑ = -Δ̂_rr”^↓↑ +∫d^3r'(h_rr'^↓↓κ_r'r”^↓↑+κ_rr'^↓↑h^*_r'r”^↑↑ +Δ_rr'^↓↑ρ^*_r'r”^↑↑+ρ_rr'^↓↓Δ_r'r”^↓↑).This set of equations must be complemented by the complex conjugated equations.We work with the Wigner transform <cit.> ofequations (<ref>). The relevant mathematical details can be found in <cit.>.We do not write out the coordinatedependence (,) of all functions in order to make the formulaemore transparent. We haveiħḟ^↑↑ = iħ{h^↑↑,f^↑↑} +h^↑↓f^↓↑-f^↑↓h^↓↑ +iħ/2{h^↑↓,f^↓↑} -iħ/2{f^↑↓,h^↓↑}- ħ^2/8{{h^↑↓,f^↓↑}} +ħ^2/8{{f^↑↓,h^↓↑}}+ κΔ^* - Δκ^*+ iħ/2{κ,Δ^*}-iħ/2{Δ,κ^*} - ħ^2/8{{κ,Δ^*}} + ħ^2/8{{Δ,κ^*}} +...,iħḟ^↓↓ = iħ{h^↓↓,f^↓↓} +h^↓↑f^↑↓-f^↓↑h^↑↓ +iħ/2{h^↓↑,f^↑↓} -iħ/2{f^↓↑,h^↑↓}- ħ^2/8{{h^↓↑,f^↑↓}} +ħ^2/8{{f^↓↑,h^↑↓}}+ Δ̅^* κ̅- κ̅^* Δ̅+ iħ/2{Δ̅^*,κ̅}-iħ/2{κ̅^*,Δ̅} - ħ^2/8{{Δ̅^*,κ̅}} + ħ^2/8{{κ̅^*,Δ̅}} +...,iħḟ^↑↓ =f^↑↓(h^↑↑-h^↓↓) +iħ/2{(h^↑↑+h^↓↓),f^↑↓} -ħ^2/8{{(h^↑↑-h^↓↓),f^↑↓}}- h^↑↓(f^↑↑-f^↓↓) +iħ/2{h^↑↓,(f^↑↑+f^↓↓)} +ħ^2/8{{h^↑↓,(f^↑↑-f^↓↓)}}+...., iħḟ^↓↑ =f^↓↑(h^↓↓-h^↑↑) +iħ/2{(h^↓↓+h^↑↑),f^↓↑} -ħ^2/8{{(h^↓↓-h^↑↑),f^↓↑}}- h^↓↑(f^↓↓-f^↑↑) +iħ/2{h^↓↑,(f^↓↓+f^↑↑)} +ħ^2/8{{h^↓↑,(f^↓↓-f^↑↑)}}+...,iħκ̇ = κ (h^↑↑+h̅^↓↓) +iħ/2{(h^↑↑-h̅^↓↓),κ} -ħ^2/8{{(h^↑↑+h̅^↓↓),κ}}+ Δ (f^↑↑+f̅^↓↓) +iħ/2{(f^↑↑-f̅^↓↓),Δ} -ħ^2/8{{(f^↑↑+f̅^↓↓),Δ}} - Δ + ..., iħκ̇^*= -κ^*(h^↑↑+h̅^↓↓)+iħ/2{(h^↑↑-h̅^↓↓),κ^*} +ħ^2/8{{(h^↑↑+h̅^↓↓),κ^*}}- Δ^*(f^↑↑+f̅^↓↓)+iħ/2{(f^↑↑-f̅^↓↓),Δ^*} +ħ^2/8{{(f^↑↑+f̅^↓↓),Δ^*}} + Δ^* +..., where the functions h, f, Δ, and κ are the Wigner transforms of ĥ, ρ̂, Δ̂, and κ̂, respectively, f̅(,)=f(,-),{f,g} is the Poisson bracket of the functions f and g and {{f,g}} is their double Poisson bracket. The dots stand for terms proportional to higher powers of ħ – afterintegration over phase space these terms disappear and we arrive to the set of exact integral equations. This set of equations must be complemented by the dynamical equations forf̅^↑↑, f̅^↓↓, f̅^↑↓,f̅^↓↑,κ̅,κ̅^*. They are obtained by the change → - in arguments of functions and Poisson brackets.So, in reality we deal with the set of twelve equations. We introduced the notation κ≡κ^↑↓ and Δ≡Δ^↑↓.Following the papers <cit.> in the next step we write above equations in terms of spin-scalarand spin-vectorfunctions.As a result, we obtain a set of twelve equations, which is solved by the method of moments in a small amplitudeapproximation. To this endall functions f(,,t) and κ(,,t) are divided into equilibrium partand deviation (variation): f(,,t)=f(,)_eq+δ f(,,t),. Then equations are linearized neglecting quadraticin δ f and δκ terms <cit.>. §.§ Model HamiltonianThe microscopic Hamiltonian of the model, harmonic oscillator withspin orbit potential plus separable quadrupole-quadrupole andspin-spin residual interactions is given byH=∑_i=1^A[_i^2/2m+1/2mω^2_i^2 -η_i_i]+H_qq+H_sswithH_qq = ∑_μ=-2^2(-1)^μ{κ̅∑_i^Z∑_j^N+κ/2[∑_i,j (i≠ j)^Z +∑_i,j (i≠ j)^N] }×q_2-μ(_i)q_2μ(_j), H_ss = ∑_μ=-1^1(-1)^μ{χ̅∑_i^Z∑_j^N+χ/2[ ∑_i,j (i≠ j)^Z +∑_i,j (i≠ j)^N] }× Ŝ_-μ(i)Ŝ_μ(j)δ(_i-_j),where N and Z are numbers of neutrons and protons, q_2μ()=√(16π/5) r^2Y_2μ(θ,ϕ) and Ŝ_μ are spin matrices <cit.>.§.§ Pair potentialThe Wigner transform of the pair potential (pairing gap) Δ(,) is related tothe Wigner transform of the anomalous density by <cit.>Δ(,)=-∫d^3p'/(2πħ)^3 v(|-'|)κ(,'),where v(p) is a Fourier transform of the two-body interaction. We take for the pairing interaction a simple Gaussian,v(p)=β e^-α p^2<cit.> with β=-|V_0|(r_p√(π))^3 and α=r_p^2/4ħ^2.The following values of parameters were used in calculations: r_p=1.9 fm,|V_0|=25 MeV.Several exceptions were done for rare earth nuclei: |V_0|=26 MeV for ^150Nd, |V_0|=26.5 MeV for ^176, 178, 180Hf and ^182, 184W,|V_0|=27 MeV for nuclei with deformation δ⩽ 0.18.§.§ Equations of motion Integrating the set of equations for the δ f_τ^ς(,,t)andδκ_τ(,,t) over phase space with the weights W ={r⊗ p}_λμ, {r⊗ r}_λμ, {p⊗ p}_λμ,1one gets dynamic equations for the following collective variables:Ł^τς_λμ(t)=∫ d(,) {r⊗ p}_λμδ f^ς_τ(,,t),^τς_λμ(t)=∫ d(,) {r⊗ r}_λμδ f^ς_τ(,,t),^τς_λμ(t)=∫ d(,) {p⊗ p}_λμδ f^ς_τ(,,t),^τς(t)=∫ d(,) δ f^ς_τ(,,t),Ł̃^τ_λμ(t)=∫ d(,) {r⊗ p}_λμδκ_τ(,,t),^τ_λμ(t)=∫ d(,) {r⊗ r}_λμδκ_τ(,,t),^τ_λμ(t)=∫ d(,) {p⊗ p}_λμδκ_τ(,,t),whereς=+, -, ↑↓, ↓↑,{r⊗ p}_λμ=∑_σ,νC_1σ,1ν^λμr_σp_ν. It is convenient to rewrite the dynamical equations in terms of isoscalar and isovector variables_λμ=_λμ^n+_λμ^p,_λμ=_λμ^n-_λμ^p,_λμ=_λμ^n+_λμ^p,_λμ=_λμ^n-_λμ^p,Ł̅_λμ=Ł_λμ^n+Ł_λμ^p,Ł_λμ=Ł_λμ^n-Ł_λμ^p.We are interested in the scissors mode with quantum number K^π=1^+. Therefore, we only need the part of dynamic equationswith μ=1. The integration yields the following set of equationsfor isovector variables <cit.>: Ł̇^+_21 = 1/m_21^+- [m ω^2 -4√(3)ακ_0R_00^ eq +√(6)(1+α)κ_0 R_20^ eq]^+_21 -iħη/2[Ł_21^- +2Ł^↑↓_22+ √(6)Ł^↓↑_20],Ł̇^-_21 = 1/m_21^- - [m ω^2+√(6)κ_0 R_20^ eq -√(3)/20ħ^2( χ-χ̅/3) (I_1/a_0^2+I_1/a_1^2)(a_1^2/A_2-a_0^2/A_1) ]^-_21 -iħη/2Ł_21^+ +4/ħ|V_0| I_rp^κΔ(r') Ł̃_21,Ł̇^↑↓_22 = 1/m_22^↑↓- [m ω^2-2√(6)κ_0R_20^ eq -√(3)/5ħ^2( χ-χ̅/3)I_1/A_2]^↑↓_22 -iħη/2Ł_21^+,Ł̇^↓↑_20 = 1/m_20^↓↑- [m ω^2 +2√(6)κ_0 R_20^ eq]^↓↑_20 +4√(3)κ_0 R_20^ eq ^↓↑_00 -iħη/2√(3/2)Ł_21^+ +√(3)/15ħ^2( χ-χ̅/3)I_1[ (1/A_2-2/A_1) _20^↓↑+ √(2)(1/A_2+1/A_1) _00^↓↑],Ł̇^+_11 =-3√(6)(1-α)κ_0 R_20^ eq ^+_21 -iħη/2[Ł_11^-+√(2)Ł^↓↑_10],Ł̇^-_11 =-[3√(6)κ_0 R_20^ eq -√(3)/20ħ^2( χ-χ̅/3) (I_1/a_0^2-I_1/a_1^2)(a_1^2/A_2-a_0^2/A_1) ]^-_21 -ħη/2[iŁ_11^+ +ħ^↓↑]+4/ħ|V_0| I_rp^κΔ(r') Ł̃_11,Ł̇^↓↑_10 =-ħη/2√(2)[iŁ_11^+ +ħ^↓↑],^↓↑ =-η[Ł_11^- +√(2)Ł^↓↑_10],^+_21 = 2/mŁ_21^+ -iħη/2[_21^- +2^↑↓_22+ √(6)^↓↑_20],^-_21 = 2/mŁ_21^- -iħη/2_21^+,^↑↓_22 = 2/mŁ_22^↑↓ -iħη/2_21^+,^↓↑_20 =2/mŁ_20^↓↑ -iħη/2√(3/2)_21^+, ^+_21 =-2[m ω^2+√(6)κ_0 R_20^ eq]Ł^+_21 +6√(6)κ_0 R_20^ eqŁ^+_11 -iħη/2[_21^-+2^↑↓_22+√(6)^↓↑_20]+3√(3)/4ħ^2χI_2/A_1A_2[(A_1-A_2) Ł_21^+ + (A_1+A_2) Ł_11^+]+4/ħ|V_0| I_pp^κΔ(r') _21,^-_21 =-2[m ω^2+√(6)κ_0 R_20^ eq]Ł^-_21 +6√(6)κ_0 R_20^ eqŁ^-_11 -6√(2)κ_0 L_10^-( eq)^+_21 -iħη/2_21^+ +3√(3)/4ħ^2χI_2/A_1A_2[(A_1-A_2)Ł_21^- + (A_1+A_2) Ł_11^-],^↑↓_22 =-[2m ω^2-4√(6)κ_0 R_20^ eq -3√(3)/2ħ^2χI_2/A_2]Ł^↑↓_22 -iħη/2_21^+,^↓↑_20 =-[2m ω^2+4√(6)κ_0 R_20^ eq]Ł^↓↑_20 +8√(3)κ_0 R_20^ eqŁ^↓↑_00 -iħη/2√(3/2)_21^+ +√(3)/2ħ^2χI_2/A_1A_2[(A_1-2A_2)Ł_20^↓↑+ √(2)(A_1+A_2) Ł_00^↓↑],Ł̇^↓↑_00 = 1/m_00^↓↑-m ω^2^↓↑_00 +4√(3)κ_0 R_20^ eq ^↓↑_20 +1/2√(3)ħ^2[( χ-χ̅/3)I_1-9/4χ I_2] [(2/A_2-1/A_1)_00^↓↑+ √(2)(1/A_2+1/A_1)_20^↓↑],^↓↑_00 = 2/mŁ_00^↓↑,^↓↑_00 =-2m ω^2Ł^↓↑_00 +8√(3)κ_0 R_20^ eq Ł^↓↑_20 +√(3)/2ħ^2χ I_2 [(2/A_2-1/A_1)Ł_00^↓↑+ √(2)(1/A_2+1/A_1)Ł_20^↓↑], _21 =-1/ħΔ_0(r') ^+_21 + 6ħακ_0 K_0 R^+_21, Ł̇̃̇_21 =-1/ħΔ_0(r') Ł^-_21,Ł̇̃̇_11 = -1/ħΔ_0(r') Ł^-_11,where A_1=√(2)R_20^ eq-R_00^ eq=Q_00/√(3)(1+4/3δ),A_2= R_20^ eq/√(2)+R_00^ eq=-Q_00/√(3)(1-2/3δ), a_-1 = a_1 = R_0( 1-(2/3)δ/1+(4/3)δ)^1/6,a_0 = R_0( 1-(2/3)δ/1+(4/3)δ)^-1/3,Q_00=3/5AR_0^2a_i are semiaxes of ellipsoid by which the shape of nucleus is approximated, δ – deformation parameter,R_0=1.2A^1/3 fm – radius of nucleus.I_1=π/4∫_0^∞drr^4(∂ n(r)/∂ r)^2,I_2=π/4∫_0^∞drr^2 n(r)^2,n(r) – nuclear density. The values L_10^-( eq), K_0, Δ_0(r'), I_rp^κΔ(r'),I_pp^κΔ(r') entering into the equations (<ref>), details of calculations relating to accounting for pair correlations and choice of parameters are discussed in the Ref. <cit.>. §.§ Energies and excitation probabilitiesImposing the time evolution via e^iΩ t for all variables one transforms (<ref>) into a set of algebraic equations.Eigenfrequencies are found as the zeros of its secular equation. Excitation probabilities are calculated with the help of the theory of linearresponse of the system to a weak external fieldÔ(t)=Ô ^-iΩ t+Ô^† e^iΩ t.A detailed explanation can be found in <cit.>.We recall only the main points. The matrix elements of the operator Ô obey the relationship <cit.>|⟨ψ_a|Ô|ψ_0⟩|^2= ħlim_Ω→Ω_a(Ω-Ω_a) ⟨ψ'|Ô|ψ'⟩^-iΩ t,where ψ_0 and ψ_a are the stationary wave functions of the unperturbed ground and excited states; ψ' is the wave function of the perturbed ground state, Ω_a=(E_a-E_0)/ħ are the normal frequencies, the bar means averaging over a time interval much larger than 1/Ω.To calculate the magnetic transition probability, it is necessary to excite the system by the following external field:Ô_λμ=μ_N(g_s/ħ-ig_l2/λ+1[×∇]) ∇(r^λY_λμ), μ_N=eħ/2mc.Here g_l^ p=1, g_s^ p=5.5856 for protons and g_l^ n=0, g_s^ n=-3.8263 for neutrons. The dipole operator in cyclic coordinates looks likeÔ_11= μ_N√(3/4π)[g_sŜ_1/ħ -g_l√(2)∑_ν,σ C_1ν,1σ^11r_ν∇_σ].Its Wigner transform is(Ô_11)_W= √(3/4π)[g_sŜ_1 -ig_l√(2)∑_ν,σ C_1ν,1σ^11r_νp_σ]μ_N/ħ. For the matrix element we have⟨ψ'|Ô_11|ψ'⟩ = √(3/2π)[-ħ/2 (g_s^ n^ n+g_s^ p^ p) -ig_l^ pŁ_11^ p+]μ_N/ħ= √(3/8π)[-1/2 [(g_s^ n-g_s^ p)^+(g_s^ n+g_s^ p)^] -i/ħg_l^ p(Ł̅_11^+- Ł_11^+)]μ_N = √(3/8π)[ 1/2(g_s^ p-g_s^ n)^↓↑ +i/ħg_l^ pŁ_11^+ +i/ħ[g_s^ n+g_s^ p-g_l^ p]Ł̅_11^+]μ_N. Deriving (<ref>) we have used the relation 2iŁ̅^+_11=-ħ^, which follows from the angular momentum conservation <cit.>.One has to add the external field (<ref>) to the Hamiltonian (<ref>).Due to the external field some dynamical equations of(<ref>) become inhomogeneous:^+_21 = …+i3/√(π)μ_N/4ħ g^ p_l R^+_20( eq) ^iΩ t,Ł̇^-_11 = …+ i√(3/π)μ_N/4ħ g^ p_l L^-_10( eq) ^iΩ t,Ł̇^_10 = … +i√(3/2π)μ_N/4ħ( g^ n_s-g^ p_s) L^-_10( eq)^iΩ t.For the isoscalar set of equations, respectively, we obtain:^+_21 = …- i3/√(π)μ_N/4ħ g^ p_l R^+_20( eq) ^iΩ t,Ł̇̅̇^-_11 = …- i√(3/π)μ_N/4ħ g^ p_l L^-_10( eq) ^iΩ t,Ł̇̅̇^_10 = … +i√(3/2π)μ_N/4ħ( g^ n_s+g^ p_s) L^-_10( eq) ^iΩ t.Solving the inhomogeneous set of equationsone can find the required in (<ref>) values of Ł_11^+ , Ł̅_11^+ and ^ and using (<ref>) calculateB(M1) factors for all excitations. 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Lane, Nuclear Theory (Benjamin, New York, 1964). | http://arxiv.org/abs/1705.09115v2 | {
"authors": [
"E. B. Balbutsev",
"I. V. Molodtsova",
"P. Schuck"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170525100133",
"title": "Experimental status of the nuclear spin scissors mode"
} |
^1Department of Physics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. ^2Institute for Structure and Function & Department of physics, Chongqing University, Chongqing 400044, P. R. China. ^3Dalian Institute of Chemical Physics,Chinese Academy of Sciences, 116023 Dalian, P. R. China ^4Department of Physics, South China University of Technology, Guangzhou 510640, P. R. China We propose that the topological semimetal features can co-exist with ferromagnetic ground state in vanadium-phosphorous-oxide β-V_2OPO_4 compound from first-principles calculations. In this magnetic system with inversion symmetry, the direction of magnetization is able to manipulate the symmetric protected band structures from a node-line type to a Weyl one in the presence of spin-orbital-coupling. The node-line semimetal phase is protected by the mirror symmetry with the reflection-invariant plane perpendicular to magnetic order. Within mirror symmetry breaking due to the magnetization along other directions,the gapless node-line loop will degenerate to only one pair of Weyl points protected by the rotational symmetry along the magnetic axis, which are largely separated in momentum space. Such Weyl semimetal phase provides a nice candidate with the minimum number of Weyl points in a condensed matter system. The results of surface band calculations confirm the non-trivial topology of this proposed compound. This findings provide a realistic candidate for the investigation of topological semimetals with time-reversal symmetry breaking, particularly towards the realization of quantum anomalous Hall effect in Weyl semimetals.Topological Weyl and Node-Line Semimetals in Ferromagnetic Vanadium-Phosphorous-Oxide β-V_2OPO_4 Compound Y. J. Jin^1^,*, R. Wang^1, 2^,*, J. Z. Zhao^1,3, Z. J. Chen^1,4, Y. J. Zhao^4, and H. Xu^1^,† Accepted .... Received ...; in original form 2017 May 17 ===========================================================================================================12 ptQuantum-Hall-effect (QHE) offered us a different pathway to achieve dissipationless current beyond superconductors <cit.>. However, the potential applications of QHE are strongly limited by the required external high-intensity magnetic fields <cit.>. An alternative option is the Quantum-spin-Hall (QSH) insulators that works independently with respect to external magnetic fields but, unfortunately, will be suppressed when the sample size become larger than a critical value due to the existence of non-elastic scattering between states with opposite directions in helical edge states<cit.>. The most promising solution of these challenges is the proposed quantum-anomalous-Hall (QAH) materials with intrinsic long-rang magnetic order that are able to be free for any sample-size or external field <cit.>.To achieve this, magnetic topological materials, including topological insulators (TIs) <cit.> and topological semimetals (TSMs)<cit.> in their quantum-well structure with time-reversal (TR) symmetry breaking, have attracted intensive attention recently. Distinguished from gaped magnetic TIs, magnetic TSMs, i.e. spin polarized Weyl and node-line semimetals, host finite numbers (Weyl) or continues distributed (node-line) band crossing points near the Fermi level in momentum space.Such crystal symmetry protected features lead exotic conductive surface states, which are Fermi arcs in Wely semimetals (WSMs) <cit.> and drumhead states in node-line semimetals (NLSs) <cit.>. Accomplishing these robust spin dependent surface states is the key step of achieving several unusual spectroscopic and transport phenomena especially the QAH in their quantum-well structure experimentally.Up to now, the non-magnetic TSMs have been studied intensively <cit.>,but only a few magnetic TSMs have been proposed. For instance, magnetic HgCr_2Se_4 has been predicted to host only two nodal points <cit.> which is defined as a Chern semimetal, not fully fit the WSM features since each crossing points possessing chiral charge of 2. So far, nontrivial properties in HgCr_2Se_4 have not been verified experimentally due to the requirement of large magnetic domains <cit.>.Very recently, Co-based magnetic Heusler alloys were also predicted to host the topology of WSMs <cit.>. However, the energy of Weyl points is much higher above the Fermi level (∼0.6 eV). Therefore, additional tuning of the energy of Weyl points relative to Fermi level is necessary <cit.> for future applications.In this work, we propose that the ferromagnetic (FM) vanadium-phosphorous-oxide β-V_2OPO_4, a potential material for lithium-ion battery <cit.>, presents either NL or WSM features that can be easily switched by different magnetization directions.The node-line band structures, in D_4h(C_4h) magnetic group with spin-orbital coupling (SOC), are protected by the mirror reflection symmetry with the reflection-invariant plane perpendicular to the spin-polarized direction. This gapless node-line loop, which lies in the reflection-invariant plane and very close to the Fermi level, will degenerate to two large separated Weyl points protected by rotational symmetry in the same plane once mirror reflection symmetry is broken due to the magnetization along other directions. In these cases, the system host the minimum number of Weyl points. Based on our first-principles calculations, this non-trivial FM β-V_2OPO_4 is a promising candidate of experimentally studying on magnetic TSMs.We perform the first-principles calculations using the Vienna ab initio Simulation Package (VASP) <cit.> based on density functional theory <cit.>. For the exchange-correlation potential we choose the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) formalism <cit.>. The core-valence interactions are treated by the projector augmented wave (PAW) method <cit.>. A plane-wave-basis set with kinetic-energy cutoff of 600 eV has been used. The full Brillouin zone(BZ) is sampled by 21×21×21 Monkhorst-Pack grid to simulate the electronic behaviors <cit.>. Due to the strongly correlated effects of 3d electrons in vanadium, GGA+U calculations is nessesary to describe the on-site Coulomb repulsion beyond the GGA pictures <cit.>. We notice that the band inversion has been confirmed in a large range of U in β-V_2OPO_4 from 1.5 eV to 10.0 eV. In this work, the effective on-site Coulomb energy U is chosen to be 3.0 eV to illustrate the band topology, which works well in fitting the properties of vanadium-oxide system <cit.>.To calculate the surface states and Fermi arcs, the tight-binding Hamiltonian is constructed by projecting the Bloch states into maximally localized Wannier functions <cit.>.As it is illustrated in Figs. <ref>(a) and (b), the vanadium-phosphorous-oxide β-V_2OPO_4 compound crystalizes in body-centered-tetragonal (BCT) structure with a space group I4_1amd (No. 141). Structural optimization obtains the calculated lattice constants which are a=5.465 Å and c=12.544 Å, in nice agreement with experimental values a=5.362 Å and c=12.378 Å <cit.>. The O atoms take two Wyckoff positions 4a (0.0, 0.0, 0.0) and 16h (0.00000, 0.23762, 0.43299). The P and V atoms are located at Wyckoff positions 4b (0.0, 0.0, 0.5) and 8c (0.250, 0.000 0.875), respectively. The BCT BZ and corresponding (001) surface BZ are shown in Fig. <ref>(c), in which high-symmetry points are marked. Our results confirm the FM groundstate of β-V_2OPO_4, which is about 86 meV lower than the antiferromagnetic state per unit cell, with magnetic moment ∼2.5 μ_B per V atom. The electronic band structures in the absence of SOC [see Fig. <ref>(a)] show that the spins and orbitals are independent and two spin channels are decoupled around the Fermi level. These two spin channels present different electronic states, i.e. a ∼3.68 eV band gap of the minority spin states and the semimetallic features of majority spin states, indicating the half-metallic properties of the system. The band crossings reveals around Γ point in the k_z = 0 plane, giving rise to a symmetry protected nodal-line loop. These two crossing bands belong to the states that has opposite eigenvalues ±1 of mirror reflection symmetry operation M_z, respectively, which protects the nodal ring in k_x-k_y plane with k_z = 0 <cit.>. The spin-polarized nodal ring shows a tiny dispersion, the maximum and minimum of which are in the Γ-X and Γ-Σ directions, respectively, i.e. ∼12 meV higher and ∼36 meV lower than the Fermi level E_F.As we report in Fig. <ref>(b), the SOC little influences on the band structures in which the half-metallic ferromagnetism remains. The spontaneous magnetization direction is determined by studying the total energy of system with magnetization along different high symmetry axis. The [001] axis is found to be the energetically most favorable magnetization direction. It is worth to mention that, results show very small energy differences, below 0.3 meV, among all magnetic configurations. This implies that the switching between each configurations by external magnetic field would be easy.In the presence of SOC, the magnetic symmetry is dependent on the direction of magnetization. We will respectively present the topological features of the two typical [001] and [110] magnetisms in the following.When magnetization is polarized along [001] direction, the system reduces to point group C_4h, the subgroup of D_4h, in which the fourfold rotation C_4^z is tensored by the inversion I, namely C_4^z⊗ I. This magnetic group contains eight irreducible symmetry operators: inversion I, fourfold rotation C_4^z, the product of time reversal T, twofold rotations of C_2 symmetry axes [100], [010], [110], [11̅0], and the product (IC_2^z) of inversion and rotation C_2^z. The group element I C_2^z is equivalent to the mirror reflection symmetry corresponding to the x-y plane, which can protect the existence of gapless nodal ring ink_z=0 plane with respect to SOC <cit.> [see Fig. <ref>(b)]. The topological invariant of the nodal ring can be viewed as the variation of the quantized Berry phase with respect to the mirror plane <cit.>, which is related to the change at the end of the one-dimensional system along a line across the ring ink_z=0 plane. As shown in Fig. <ref>(a), the Berry phase of β-V_2OPO_4 shows the jump across the ring, further confirming the topological features of the nodal ring perpendicular to the [001] magnetization direction. Our calculations suggest that the magnetization along other directions is energetically very close to the [001]. When the magnetization is deviated from [001] direction, the mirror reflection symmetry is broken.Here, we take the case of [110] magnetization as an example since the symmetry analysis for the rest cases are essentially the same. The group elements of the corresponding magnetic space group C_2h remains: I, C_2^110 and TC_2^z, TC_2^11̅0. The vanishing of mirror reflection symmetry makes nodal line gapped. However, the anti-unitary symmetry TC_2^z allows the existence of Weyl points ink_z=0 plane. A pair of Weyl points protected by the C_2^110 rotation is present on k_x=k_y axis. We also calculate the parities of inversion eigenvalues at time reversal invariant momenta (TRIM) points. The product of the occupied bands running over all TRIM points is -1, confirming the presence of odd number of pairs of Weyl points <cit.>.As shown in Fig. <ref>(b), the two crossing bands along Γ-X (or [110]) belong to eigenvalues ± i of C_2^110, respectively. The chirality of Weyl point can be determined by the evolution of the average position of Wannier centers, and the Wilson-loop method applied on a sphere around a Weyl point is used <cit.> [see Fig. <ref>(b)].The Weyl point with Chern number C=+1 is located at (0.46Å^-1, 0.46Å^-1, 0) in momentum space, while the Weyl point with Chern number C=-1 related by I symmetry located at same axis [see Fig. <ref>(c)]. The Weyl points only existed on k_x=k_y axis can be further verified by the Berry curvature. As shown in Fig. <ref>(c), the Weyl points with positive and negative chirality are regarded as the "source" and "sink" of Berry curvature in momentum space. As it is discussed above, our results indicate the existence of either the topological NLs or WSMs in FM β-V_2OPO_4 compound, depending on the magnetization direction. The NL features are protected by the mirror reflection symmetry in the case of [001] magnetization, while the WSM features can arise with magnetization direction along other high symmetry axis. The Weyl points would be protected by C_2^m rotational symmetry, where m represents a symmetry axis in x-y plane. In the cases of which only a pair of Weyl points, the minimum number of Weyl points in condensed matter systems, with large separation in momentum. Considering the energy of Weyl points in FM β-V_2OPO_4 is very close to Fermi level, e.g. 12 meV when [110] magnetization, we would suggest β-VOPO_4 compound to be a excellent experimental candidates for the observation of the nontrivial properties in FM WSMs. One hallmark of nontrivial semimetals is the existence of topologically protected surface state, which arises from the inversion of TRIM parities. In the WSM states, topological surface bands connect the valence and conduction bands. The drumhead surface states of NLSs will appear when the gap of bulk band is closed forming a nodal ring. Since the gap size of WSM state is very small, which make only tiny difference of surface states between NL and WSM phases. Here, we only present the surface states and Fermi arcs in the [110] magnetic configuration. To obtain the surface states, we constructed a tight-binding (TB) Hamiltonian with basis of maximally localized Wannier functions <cit.> in which the Green's function method <cit.> is employed.The calculated local density of states (LDOS) and Fermi surface projected on (001) surface are shown in Figs. <ref>(a) and (b), respectively. The surface states are clearly observed in Fig. <ref>(a). On the (001) surface, the anti-unitary magnetic symmetry TC_2 leads to different behavior for the surface bands around the X̅ and Y̅.Although some trivially residual bands would project on (001) surface of this compound, the Fermi arc states are quite clear as it is shown in Fig. <ref>(b).In conclusion, using first-principles calculations, we proposed that FM vanadium-phosphorous-oxide β-V_2OPO_4 can possess the nontrivial properties of TSMs, either NLs or WSMs, that can be switched between each other by different magnetization directions. When the spin is polarized along [001] direction, system belongs to D_4h(C_4h) point group and present the gapless nodal ring, protected by the mirror reflection symmetry, close to the Fermi level at k_x - k_y plane with k_z=0 in momentum space. When the magnetization directions deviate from [001], the nodal ring will degenerate to a pair of Weyl points due to the vanishing of mirror reflection symmetry. These two Weyl points, protected by twofold rotational symmetry along magnetization directions, lie in k_x - k_y plane with k_z=0, which are largely separated in momentum. All the non-trivial band-crossing points are very close to the Fermi level. The topology of the system is confirmed by the existence of non-trivial surface states. 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Supplemental Material for “Topological Semimetals in Ferromagnetic Vanadium-Phosphorous-Oxide β-V_2OPO_4 Compound" § TWO-BAND 2× 2K· PHAMILTONIAN In the absence of SOC, a node-line around the Γ point in k_z = 0 plane is present. Generally, the nodal ring around the Γ point can be modeled by a two-band k· p theory, and the Hamiltonian isH=∑_i=x,y,zd_i(𝐤)σ_i, where d_i(𝐤) are real functions and 𝐤=(k_x,k_y,k_z) are three components of the momentum 𝐤 relative to the Γ point. In Eq. (<ref>), we have ignored the kinetic term proportional to the identity matrix, since it is irrelevant in studying the band crossing. The Pauli matrix σ_i areσ_x=([ 0 1; 1 0; ]) , σ_y=([0 -i;i0;]),σ_z=([10;0 -1;])Within SOC, two spin channels couple together and symmetries can decreasing depending on the direction of the spontaneous magnetization. In spin representation, an any three dimensional (3D) rotation R(α, β, γ) has to correspond a two-dimensional (2D) unitary matrix u(α, β, γ) asu(α, β, γ)=([e^-iα+γ/2cosβ/2 -e^-iα-γ/2sinβ/2; e^iα-γ/2sinβ/2 e^iα+γ/2cosβ/2;]), where α, β, γ are Euler angle of 3D rotation. § THE NODAL RING DEPENDING ON MAGNETIC SPACE GROUP D_4H(C_4H) IN THE PRESENCE OF SPIN-ORBITAL-COUPLINGWhen all spins are oriented along [011](or z) direction, the C_4h subgroup of D_4h is just the fourfold rotation group C_4^z with respect to z coordinate axis tensored by the inversion I, namely C_4^z⊗ I. With [100] magnetization, the group elements of the corresponding magnetic space group D_4h(C_4h) remains: I, C_4x, C_2^z· I, C_2^010· T, and C_2^100· T. It is important note that the product C_2^z· I of twofold rotation C_2^z and inversion I is mirror-reflection symmetry M_z with respect to (001) plane. The matrix form of M_z in spin representation isC_2^z· I = ([e^iπ/2 0; 0 e^-iπ/2; ])=iσ_z.The mirror reflection indicates the Hamiltonian in xy [or (001)] plane asH(k_x,k_y,0)=σ_z H(k_x,k_y,0)σ_zord_x(k_x,k_y,0)=-d_x(k_x,k_y,0)≡ 0,d_y(k_x,k_y,0)=-d_y(k_x,k_y,0)≡ 0,The energy dispersion of the two-band Hamiltonian withC_2^z· I symmetry isE=± |d_z(k_x,k_y,0)|.Generically, the band crossing means d_z(k_x,k_y,0)=0, which has codimension one, i.e., a nodal loop solution in xy plane. § THE WEYL POINTS DEPENDING ON MAGNETIC SPACE GROUP D_4H(C_2H) IN THE PRESENCE OF SPIN-ORBITAL-COUPLINGWe show that the Weyl points can arise with magnetization direction along high symmetry axis, as long as the magnetic space group C_2h symmetry remains. The Weyl points are protected by C_2^m rotation, where m represents a symmetry axis in xy [or (001)] plane. As a example, we mainly show the case of magnetization along [100] axis.With [100] magnetization, the group elements of the corresponding magnetic space group D_4h(C_2h) remains: I, C_2^z· T, C_2^010· T, and C_2^100. We now analyze the group elements on different axes to find out if Weyl points can arise generically on those axes in (001) plane. Firstly, we prove the product C_2^z · T of time reversal T=-iσ_y K and rotation C_2^z allowing for the existence of Weyl points in the (110) plane. The anti-unitary C_2^z · T with matrix representation isC_2^z · T= i([ e^-iπ/2 0; 0e^iπ/2; ])([0 -i;i0;]) K = iσ_xK,with complex conjugation K. The anti-unitary C_2^z · T requires the commutation relation as[H, C_2^z · T]=[H, iσ_xK]=0,i.e.,d_x (k_x,k_y,0)[σ_x,iσ_xK]+d_y (k_x,k_y,0)[σ_y,iσ_xK]+d_z(k_x,k_y,0)[σ_z,iσ_xK]=0,which givesd_z (k_x,k_y,k_z)≡ 0.Now the Hamiltonian in (001) plane isH=d_x(k_x,k_y,0)σ_x+d_y(k_x,k_y,0)σ_y.It is noted that the Hamiltonian contains two parameters and two momenta. Hence the crossing points can generically exist in the (001) plane.In the following, we show that the Weyl points can arise from the unitary element C_2^100 with [100] magnetization axis. This symmetry requires the constraints:[H,C_2^100]=0. In spin representation, the unitary matrix C_2^100 can be obtained Eq. (<ref>), asC_2^100=( [ 0 i; i 0; ])=iσ_x.Eq. (<ref>) givesH(k_x,k_y,0)=σ_x H(k_x,-k_y,0)σ_xord_x(k_x,k_y,0)=d_x(k_x,-k_y,0),d_y(k_x,k_y,0)=-d_y(k_x,-k_y,0).In k_y =0 axis, i.e. [100] direction, Eq. (<ref>) must requiresd_y(k_x,0,0)≡ 0.The energy dispersion of the two-band Hamiltonian after considering C_2^100 symmetry isE =±√(d_x(k_x,k_y,0)^2+d_y(k_x,k_y,0)^2)=± |d_x (k_x,0,0)|.In low-energy case, we haved_x(k_x,0,0)=A+B k_x +C k_x^2+…,and then the zero energy mode can require a pair of Weyl points locate at k_x=± k_x^c when B=0 and AC<0. Two bands with eigenvalues ± i of C_2^100 can cross on this axis. Along other direction with k_y = a k_x through Γ point on the xy plane, the Hamiltonian contains two function d_x and d_y, but one momentum k_x, indicating that there are no Weyl points on the k_y = a k_x line. Hence, the crossing would splits away from [100] axis. Similarly, when magnetization is along [110], [010], and [11̅0] directions, the Weyl points also arise due toC_2^m rotation of magnetic space group D_4h(C_2h). The Weyl points are only allowed in m axis. | http://arxiv.org/abs/1705.09234v1 | {
"authors": [
"Y. J. Jin",
"R. Wang",
"J. Z. Zhao",
"Z. J. Chen",
"Y. J. Zhao",
"H. Xu"
],
"categories": [
"cond-mat.mtrl-sci",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170525154212",
"title": "Topological Weyl and Node-Line Semimetals in Ferromagnetic Vanadium-Phosphorous-Oxide $β$-V$_2$OPO$_4$ Compound"
} |
AnEfficient Keyless Fragmentation Algorithm for Data Protection Katarzyna Kapusta, Gerard Memmi, and Hassan NouraTélécom ParisTech, Université Paris-Saclay23, avenue d'Italie, 75013 Paris, France Email: {firstname.lastname}@telecom-paristech.frDecember 30, 2023 ================================================================================================================================================================================================= We propose a method for recognizing moving vehicles, using data from roadside audio sensors.This problem has applications ranging widely, from traffic analysis to surveillance.We extract a frequency signature from the audio signal using a short-time Fourier transform, and treat each time window as an individual data point to be classified.By applying a spectral embedding, we decrease the dimensionality of the data sufficiently for K-nearest neighbors to provide accurate vehicle identification.§ INTRODUCTION Classification and identification of moving vehicles from audio signals is of interest in many applications, ranging from traffic flow management to military target recognition.Classification may involve differentiating vehicles by type, such as jeep, sedan, etc.Identification can involve distinguishing specific vehicles, even within a given vehicle type.Since audio data is small compared to, say, video data, multiple audio sensors can be placed easily and inexpensively.However, there are certain obstacles having to do with both hardware and physics.Certain microphones and recording devices have built-in features, for example, damping/normalizing that may be applied when the recording exceeds a threshold.Microphone sensitivity is another equipment problem: the slightest wind could give disruptive readings that affect the analysis.Ambient noise is a further issue, adding perturbations to the signal.Physical challenges include the Doppler shift, where the sound of a vehicle approaching differs from the sound of it leaving, so trying to relate these two can prove difficult.The short-time Fourier transform (STFT) is often used for feature extraction in audio signals.We adopt this approach, choosing time windows large enough that they carry sufficient frequency information but small enough that they allow us to localize vehicle events.Afterwards, weuse spectral embedding as a dimension reduction technique, reducing from thousands of Fourier coefficients to a small number of graph Laplacian eigenvectors. We then cluster the low-dimensional data using , establishing an unsupervised spectral clustering baseline prediction.Finally, we improve upon this by using as a simple but highly effective form of semi-supervised learning, giving an accurate classification without the need for large quantities of training data required by frequently used supervised approaches such as deep learning.In this paper, we apply these methods to audio recordings of passing vehicles.In Section 2, we provide background on vehicle classification using audio signals. In Section 3, we discuss the characteristics of the vehicle data that we use.Section 4 describes our feature extraction methods.Section 5 discusses our classification methods.Section 6 presents our results. We conclude in section 7 with a discussion of our method's strengths and limitations, as well as future directions.§ BACKGROUND The vast majority of the literature in audio classification is devoted to speech and music processing, with relatively few papers on problems of vehicle identification and classification. The most closely related work has included using principle component analysis for classifying car vs.motorcycle <cit.>, using an ϵ-neighborhood to cluster Fourier coefficients to classify different vehicles <cit.>, and using both the power spectral density and wavelet transform with K-nearest neighbors and support vector machines to classify vehicles <cit.>. Our study takes a graph-based clustering approach to identifying different individual vehicles from their Fourier coefficients.Analyzing audio data generally involves the following steps:* Preprocess raw data.* Extract features in data.* Process extracted data.* Analyze processed data. The most common form of preprocessing on raw data is ensuring that it has zero mean, by subtracting any bias introduced in the sound recording <cit.>.Another form of preprocessing is applying a weighted window filter to the raw data.For example, the Hamming window filter is often used to reduce the effects of jump discontinuity when applying the short-time Fourier transform, known as the Gibbs' effect <cit.>.The final preprocessing step deals with themanipulation of data size, namely how to group audio frames into larger windows.Different window sizes have been used in the literature, with no clear set standard.Additionally, having some degree of overlap between successive windows can help smooth results <cit.>.The basis for these preprocessing steps is to set up the data to allow for better feature extraction.STFT is frequently used for feature extraction <cit.>.Other approaches include the wavelet transform <cit.> and the one-third-octave filter bands <cit.>.All of these methods aim at extracting underlying information contained within the audio data.After extracting pertinent features, additional processing is needed.When working with STFT, the amplitudes for the Fourier coefficients are generallynormalized before analysis is performed <cit.>.Another processing step applied to the extracted features is dimension reduction <cit.>.The Fourier transform results in a large number of coefficients, giving a high-dimensional description of the data.We use a spectral embedding to reduce the dimensionality of the data <cit.>.The spectral embedding requires the use of a distance function on the data points: by adopting the cosine distance, we avoid the need for explicit normalization of the Fourier coefficients.Finally, the analysis of the processed data involves the classification algorithm.Methods used for this have included the following: * K-means and K-nearest neighbors <cit.>* Support vector machines <cit.>* Within ϵ distance <cit.>* Neural networks <cit.>K-means and K-nearest neighbors are standard techniques for analyzing the graph Laplacian eigenvectors resulting from spectral clustering <cit.>.They are among the simplest methods, but are also well suited to clustering points in the low-dimensional space obtained through the dimensionality reduction step.§ DATAOur dataset consists of recordings, provided by the US Navy's Naval Air Systems Command <cit.>, of different vehicles moving multiple times through a parking lot at approximately 15mph. While the original dataset consists of MP4 videos taken from a roadside camera, we extract the dual channel audio signal, and average the channels together into a single channel.The audio signal has a sampling rate of 48,000 frames per second.Video information is used to ascertain the ground truth (vehicle identification) for training data.The raw audio signal already has zero mean.Therefore, the only necessary preprocessing is grouping audio frames into time windows for STFT.We found that with windows of 1/8 of a second, or 6000 frames, there is both a sufficient number of windows and sufficient information per window.This is comparable to window sizes used in other studies <cit.>.We use two different types of datasets for our analysis.The first is a single audio sequence of a vehicle passing near the microphone, used as a test case for classifying the different types of sounds involved, differentiating background audio from vehicle audio. This sequence, whose raw audio signal is shown in Figure <ref>, involves the vehicle approaching from a distance, becoming audible after 5 or 6 seconds, passing the microphone after 10 seconds, and then leaving. The second sequence, shown in Figure <ref>, is a compilation formed from multiple passages of three different vehicles (a white truck, black truck, and jeep).We crop the two seconds where the vehicle is closest to the camera, having the highest amplitude, and then combine these to form a composite signal.The goal here is test the clustering algorithm's ability to differentiate the vehicles.§ FEATURE EXTRACTION §.§ Fourier coefficientsIn order to extract relevant features from our raw audio signals, we use the short-time Fourier transform.With time windows of 1/8 of a second, or 6000 frames, the Fourier decomposition contains 6000 coefficients.These are symmetric, leaving 3000 usable coefficients.Figure <ref> shows the first 1000 Fourier coefficients for a time window representing background noises.Note that much of the signal is concentrated within the first 200 coefficients. §.§ Fourier reconstructions Given the concentration of frequencies, we hypothesize that we can isolate specific sounds by selecting certain ranges of frequency.To test this, we perform a reconstruction analysis of the Fourier coefficients.After performing the Fourier transform, we zero out a certain range of Fourier coefficients and then perform the inverse Fourier transform.This has the effect of filtering out the corresponding range of frequencies.Figure <ref> shows the results of the reconstruction on an audio recording exhibiting strong wind sounds for the first 12 seconds, before the arrival of the vehicle at second 14.In a) the raw signal is shown.In b) we keep only the first 130 coefficients, in c) we keep only the next 130 coefficients, and in d) we keep all the rest of the coefficients.We see in the reconstruction that the first 130 Fourier coefficients contain most of the background sounds, including the strong wind that corresponds to the large raw signal amplitudes in the first 12 seconds.The remaining Fourier coefficients are largely insignificant during this time.When the vehicle becomes audible, however, the second 130 and the rest of the coefficients exhibit a significant increase in amplitude.By listening to the audio of the reconstructions b) through d), one can confirm that the first 130 coefficients primarily represent the background noise, while the second 130 and the rest of the audio capture most of the sounds of the moving vehicle. This suggests that further analysis into the detection of background frame signatures could yield a better method for finding which frequencies tofilter out, in order to yield better reconstructed audio sequences.§.§ Vehicle comparisonsThe goal of feature extraction is to detect distinguishing characteristics in the data.As a further example of why Fourier coefficients form a suitable set of features for vehicle identification,Figure <ref> shows Fourier coefficients for a sedan and for a truck, in both cases for time windows where the vehicle is close to the microphone.A moving mean of size 5 is used to smooth the plots, and coefficients are normalized to sum to 1 in this figure, to enable a comparison that is affected by different microphone volumes. There is a clear distinction between the two frequency signatures, particularly at lower frequencies.Therefore, in order to distinguish between different vehicles, we focus on effective ways of clustering these signatures. § SPECTRAL EMBEDDINGTo differentiate background sounds from vehicle sounds, and to identify individual vehicles, we apply a spectral embedding and then use both K-means and K-nearest neighbors as the final classification step.We treat each time window as an independent data point to be classified: for window i, let 𝐱_𝐢∈ℝ^m represent the set of m Fourier coefficients associated with that window.A spectral embedding requires a distance function for comparing the different data points.Given that we use a large number of Fourier coefficients (dimensionality m), many of which may be relatively insignificant, we use the cosine distance so as to decrease our sensitivity to these small coefficient values. The distance is given byd_ij = 1-𝐱_𝐢·𝐱_𝐣/𝐱_𝐢 𝐱_𝐣 The goal of a spectral embedding is to reduce the dimensionality of the data coming from our feature extraction step. The method, described in Algorithm <ref>, involves associating data points with vertices on a graph, and similarities (Gaussian function of distance) with edge weights.By considering only the leading eigenvectors of the graph Laplacian, we obtain a description of the data points in a low-dimensional Euclidean space, where they may be more easily clustered.This approach allows for greater control than other dimensionality reduction methods for vehicle audio recognition, such as PCA <cit.>.Note that our algorithm usesan adaptive expression for the Gaussian similarities, with the variance set according the distance to a point's Nth neighbor, where N is a specified parameter.We also set all similarities beyond the Nth neighbor to be zero, to generate a sparser and more easily clustered graph. § RESULTS We used the following parameters and settings:* 6000 frames per time window, resulting in 6000 possible Fourier coefficients.We use only the first m=1500 coefficients for spectral clustering, since these coefficients represent 98% of our data. * Standard box windows, with no overlap.We found no conclusive benefit when introducing weighted window filters or overlapping time windows.* N=15 for spectral embedding: each node in the graph has 15 neighbors, and the distance to the 15th neighbor is used to establish the variance in the Gaussian similarity. §.§ EigenvectorsFigure <ref> shows the eigenvalues of the Laplacian resulting from the spectral embedding of the composite(multiple-vehicle) dataset.This spectrum shows gaps after the third eigenvalue and the fifth eigenvalue, suggesting that a sufficient number of eigenvectors to use may be either three or five <cit.>.In practice, we find that five eigenvectors give good K-means and K-nearest neighbor clustering results. .8./fig/evals2Spectrum of SNGL for composite dataset, with N=15.Note gaps after third and fifth eigenvalue.fsc31§.§ Spectral clustering The spectral clustering method applies K-means to theleading eigenvectors of the Laplacian.Figure <ref> shows results for the single-vehicle data, for K=2,3,4.We show “best” results over 1000 randomized initializations: we select the clustering result with the smallest sum of squared distances between data points and associated cluster centers.K=3 gives a relatively clear separation of the signal into background noise, approaching vehicle, and departing vehicle (the latter two sounds differentiated by Doppler shift). K=2 and K=4 are less satisfactory, either clustering the vehicle approach together with background or subdividing the background cluster.Figure <ref> shows the results of K-means on the composite data. Given that there are 3 distinct vehicles, K=3 is chosen in an attempt to classify these, and is also consistent with the largest eigengap in Figure <ref> falling after the third eigenvalue. While K-means accurately clusters the majority of the data, many individual data points are misclassified.To improve these results, we instead turn to a semi-supervised classification method..9./fig/singlekmEUCon eigenvectors from spectral clustering of single-vehicle data. Raw signal, followed by results for K=2,3,4. fsc12 .9./fig/multikmEucon eigenvectors from spectral clustering of composite (multiple-vehicle) data.Raw signal, followed by results for K=3.fsc32 §.§ Spectral embedding with K-nearest neighbors We now test K-nearest neighbors on the eigenvectors for the composite data.We use this as a semi-supervised classification method, training using one entire audio sample from each of the three different vehicles. In this way,our method reflects an actual application where we might have known samples of vehicles.The results for K=16 are shown in Figure <ref>. The corresponding confusion matrix is given in Table <ref>. We allow for training points to be classified outside of their own class (seen in the case of vehicle 3), allowing for a better evaluation of the method's accuracy. While a few data points are misclassified, the vast majority (88.2%) are correct.Training on an entire vehicle passage appears sufficient to overcome Doppler shift effects in our data: the approaching sounds and departing sounds of a given vehicle are correctly placed in the same class..9./fig/knnEucK-nearest neighbors on eigenvectors from spectral embedding of composite (multiple-vehicle) data, for K=15. Training points are shown with red circles. Shaded regions show correct classification.fsc34 § CONCLUSIONS Identifying moving vehicles from audio recordings is a challenging and broadly applicable problem.We have demonstrated an approach that classifies frequency signatures, applying the short-time Fourier transform (STFT) to the audio signal and describing the sound at each 1/8-second time window using 1500 Fourier coefficients.Using a spectral embedding, we reduce the dimensionality of the data from 1500 to 5, corresponding to the five eigenvectors of the graph Laplacian.K-nearest neighbors then associates vehicle sounds with the correct vehicle in 88.2% of the time windows in our test data.Our analysis treats time windows as independent data points, and therefore ignores temporal correlations.It is possible that we could improve results by explicitly incorporating time information into our classification algorithm.For instance, one straightforward approach could be to use as data points a sliding window of larger width.In some cases, however, ignoring time information could actually help our method, for instance by helping the classifier correctly associate the Doppler-shifted sounds of a given vehicle approaching and departing.A limitation of our study is that our audio samples only involve single vehicles, under relatively tightly controlled conditions.The presence of multiple vehicles, or significant external noise such as in an urban environment, would pose a challenge to our feature extraction method.While the STFT is standard in audio processing, the use of time windows imposes a specific time scale that may not always be appropriate.Furthermore, the Fourier decomposition may be insufficiently sparse, with too many distinct Fourier components present in vehicle audio signals.To overcome these difficulties, one could use multiscale techniques such as wavelet decompositions that have been proposed for vehicle detection and classification <cit.>.More recently developed sparse decomposition methods may also be of use, as they implicitly learn a good choice of basis functions from the data <cit.>.An additional area for improvement is our clustering algorithm.More sophisticated methods than K-means and K-nearest neighbors may allow for vehicle identification under less tightly controlled conditions than those in our experiments, or possibly for identifying broad types of vehicles such as cars or trucks.Such semi-supervised methods would preserve the chief benefit of our approach, namely its applicability in cases where only very limited training data are available. IEEEtran | http://arxiv.org/abs/1705.09869v2 | {
"authors": [
"Justin Sunu",
"Allon G. Percus"
],
"categories": [
"stat.ML",
"cs.LG",
"physics.data-an"
],
"primary_category": "stat.ML",
"published": "20170527213947",
"title": "Dimensionality reduction for acoustic vehicle classification with spectral embedding"
} |
Is It Reasonable to Substitute Discontinuous SMC by Continuous HOSMC? Comments to Discussion Paper by V. Utkin Ulises Pérez Ventura Posgrado de Ingeniería Facultad de Ingeniería, UNAM, MéxicoEmail: [email protected] Leonid Fridman Departamento de Ingeniería de Control y Robótica Facultad de Ingeniería, UNAM, MéxicoEmail: [email protected] December 30, 2023 ==================================================================================================================================================================================================================================================== Professor Utkin in his discussion paper proposed an example showing that the amplitude of chattering caused by the presence of parasitic dynamics in systems governed by First-Order Sliding-Mode Control (FOSMC) is lower than the obtained using Super-Twisting Algorithm (STA). This example served to motivate this research reconsidering the problem of comparison of chattering magnitude in systems governed by FOSMC that produces a discontinuous control signal and by STA that produces a continuous one, using Harmonic Balance (HB) methodology. With this aim the Averaged Power (AP) criteria for chattering measurements is revisited. The STA gains are redesigned to minimize amplitude or AP of oscillations predicted by HB. The comparison of the chattering produced by FOSMC and STA with redesigned gains is analyzed taking into account their amplitudes, frequencies and values of AP allowing to conclude that: (a) for any value of upperbound of disturbance and Actuator Time Constant (ATC) there exist a bounded disturbance for which the amplitude and AP of chattering produced by FOSMC is lower than the caused by STA; (b) if the upperbound of disturbance and upperbound of time-derivative disturbance are given, then for all sufficiently small values of ATC the amplitude of chattering and AP produced by STA will be smaller than the caused by FOSMC; (c) critical values of ATC are predicted by HB for which the parameters, amplitude of chattering and AP, produced by FOSMC and STA are the same. Also the frequency of self-exited oscillations caused by FOSMC is always grater than the produced by STA.§ INTRODUCTIONSliding-Mode Control (SMC) is an efficient technique used for bounded matched uncertainty compensation <cit.>. The First-Order Sliding-Mode Control (FOSMC) keeps a desired constraint σ of relative degree one, by means of theoretically infinite-frequency switching control. However, infinite-frequency switching control is not feasible due to the presence of parasitic dynamics as actuators and sensors <cit.> hysteresis effects <cit.>, <cit.>, and other non-idealities. Hence, the sliding set converges to a real sliding motion with finite (high) frequency, this effect is well-known as chattering effect and it is the main drawback of the sliding-mode control theory.Higher-Order Sliding Mode Control (HOSMC) algorithms were proposed as an attempt to adjust the chattering by substituting (intuitively) discontinuous control inputs by continuous ones <cit.>, <cit.>, <cit.>. One of the most efficient algorithms is the Super-Twisting Algorithm (STA) compensating theoretically exactly matched Lipschitz uncertainties in finite-time <cit.>, <cit.>. This idea was very attractive but in the paper <cit.> was shown that in systems driven by STA (as well as by any other controller with infinite gain at the origin) the chattering also appears. Moreover, Professor Utkin <cit.>, <cit.>, <cit.>, presented some examples showing that some systems governed by STA exhibit bigger amplitude of chattering that the systems driven by First-Order Sliding Mode Control (FOSMC), when parasitic dynamics affects the SMC/HOSMC closed-loop.In this paper we will try to answer the question: Is It Expedient to Substitute Discontinuous SMC by Continuous HOSMC, when parasitic dynamics are presented in the control loop? For that, we analyze the amplitude and frequency of possible self-excited oscillations and its effect on the Average Power (AP). Harmonic Balance (HB) is widely used to estimate the amplitude and frequency of possible oscillations for dynamically perturbed SMC/HOSMC systems <cit.>, <cit.>, where a Describing Function (DF) characterizes the non-linearities effects on the parameters of periodic motions <cit.>, <cit.>.The contributions of this work are listed below: * We confirm the hypothesis of Professor V. Utkin: for any value of the actuator time-constant (ATC) there exist a bounded disturbance for which the amplitude of possible oscillations produced by FOSMC is lower than the obtained applying STA. * Given the upperbound of disturbance and the upperbound of time-derivative disturbance, for a sufficiently small value of ATC the amplitude of chattering and AP produced by STA will be smaller than the caused by FOSMC.With this aim, we propose the following: 1. Reformulation of A. Levant “energy like" criterion to compute the AP based on HB methodology.2. Selection of STA gains to minimize the amplitude ofpossible oscillations.3. Selection of STA gains to minimize the AP.The structure of the paper is as follows: a motivation example is discussed in section II; section III contains the preliminaries about HB approach for FOSMC and STA; chattering parameters obtained by HB are analyzed in section IV; comparison examples in section V; and section VI presents the conclusions about the obtained results.§ MOTIVATION EXAMPLEConsider the disturbed first-order system shown in the Figure 1, the plant can be modeled asẋ(t) = u̅(t) + F(t),where x∈ is the output and u̅ ∈ is the control input. The disturbance term F has the formF=αsin (Ω t) ⇒{[ |F| ≤δ = α; |Ḟ| ≤Δ = α Ω ]. Let us apply two control laws:* Discontinuous FOSMC <cit.>: u = - M (x),where the control gain is chosen M = 1.1δ from the upperbound of disturbance (<ref>), ensuring the global finite-time convergence to the first order sliding-mode (∃t_r: x(t)=0, ∀t ≥ t_r) when the actuator dynamics is fast enough. * Continuous STA <cit.>, <cit.>: [u= - k_1 |x|^1/2 (x) + v,; v̇=-k_2 (x), ]where the control gains are chosen k_1 = 1.5√(Δ), k_2 = 1.1Δ from the upperbound of time-derivative disturbance (<ref>), ensuring the global finite-time convergence to the second order sliding-mode (∃t_r: x(t)=ẋ(t)=0, ∀t ≥ t_r) when the actuator dynamics is fast enough. FOSMC can reject bounded disturbances but STA can compensate Lipschitz disturbances (not necessarily bounded). In order to compare both algorithms we consider a bounded and Lipschitz disturbance (<ref>). Following <cit.>, we consider the same actuator model as Professor V. Utkin which consists of a 2^nd-order linear system[ ż(t)= [01; -1/μ^2 -2/μ ]z(t) + [ 0; 1/μ^2 ]u(t),;u̅(t)= [ 1 0 ]z(t), ]where μ>0 is the actuator time-constant (ATC). Thus, the effects of parasitic dynamics can be parameterized through ATC. Let us notice that FOSMC and STA were designed for the system (<ref>) with relative degree one but the presence of actuator (<ref>) increases the relative degree of the system (<ref>).Table <ref> presents the simulation results of the system conformed by the plant (<ref>) and the actuator dynamics (<ref>) in closed loop with FOSMC (<ref>) and STA (<ref>). Chattering magnitude of the output x is compared taking into account several values of ATC μ and disturbance frequency Ω (fixing α = 1). It can be seen that the amplitude of chattering has the same order for Ω = 1 when μ = 10^-1, and it is lower for STA when μ = 10^-2 and μ = 10^-3. For Ω = 10 the results change, when μ = 10^-1 the amplitude of oscillations for FOSMC is lower than for STA, when μ = 10^-2 they have the same order, but when μ = 10^-3 the amplitude of chattering in the system governed by STA is lower that ones in the system with FOSMC. The third column of Table <ref> shows that the amplitude of oscillations for FOSMC and STA have the same order only for μ=10^-3 and for bigger values of ATC the amplitude generated by FOSMC is lower than the produced by STA.§.§.§ Conclusions* Table <ref> confirms the hypothesis of Professor V. Utkin: for any value of ATC there exist a bounded disturbance for which the amplitude of possible oscillations produced by FOSMC is lower than the obtained applying STA.* It should exists a value of ATC for which the amplitude of chattering produced by FOSMC and STA are the same.* For any bounded disturbance, the amplitude of possible oscillations produced by STA may be less than the obtained using FOSMC if the actuator dynamics is fast enough (μ→ 0). § CHATTERING ANALYSIS OF FOSMC AND STA USING HARMONIC BALANCETaking into account the control scheme shown in Figure 1, FOSMC (<ref>) and STA (<ref>) are analyzed in frequency domain using the HB methodology to understand how the parasitic dynamics (<ref>) may degrade the accuracy, when the control gains are selected to reject the matched disturbance F.Consider the nominal case (F=0), the dynamically perturbed system (<ref>)-(<ref>) has the transfer function W(s) = G_a(s)G(s) =1s(μ s+1)^2 ,whose relative degree is r=3. Due the parasitic dynamics (<ref>), the output of the system converges to a periodic solution <cit.>, <cit.>, which can be approximated by its first-harmonic,[ x(t) = A sin(ω t),; ẋ(t) = A ωcos(ω t) . ]where A and ω are the chattering parameters, amplitude and frequency, respectively. §.§ Amplitude and Frequency EstimationLet us apply the DF method <cit.>, <cit.> to predict periodic oscillations. Parameters of a possible limit cycle may be found as an intersection point of the actuator-plant dynamics W(s) Nyquist plot and the negative reciprocal DF -N^-1(A,ω) of the SMC/HOSMC algorithm, which corresponds to the Harmonic Balance equation (HBE)N(A,ω) W(jω) = -1,whose solution is an estimate of chattering parameters: amplitude A and frequency ω.§.§ Averaged Power CriteriaIn the paper by A. Levant <cit.> an “energy like" criteria for chattering measurement are presentedE = ( ∫_0^Tẋ^p(τ) dτ)^1/p ,inspired by L_p norm. Unfortunately, it is only a qualitative criterion to understand the chattering effects because it has no physical sense and require information of ẋ (knowledge of disturbance) to compute it.HB approach allows to compute the Averaged Power (AP) of the steady-state behavior of the system,P = ω2π∫_0^2π/ω( Aωcos(ω t) )^2dt = A^2ω^22 ,due to the output x and its time-derivative ẋ are assumed of the form (<ref>). §.§ HB Analysis of First Order Sliding Mode ControlThe describing function of the non-linearity (<ref>) has the form <cit.>N(A) = 4 Mπ A .The HBE (<ref>) can be separated as real and imaginary parts[ 4 Mπ A=2μ ω^2,;0= ω(μ^2ω^2-1), ]whose analytic solution is <cit.>A =2 M μ/π,ω=1/μ.Substituting the limit cycle parameters (<ref>), (<ref>) in the expression (<ref>), the AP has the formP = 2M^2π^2 .§.§ HB Analysis of Super-Twisting AlgorithmThe describing function of the non-linearity (<ref>) has the form <cit.>N(A,ω) = 1.1128 k_1A^1/2 - j 4 k_2π A ω .Once again, the HBE (<ref>) can be separated as real and imaginary parts[ 1.1128k_1A^1/2=2μ ω^2,;4k_2π A ω= ω(1-μ^2ω^2), ]whose analytic solution is <cit.>A =μ^2 𝕂__A,ω=𝕂_ω/μ, where[ 𝕂_A = ( 1/2·(1.1128k_1)^2 + 16/πk_2/1.1128k_1)^2 = ( 1.1128k_1/2𝕂_ω^2)^2,; 𝕂_ω = ( (1.1128k_1)^2/(1.1128k_1)^2 + 16/πk_2) ^1/2 . ]Substituting the limit cycle parameters (<ref>), (<ref>) in the expression (<ref>), the AP has the formP = μ^232·( (1.1128k_1)^2 + 16/πk_2)^3(1.1128k_1)^2 . §.§ Design of STA Gains to Minimize the Amplitude of Possible OscillationsThe expression (<ref>) allows to obtain the values of STA gains, k_1 and k_2, such that the amplitude is minimized. Consider the following normalization Aμ^2 = ( 1/2·(1.1128k_1)^2 + 16/πk_2/1.1128k_1)^2,it is possible to compute the value of the gain k_1 which minimizes the amplitude (<ref>) for each k_2 > Δ,k_1 = ( 16k_2π(1.1128)^2)^1/2 = 2.028√(k_2) .The selection of STA gains that minimize the amplitude of possible oscillations based on HB approach isk_1 = 2.127√(Δ) ,k_2 = 1.1 Δ .For these STA gains, the chattering parameters becomeA = 5.6023Δμ^2, ω = 1/μ√(2) , P = 7.8464Δ^2μ^2.Figure 2 shows the critical value of the amplitude normalization (<ref>) for the STA gains (<ref>) with Δ = 1. Table <ref> contains the simulation results for Δ = 10 and μ = 10^-2, the STA gains are selected to compare the amplitude of oscillations and confirms the selection criterion (<ref>).§.§ Design of STA Gains to Minimize the Averaged PowerThe expression (<ref>) allows to obtain the values of STA gains, k_1 and k_2, such that the AP is minimized. Consider the following normalizationPμ^2 = 132·( (1.1128k_1)^2 + 16/πk_2)^3(1.1128k_1)^2 ,it is possible to compute the value of the gain k_1 which minimizes the AP for each k_2 > Δ,k_1 = ( 8k_2π(1.1128)^2)^1/2 = 1.434 √(k_2) .The selection of STA gains that minimize the AP based on HB approach isk_1 = 1.504√(Δ) ,k_2 = 1.1 Δ .For these STA gains, the parameters of periodic motion becomeA = 6.3025Δμ^2, ω = 1/μ√(3) , P = 6.6203Δ^2μ^2.Figure 2 shows the critical value of the AP normalization (<ref>) with Δ = 1. Table <ref> contains the simulation results for Δ = 10 and μ = 10^-2, the STA gains are selected to compare the AP and confirms the selection criterion (<ref>).Sufficient conditions to guarantee finite-time stability are presented in <cit.>, where the STA gains have to satisfy k_1 > 1.414 √(k_2) ,k_2 > Δ ,where Δ is the upperbound of the time-derivative disturbance (<ref>). STA proposed gains in the expressions (<ref>) and (<ref>) ensure finite-time stability when the actuator dynamics is fast enough. § ANALYSIS OF CHATTERING PARAMETERS ESTIMATED BY HARMONIC BALANCE§.§ Amplitude AnalysisIn order to analyze the chattering parameters with respect to the ATC, consider δ = Δ = 1 and the control gain M = 1.1δ for FOSMC and the selection proposed in (<ref>) for STA. Figure 3 shows the chattering parameters for several values of ATC, there is a value of the ATC such that the amplitude of the output is the same<cit.>, despite the use of FOSMC (<ref>) or STA (<ref>) on the dynamically perturbed system (<ref>),μ^∗ =8 M (1.1128k_1)^2π( (1.1128k_1)^2+ 16/π k_2)^2.for this example, the value of ATC to have the same amplitude isμ^∗=0.125δΔ . Figure 3 confirms that to substitute FOSMC by STA, we should consider that the amplitude of oscillations may be greater(lower) when the ATC, [ μ > μ^∗ ⇒ A_FOSMC < A_STA ,; μ < μ^∗ ⇒ A_FOSMC > A_STA . ] §.§ Frequency Analysis The order of high-frequency oscillations is O(1/μ) when it is applied FOSMC (<ref>) or STA (<ref>). However, the frequency is always lower for the STA than the obtained using FOSMC, as it is shown in the graphical solution of the HB equation (<ref>) of Figure 4. §.§ Averaged Power AnalysisIn order to analyze the chattering parameters with respect to the ATC, consider δ = Δ = 1 and the control gain M = 1.1δ for FOSMC and the selection proposed in (<ref>) for STA. Figure 5 shows the chattering parameters for several values of ATC, there is a value of ATC such that the AP is the same despite the use of FOSMC (<ref>) or STA (<ref>) on the dynamically perturbed system (<ref>),μ^⋆ = 8M(1.1128k_1)π( (1.1128k_1)^2 + 16/πk_2)^3/2 . for this example, the value of ATC to have the same AP isμ^⋆ = 0.1924δΔ . Figure 5 confirms that to substitute FOSMC by STA, we should consider that the AP may be greater(lower) when the ATC,[ μ > μ^⋆ ⇒ P_FOSMC < P_STA ,; μ < μ^⋆ ⇒ P_FOSMC > P_STA . ] § COMPARISON EXAMPLES §.§ High-Frequency DisturbancesThe previously simulation examples are for the nominal case (F=0) but the control gains are selected according to the upperbound of disturbance in the case of FOSMC (<ref>), or the upperbound of time-derivative disturbance for the STA (<ref>), then * FOSMC (<ref>):[M= 1.1δ= 1.1α . ] * STA (<ref>):[ k_1 = 2.127√(Δ) = 2.127√(α Ω) ,; k_2 =1.1Δ =1.1α Ω . ] The value of ATC predicted by HB for which the amplitude of chattering is the same despite the use of discontinuous FOSMC (<ref>) or continuous STA (<ref>), on the dynamically perturbed system (<ref>) isμ^∗ = 0.125 δΔ = 0.125 1Ω .In order to compare the system behavior for some values of the ATC, considerμ_1 = 0.25 1Ω , μ_2 = 0.0833 1Ω .Table <ref> contains the sliding-mode output accuracy for some values of disturbance frequency Ω, taking into account the critical value of ATC (<ref>) and μ_1>μ^∗, μ_2<μ^∗ from (<ref>). Simulation results confirm that for any disturbance frequency Ω should be a critical value of ATC μ^∗ for which the magnitude of chattering is the same when FOSMC or STA are applied. If ATC is greater than μ^∗ (for example μ_1) the amplitude of oscillations is lower using FOSMC than the obtained applying STA. But if ATC is lower than μ^∗ (for example μ_2) the amplitude of oscillations is higher using FOSMC than the obtained applying STA.§.§ Professor V. Utkin ExampleThe following example was taken from the paper <cit.>, they propose that the upperbound of disturbance and the upperbound of time-derivative disturbance have the same value δ = Δ = 60. Taking into account the FOSMC (<ref>) gain M = 1.1δ and the STA (<ref>) proposed gains (<ref>), the following chattering parameters are obtained by HB: * FOSMCA = 42.017μ , ω = 1/μ , P = 882.7102. * STAA = 336.135μ^2, ω = 1/μ√(2) , P = 28246.93μ^2.Hence the critical values of ATC becomeμ^∗ = 0.125, μ^⋆ = 0.1768,for same amplitude and same AP, respectively. Table V shows the chattering parameters obtained in simulation for some values of ATC and the critical values (<ref>). Note that when the ATC μ>μ^∗ the amplitude of oscillations generated by FOSMC is lower than the produced by STA, this situation is reversed when μ<μ^∗. On the other hand, when the ATC μ>μ^⋆ the AP generated by FOSMC is lower than the produced by STA, and when μ<μ^⋆ the AP caused by FOSMC is grater than the produced by STA.§ CONCLUSIONSA methodology for analysis based on Harmonic Balance approach is proposed to study the chattering for dynamically perturbed systems driven by FOSMC and STA. HB approach and simulation results allows to confirm the Professor V. Utkin hypothesis, given the value of ATC and the upperbound of disturbance there exist a bounded disturbance for which the amplitude of possible oscillations produced by FOSMC is lower than the obtained applying STA. On the other hand, given the upperbound of disturbance and the upperbound of time-derivative disturbance there exist an actuator fast enough for which the STA amplitude of oscillations or(and) AP are lower than the generate by FOSMC. Selection criteria of STA gains are presented to adjust the chattering effects on the amplitude of possible oscillations and AP. § ACKNOWLEDGMENTThe authors are grateful for the financial support of CONACyT (Consejo Nacional de Ciencia y Tecnología): CVU 631266; PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) IN 113612.*20Atherton75 D.P. 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Utkin, Discussion Aspects of High Order Sliding Mode Control, Automatic Control, IEEE Transactions on, vol. 61, no. 3, pp. 829–833, IEEE, 2016.Utkin16 V. Utkin, Divergence theorem for super twisting control, 2016 14th International Workshop on Variable Structure Systems (VSS), pp. 178–181, IEEE, 2016. | http://arxiv.org/abs/1705.09711v1 | {
"authors": [
"Ulises Pérez-Ventura",
"Leonid Fridman"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170526204142",
"title": "Is It Reasonable to Substitute Discontinuous SMC by Continuous HOSMC?"
} |
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